{
"built": "2026-11-01 17:50",
"version": "0.1.0",
"tiers": {
"A": "proof or exhaustive computation; agents could finish",
"B": "open empirical question a pre-registered data test can settle",
"C": "hard and open, where these agents have an unusual angle"
},
"registry": null,
"problems": [
{
"id": "rule110-universality-machine-checked",
"title": "Machine-check the universality of Rule 110",
"field": "mathematics",
"tier": "A",
"question": "Can Cook's proof that elementary cellular automaton Rule 110 is computationally universal be formalized and checked in Lean 4 with no `sorry` and no added axioms?",
"known_state": "Cook (2004) proved universality by emulating cyclic tag systems with glider collisions; Neary and Woods (2006) sharpened the emulation to polynomial time. Neither proof has been machine-checked. The repo records the claim as inconclusive (wolfram2002-ch11-14) because reproduction tests can only exercise the construction, not verify it.",
"acceptance_criteria": [
"A Lean 4 module whose main theorem states that for every cyclic tag system and input there is an initial configuration of Rule 110 whose evolution encodes the computation, compiled with no sorry and only Lean's standard axioms.",
"The glider catalog, collision table and the periodic background are each a checked lemma rather than an assumption."
],
"falsifiers": [
"A gap in the collision table that cannot be closed (a required collision that does not behave as Cook's table says), which would also be a finding against the published proof."
],
"required_capabilities": [
"lean4",
"cellular-automaton-simulation",
"proof-engineering"
],
"data_access": "No data needed; Rule 110 is fully specified. Cook's glider tables are in the open-access Complex Systems paper.",
"scales": [],
"frontier_problem": null,
"related_claims": [
{
"id": "wolfram2002-ch11-14",
"label": "Elementary cellular automaton rule 110 is universal",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": "inconclusive",
"theorem": false
},
{
"id": "wolfram2002-ch11-27",
"label": "A 2-state 5-color Turing machine is universal via rule 110",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": "supports",
"theorem": false
},
{
"id": "wolfram2002-ch01-07",
"label": "Universal computation is possible in systems with one-sentence rules",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": "supports",
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.15.1.1",
"title": "Universality in Elementary Cellular Automata",
"year": 2004,
"note": "Cook 2004, Universality in elementary cellular automata"
},
{
"url": "https://doi.org/10.1007/11786986_13",
"title": "P-completeness of Cellular Automaton Rule 110",
"year": 2006,
"note": "Neary and Woods 2006, P-completeness of cellular automaton Rule 110"
}
],
"nanopub": null,
"approaches": [
{
"id": "rule110-universality-machine-checked--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Modular Lean proof of Cook's construction via a light-cone splicing lemma",
"kind": "formalization",
"method": "Formalize Rule 110 as a map on configurations ℤ → Bool and prove Cook's emulation of cyclic tag systems (CTS) as a stack of independent layers. The load-bearing general lemma is locality (light-cone splicing): the restriction of step^[t] x to the interval [a+t, b−t] depends only on x restricted to [a, b]. With it, any finite pattern embedded in the periodic ether can be checked on a finite window by kernel computation (`decide` or `rfl` on List Bool, never `native_decide`), and the result transfers to the infinite configuration. Layers: (1) the ether is spacetime-periodic: it returns to a spatial shift of itself after its temporal period (Cook gives horizontal period 14 and vertical period 7); (2) each glider Cook uses (A through H and the variants in his construction) is a finite word that, spliced into ether, reappears translated after its period; (3) each collision in Cook's table, at each phase and offset the construction uses, produces the stated outgoing gliders, checked on a finite window; (4) a glider calculus: an abstract configuration (a list of glider type, phase and gap triples), an interpretation map into Rule 110 configurations, and a simulation theorem saying that for gaps above a stated threshold the CA evolution realizes the abstract collision rules; (5) Cook's encoding of CTS tape, appendants and the periodic right-hand program stream, with a theorem that one CTS step corresponds to a bounded number of abstract collision rounds.",
"steps": [
"Define `rule110`, `step`, shift, and prove the light-cone splicing lemma by induction on t.",
"Encode the ether word and prove its spacetime periodicity on a finite window; lift to ℤ with the splicing lemma.",
"Transcribe Cook's glider catalogue and collision table from the open-access Complex Systems paper into Lean data; prove each entry by bounded kernel computation.",
"Define the abstract glider-list semantics and prove the simulation theorem for well-separated configurations.",
"Define cyclic tag systems in Lean and Cook's encoding; prove the one-step correspondence and the main theorem by induction on CTS steps.",
"Check `#print axioms` lists only propext, Classical.choice and Quot.sound; write a report in the format of the Thompson Lean reports."
],
"decisive_result": "A Lean 4 module whose main theorem states: for every cyclic tag system and input word there is an initial Rule 110 configuration (ether plus a finite perturbation on the left and a periodic glider stream on the right) and a decoding map such that the CA evolution encodes every CTS step. It must compile with no sorry and only the three standard axioms, with every glider and collision fact a proved lemma.",
"failure_signature": "A collision required by the encoding that the kernel check refutes at the phase or gap the construction needs (a gap in Cook's table, itself a publishable finding), or kernel computations on the required windows that are too large to complete without native_decide, forcing the project to stall at layer 3.",
"tools": [
"lean4",
"mathlib",
"cellular-automaton-simulation",
"proof-engineering"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.15.1.1",
"title": "Universality in Elementary Cellular Automata",
"year": 2004,
"note": "Cook 2004, Universality in elementary cellular automata"
},
{
"url": "https://doi.org/10.1007/11786986_13",
"title": "P-completeness of Cellular Automaton Rule 110",
"year": 2006,
"note": "Neary and Woods 2006, P-completeness of cellular automaton Rule 110"
},
{
"url": "https://github.com/leanprover-community/mathlib4",
"title": null,
"year": null,
"note": "Mathlib 4; Mathlib.Computability provides Turing machine and partial-recursive definitions to which CTS universality can later be connected"
}
],
"difficulty": "long-shot",
"missing": "Proof-engineering volume rather than new mathematics: Cook's construction has many glider spacings and timing constraints, and kernel-only evaluation of large windows may be slow. A month is likely enough for layers 1 to 3, not the full chain.",
"nanopub": null
},
{
"id": "rule110-universality-machine-checked--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Independent exhaustive audit of Cook's glider and collision table",
"kind": "computation",
"method": "Before or alongside formalization, re-derive from scratch every fact Cook's construction depends on, using code written without reference to Cook's tables except for the final comparison. Enumerate all gliders on the ether background up to period 200 and width 100 using de Bruijn-diagram search for spacetime-periodic patterns. For every ordered pair of gliders used by the construction, simulate the collision at every relative phase and every integer gap in the range the construction allows, and record the outgoing glider multiset with phases. Then run the full construction for small cyclic tag systems (including Cook's worked example and randomly generated CTS with up to 6 appendants of length up to 6) for at least 10^7 generations with bit-parallel simulation, and decode the CTS tape from the glider stream at each emulated step, comparing with a direct CTS interpreter.",
"steps": [
"Implement a bit-sliced Rule 110 simulator and a de Bruijn-diagram glider finder; verify against the published list of Rule 110 gliders.",
"Produce a machine-readable collision table (JSON) over all phases and gaps used by the construction, with SHA-256 of the generating code and output.",
"Diff the table against a transcription of Cook's table; classify every discrepancy as transcription error, phase-convention difference or genuine mismatch.",
"Run end-to-end emulations of at least 100 random CTS and compare decoded tapes step by step with a reference interpreter."
],
"decisive_result": "Either a complete match (every collision the construction needs behaves as stated at every allowed gap, and all end-to-end emulations decode correctly for 10^7 generations), which hands layer 3 of a formal proof a verified target, or a specific (glider pair, phase, gap) triple whose outcome differs from Cook's table and breaks an emulation.",
"failure_signature": "The audit cannot settle universality either way by itself; it fails as an approach if the gap ranges the construction needs cannot be extracted unambiguously from the paper, so the audit checks a reconstruction rather than Cook's proof.",
"tools": [
"large-scale-computation",
"cellular-automaton-simulation",
"de-bruijn-diagrams"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.15.1.1",
"title": "Universality in Elementary Cellular Automata",
"year": 2004,
"note": "Cook 2004, source of the glider catalogue and collision table to audit"
},
{
"url": "https://doi.org/10.1007/11786986_13",
"title": "P-completeness of Cellular Automaton Rule 110",
"year": 2006,
"note": "Neary and Woods 2006, gives an alternative account of the construction's timing"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "rule110-universality-machine-checked--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Verified certificate checker: reduce the proof to a finite spacetime tile set",
"kind": "mixed",
"method": "Instead of proving Cook's construction lemma by lemma, prove once in Lean that a small checker is sound, and let external search produce the certificate. The spacetime diagram of Cook's construction for an arbitrary CTS is assembled from finitely many rectangular spacetime tiles (ether blocks, glider segments, collision blocks), and the program side is periodic while the data side varies only through which tiles are chosen. Define in Lean a certificate as a finite tile set with edge labels (boundary words) plus a context-free or regular grammar describing admissible tile arrangements for any CTS. The checker verifies (a) each tile is a valid Rule 110 spacetime patch, by kernel computation, (b) adjacent tiles agree on shared boundary words, (c) the grammar generates, for each CTS step, an arrangement whose decoded output equals the CTS successor. Soundness theorem: if the checker accepts, the universality statement holds. This fails differently from approach 1: the human-proof burden shrinks to the soundness lemma and the grammar semantics, while the computational burden moves to certificate generation outside Lean.",
"steps": [
"Specify tiles, boundary words and the arrangement grammar in Lean; prove the soundness theorem using the light-cone splicing lemma.",
"Write an external generator that runs the construction on many CTS, cuts the spacetime diagram into tiles, deduplicates them and infers the arrangement grammar.",
"Run the checker inside Lean by kernel reduction on the certificate; iterate on tile size if kernel cost is prohibitive.",
"Prove the grammar-to-CTS correspondence by induction on CTS steps."
],
"decisive_result": "Lean accepts the certificate, and the soundness theorem plus the checker run yields the universality theorem with only standard axioms. A secondary artifact is the tile count, a compact measure of the construction's complexity.",
"failure_signature": "The tile decomposition does not close: arrangements depend on unbounded gap values in a way no regular or context-free grammar captures, or the tile set needed exceeds what kernel reduction can check in reasonable time.",
"tools": [
"lean4",
"mathlib",
"large-scale-computation",
"certificate-checking"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.15.1.1",
"title": "Universality in Elementary Cellular Automata",
"year": 2004,
"note": "Cook 2004, the construction to be tiled"
},
{
"url": "https://github.com/ccz181078/Coq-BB5",
"title": null,
"year": null,
"note": "Coq proof of BB(5), a precedent for verified deciders checking large externally generated case analyses in a proof assistant"
}
],
"difficulty": "long-shot",
"missing": "No existing tile-level description of Cook's construction; the grammar-inference step is research.",
"nanopub": null
},
{
"id": "rule110-universality-machine-checked--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Refinement proof from cyclic tag systems to Rule 110",
"kind": "formalization",
"method": "Define finite Rule 110 windows with a periodic ether outside, cyclic tag systems, and a decoding relation in Lean 4. Prove a one-step simulation theorem by decomposing each compiled tag step into a finite sequence of glider phases; discharge each collision lemma with a kernel-evaluated finite trace and compose them by induction on tag-system runs.",
"steps": [
"Formalize Rule 110 evolution on integers and prove locality and light-cone lemmas.",
"Encode Cook's ether phases, gliders and compiler from cyclic tag words.",
"Generate every required collision trace and prove its decoded precondition and postcondition by decide.",
"Prove phase synchronization, noninterference and the global simulation theorem without sorry."
],
"decisive_result": "A compiling Lean module whose theorem maps every finite cyclic-tag execution of length k to a Rule 110 execution decoding the same k transitions, with all collision and ether lemmas proved and only standard axioms.",
"failure_signature": "A required compiler collision has no finite spacetime trace with Cook's advertised phase and separation constraints, or collision lemmas cannot be composed because debris enters another light cone.",
"tools": [
"lean4",
"cellular-automaton-simulation",
"large-scale-computation"
],
"datasets": [],
"references": [],
"difficulty": "long-shot",
"missing": "A machine-readable transcription of Cook's complete glider phase table and substantial Lean proof engineering.",
"nanopub": null
},
{
"id": "rule110-universality-machine-checked--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Certified collision-table reconstruction",
"kind": "computation",
"method": "Independently reconstruct Cook's construction before formalizing it. Enumerate all ether-relative glider phases and pair/triple collisions inside exact light cones, canonicalize traces modulo translation and ether period, and search for a closed macro-transition system implementing the cyclic tag productions. Emit proof certificates containing initial words, horizons and decoded products for Lean replay.",
"steps": [
"Enumerate minimal periodic-ether defects up to width 80 and classify velocities over 500 steps.",
"Enumerate phase-separated collisions up to distance 400 and retain stable products.",
"Model compilation as a constraint problem over collision phases and spacings.",
"Replay every selected certificate in Lean and compare the reconstructed table with the published construction."
],
"decisive_result": "A closed, independently generated set of certified macro-transitions sufficient for both cyclic-tag productions and all routing operations, with every trace replayed by Lean.",
"failure_signature": "Exhaustive bounded search finds that at least one required macro-transition has no phase-consistent realization within the stated bounds, leaving no complete emulation certificate.",
"tools": [
"cellular-automaton-simulation",
"large-scale-computation",
"lean4"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Completeness beyond finite phase and spacing bounds would still require a mathematical argument.",
"nanopub": null
},
{
"id": "rule110-universality-machine-checked--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Kernel-check Cook's ether, gliders and collision table",
"kind": "formalization",
"method": "In the repo's Lean 4.35.0 and Mathlib 4.35.0, encode Rule 110 as the Boolean function on (Bool × Bool × Bool) given by Wolfram's 8-bit table number 110. Do not use the formula left XOR (center OR right); that formula is Rule 30. Transcribe the periodic background word and every glider word from Cook 2004, without rewriting them. Prove the background is invariant under the spacetime shift Cook states. For each published collision, evolve the finite word with a pure Lean function and prove equality to Cook's product word by kernel reduction (rfl or decide on a Bool), never native_decide, sorry, or an extra axiom. Split any evolution the kernel cannot reduce in one pass into blocks of a few hundred steps with an intermediate word that is itself a lemma. The glider catalog, the background and the collision table are separate lemmas, as the acceptance criteria require.",
"steps": [
"Define rule110 : Bool × Bool × Bool → Bool from the 8-bit table and prove it matches Wolfram number 110.",
"Transcribe Cook's periodic background and prove the invariance he claims.",
"Transcribe each glider as a finite word on that background.",
"Discharge each collision lemma by blocked kernel reduction of the pure step function.",
"Record #print axioms output and reject the module if any axiom other than propext, Classical.choice or Quot.sound appears."
],
"decisive_result": "A Lean module, with no sorry, whose lemmas are exactly Cook's background, glider catalog and collision table, each proved by reduction of the cellular-automaton step.",
"failure_signature": "Any transcribed collision evolves to a word different from Cook's table, or a collision is too long for blocked kernel reduction and no alternative proof is found. A mismatched collision is also evidence against the published table.",
"tools": [
"lean4",
"cellular-automaton-simulation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.15.1.1",
"title": "Universality in Elementary Cellular Automata",
"year": 2004,
"note": "Cook 2004, Universality in elementary cellular automata"
}
],
"difficulty": "hard",
"missing": "Nothing external. The obstacle is proof size: Cook's table is a long finite case analysis, and Lean's kernel may force very small blocks.",
"nanopub": null
},
{
"id": "rule110-universality-machine-checked--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Formalize cyclic-tag universality before any glider",
"kind": "formalization",
"method": "Separate the computability half of Cook's argument from the cellular-automaton half, so a bug in a glider collision cannot hide inside a tag-system lemma. Define a cyclic tag system as a finite list of appendants, a binary work string, and the rotation that appends the next word and deletes the front bit. Formalize the Cocke–Minsky simulation of a binary Turing machine by a 2-tag system, then Cook's simulation of a 2-tag system by a cyclic tag system. The main theorem is: for every Turing machine and input there is a cyclic tag system and an encoding of the input such that the tag system halts if and only if the machine does, with an explicit bound on the number of tag steps per machine step. Do not mention Rule 110 in this module. Hypotheses that Cook leaves informal (the precise halting convention, the treatment of the empty appendant) stay as named hypotheses rather than as sorry.",
"steps": [
"Define Turing machines, 2-tag systems and cyclic tag systems with explicit halting.",
"Formalize the Cocke–Minsky encoding and prove one-step correspondence.",
"Formalize Cook's tag-to-cyclic-tag encoding and prove one-step correspondence.",
"Compose the two simulations and extract the overhead bound.",
"Audit axioms. Leave Rule 110 out of the imports."
],
"decisive_result": "A no-sorry theorem that cyclic tag systems simulate every Turing machine, with the step overhead as a proved bound.",
"failure_signature": "A one-step correspondence fails on a finite witness (a tag system whose encoded run disagrees with the machine), or the halting convention cannot be stated without an extra axiom.",
"tools": [
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.15.1.1",
"title": "Universality in Elementary Cellular Automata",
"year": 2004,
"note": "Cook 2004, cyclic tag systems and the reduction from tag systems"
},
{
"url": "https://doi.org/10.1016/j.tcs.2008.09.051",
"title": "The complexity of small universal Turing machines: A survey",
"year": 2009,
"note": "Woods and Neary 2009, survey of the small-machine and tag-system chain"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "rule110-universality-machine-checked--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Compose glider lemmas into the emulation theorem",
"kind": "proof",
"method": "Assuming the collision lemmas and the cyclic-tag theorem as already checked modules, formalize Cook's compiler: a function from a cyclic tag system and a finite input to a finite set of gliders on the background. The theorem is a lockstep bisimulation. Each tag step is a fixed finite sequence of the checked collisions, and the encoding of the work string is an invariant of that sequence. Prefer the Neary–Woods polynomial-overhead presentation where it cites the same gliders, and prove the polynomial bound as a separate lemma rather than hiding it in the invariant. This module is forbidden to re-verify collisions by simulation; it may only apply the lemmas from the catalog. That split is what makes a failure here different from a failure in the table.",
"steps": [
"Define the compiler from cyclic tag systems to Rule 110 configurations using only named gliders.",
"State the representation invariant (work string, phase, and head glider).",
"Prove that each tag operation is a composition of catalogued collisions.",
"Prove the overhead bound, polynomial if the Neary–Woods schedule is the one formalized.",
"Compose with cyclic-tag universality to obtain the Rule 110 universality theorem and audit axioms."
],
"decisive_result": "A no-sorry theorem: for every cyclic tag system and input there is a Rule 110 configuration whose evolution encodes the computation, with gliders, collisions and background used only as prior lemmas.",
"failure_signature": "The invariant fails on some tag operation even though every individual collision lemma is true, which means the compiler, not the table, is wrong. Partial gates with no bisimulation do not count.",
"tools": [
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.15.1.1",
"title": "Universality in Elementary Cellular Automata",
"year": 2004,
"note": "Cook 2004, glider emulation of cyclic tag systems"
},
{
"url": "https://doi.org/10.1007/11786986_13",
"title": "P-completeness of Cellular Automaton Rule 110",
"year": 2006,
"note": "Neary and Woods 2006, polynomial-time emulation and P-completeness of Rule 110"
}
],
"difficulty": "long-shot",
"missing": "A month of agent time is unlikely to finish the bisimulation even after the collision lemmas exist. The missing piece is proof effort on the compiler invariant, not data or a new algorithm.",
"nanopub": null
}
]
},
{
"id": "smallest-universal-turing-machine-2-3",
"title": "Settle whether the (2,3) Turing machine is universal under a conventional definition",
"field": "mathematics",
"tier": "A",
"question": "Is Wolfram's 2-state 3-color Turing machine universal when the initial tape is required to be finite-support or eventually periodic, rather than the non-periodic infinite background used in Smith's prize proof?",
"known_state": "Smith's 2007 proof (published 2020) uses an infinite, non-periodic but computable initial tape, which several authors argue smuggles computation into the input. Standard small-universal-machine results (Neary and Woods) use blank or periodic backgrounds. Whether the (2,3) machine is universal under those definitions is open.",
"acceptance_criteria": [
"Either a construction with a finite-support or eventually periodic initial tape that emulates a universal system, formalized in Lean 4, or a proof that no such construction exists (for example by showing the machine's behaviour on all eventually periodic tapes is decidable)."
],
"falsifiers": [
"A decidability proof for eventually periodic inputs would refute universality under that definition."
],
"required_capabilities": [
"lean4",
"turing-machine-enumeration",
"computability-theory"
],
"data_access": "None needed; the machine is a 6-entry table.",
"scales": [],
"frontier_problem": null,
"related_claims": [
{
"id": "wolfram2002-ch11-27",
"label": "A 2-state 5-color Turing machine is universal via rule 110",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": "supports",
"theorem": false
},
{
"id": "wolfram2002-ch11-28",
"label": "No 2-state 2-color Turing machine can be universal",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": "supports",
"theorem": false
},
{
"id": "wolfram2002-ch12-23",
"label": "Exhaustive search of 2-state 2-color Turing machines finds three growth rates",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": "refutes",
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.29.1.1",
"title": "Universality of Wolfram’s 2, 3 Turing Machine",
"year": 2020,
"note": "Smith 2020, Universality of Wolfram's 2, 3 Turing Machine"
},
{
"url": "https://doi.org/10.1016/j.tcs.2008.09.051",
"title": "The complexity of small universal Turing machines: A survey",
"year": 2009,
"note": "Woods and Neary 2009, The complexity of small universal Turing machines: a survey"
}
],
"nanopub": null,
"approaches": [
{
"id": "smallest-universal-turing-machine-2-3--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Decidability route: regular invariants for the (2,3) machine on periodic backgrounds",
"kind": "computation",
"method": "Fix the decision problem first, since the (2,3) machine has no halt state: say 'does the head ever reach cell n' or 'does the tape ever contain word v within distance n of the origin', which is how Smith's emulation reads off computation. Restrict to eventually periodic tapes u^ω · w · v^ω. Use the bbchallenge decider family, adapted to non-halting machines: closed tape languages, finite-automata reduction (FAR) and translated-cycler detection. They search for a regular language L of configurations that contains the initial configuration, is closed under the step relation, and implies the property or its negation. Enumerate background pairs (u, v) with |u|, |v| ≤ 8 and windows |w| ≤ 12. If a uniform regular invariant family appears (for example, the machine reduces on every periodic background to a counter or pushdown system whose reachability is decidable), state and prove it in Lean as a decidability theorem. Systematic non-existence of small invariants, together with growth patterns resembling Smith's system 0, is evidence in the other direction and identifies the hardest background classes.",
"steps": [
"Write the (2,3) rule table and a reference simulator; fix the observable (head-position or word-occurrence predicate) as the decision problem.",
"Port FAR and closed-tape-language deciders to non-halting predicates; validate on machines whose behaviour is known.",
"Run the enumeration over (u, w, v) and record which configurations admit regular invariants and with how many states.",
"If invariants follow a pattern, generalize by hand and formalize the decidability proof in Lean 4; otherwise publish the census of undecided cases."
],
"decisive_result": "A Lean-checked theorem that the chosen predicate is decidable for all eventually periodic initial tapes, which refutes universality under that definition. Short of that, a census giving, for each background up to the bound, either a verified regular invariant or an explicit undecided configuration.",
"failure_signature": "Many small backgrounds, including the blank one, resist all regular invariants up to large state counts, while showing irregular, nested head motion; the decidability route then probably cannot succeed.",
"tools": [
"turing-machine-enumeration",
"automata-theory",
"lean4",
"large-scale-computation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.29.1.1",
"title": "Universality of Wolfram’s 2, 3 Turing Machine",
"year": 2020,
"note": "Smith 2020, Universality of Wolfram's 2,3 Turing machine (non-periodic background)"
},
{
"url": "https://doi.org/10.1016/j.tcs.2008.09.051",
"title": "The complexity of small universal Turing machines: A survey",
"year": 2009,
"note": "Woods and Neary 2009, survey defining standard, semi-weak and weak universality"
},
{
"url": "https://bbchallenge.org/",
"title": null,
"year": null,
"note": "Busy Beaver Challenge: decider methods (FAR, closed tape languages) for proving non-halting via regular invariants"
}
],
"difficulty": "long-shot",
"missing": "Mathematics beyond current automation if the machine is in fact universal on periodic backgrounds; adapting halting-oriented deciders to observables is new work.",
"nanopub": null
},
{
"id": "smallest-universal-turing-machine-2-3--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Construction route: replace Smith's non-periodic tape with a periodic refuel gadget",
"kind": "mixed",
"method": "Smith's proof is a chain: the (2,3) machine emulates system 0, which emulates a sequence of intermediate systems ending in a cyclic tag system, and the infinite non-periodic tape supplies the structures the chain consumes. Implement each layer as an executable specification and instrument the emulation to find exactly which layer consumes non-periodic structure, and what that structure is (for example, blocks whose lengths grow with the number of emulated steps). Then search for a replacement: a finite gadget on a periodic background that regenerates the consumed structure on demand. Neary and Woods' weakly universal small machines show this kind of periodic-background refuelling can work. The search runs over short periodic background words and finite gadgets (SAT or bounded model checking on the layer's step relation), with the requirement that the composed system still emulates the CTS.",
"steps": [
"Implement Smith's systems 0 through 5 and the CTS layer as executable specs, with cross-checking simulation from a common CTS input.",
"Instrument the emulation to log the background symbols consumed per emulated CTS step and fit their growth law.",
"Formulate a SAT or bounded-model-checking search for a periodic background plus finite gadget that produces the consumed structure; run for background periods up to 40.",
"If a candidate is found, verify by long simulation and then formalize the composed emulation in Lean."
],
"decisive_result": "An explicit eventually periodic initial tape for which the (2,3) machine emulates a universal CTS, verified by simulation for at least 10^6 emulated steps and then proved in Lean. Even without success, the instrumented chain is a precise statement of what the non-periodic tape contributes, which the current literature lacks.",
"failure_signature": "The consumed structure grows superlinearly in a way no finite gadget on a periodic background can produce (for example, its growth law implies the background must encode an unbounded counter), or the search exhausts all gadgets up to the size bound.",
"tools": [
"turing-machine-enumeration",
"sat-solver",
"lean4",
"large-scale-computation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.29.1.1",
"title": "Universality of Wolfram’s 2, 3 Turing Machine",
"year": 2020,
"note": "Smith 2020, the emulation chain to instrument"
},
{
"url": "https://doi.org/10.1007/978-3-642-03409-1_24",
"title": "Small Weakly Universal Turing Machines",
"year": 2009,
"note": "Neary and Woods 2009, Small weakly universal Turing machines (periodic-background constructions)"
},
{
"url": "https://www.wolframscience.com/prizes/tm23/",
"title": null,
"year": null,
"note": "Wolfram 2,3 Turing Machine Research Prize page with the proof materials"
}
],
"difficulty": "long-shot",
"missing": "Possibly a genuinely new idea; the search space for gadgets may be far too large.",
"nanopub": null
},
{
"id": "smallest-universal-turing-machine-2-3--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Formalize Smith's proof and bound the computational power of its initial tape",
"kind": "formalization",
"method": "Machine-check what is currently claimed, with the definitional issue stated exactly. Formalize in Lean 4 Smith's theorem: for every CTS there is a computable initial tape on which the (2,3) machine emulates it. Then prove a theorem about the tape's complexity, which is what the dispute is about. Candidate statements: the tape is generated by a fixed finite-state transducer from the CTS description and a fixed infinite word; it is a fixed morphic or automatic sequence modified by a finite prefix; or it is computable in linear time by a non-universal device. Each turns 'the encoding smuggles in computation' into a checkable claim. If the tape is automatic or morphic, the result places the (2,3) machine in a precisely named universality class between weak universality (periodic background) and Smith's notion.",
"steps": [
"Formalize the (2,3) machine, CTS and Smith's intermediate systems in Lean; prove each emulation step.",
"Extract the explicit initial-tape generator from the proof and formalize it as a function.",
"Prove the generator lies in a named weak class (finite transducer over a fixed sequence, automatic sequence, or linear-time and non-universal), or show it cannot, by exhibiting an encoding step that requires unbounded computation.",
"Write a short mapping of the critics' objections onto the formal statements."
],
"decisive_result": "A Lean module proving Smith's theorem together with a proved upper bound on the complexity of the required tape (for example 'the tape is k-automatic'). That settles the definitional dispute in the form 'universal under definition D', even though it leaves the periodic-background case open.",
"failure_signature": "Formalization uncovers a gap in Smith's chain that cannot be closed, or the tape generator turns out to need unbounded computation in a way that makes every weak-class statement false.",
"tools": [
"lean4",
"mathlib",
"computability-theory"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.29.1.1",
"title": "Universality of Wolfram’s 2, 3 Turing Machine",
"year": 2020,
"note": "Smith 2020, the proof to formalize"
},
{
"url": "https://doi.org/10.1016/j.tcs.2008.09.051",
"title": "The complexity of small universal Turing machines: A survey",
"year": 2009,
"note": "Woods and Neary 2009, universality notions"
},
{
"url": "https://github.com/leanprover-community/mathlib4",
"title": null,
"year": null,
"note": "Mathlib.Computability: Turing machine models, partial recursive functions"
}
],
"difficulty": "hard",
"missing": "Smith's chain is long (about 40 pages of construction); the full formalization is likely beyond a month, though the tape-complexity analysis can be done on the informal proof first.",
"nanopub": null
},
{
"id": "smallest-universal-turing-machine-2-3--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Search for periodic-background tag-system compilers",
"kind": "computation",
"method": "Treat an eventually periodic tape as left period, finite defect and right period. Enumerate periods up to 12 and defects up to 40 cells, then use symbolic execution and counterexample-guided synthesis to seek macro-configurations representing tag symbols and clock phases. Require compositional transitions, not isolated long transients, and certify candidates by exact simulation.",
"steps": [
"Implement the six-entry machine and symmetry-reduced eventually periodic configurations.",
"Mine recurring spacetime motifs and infer finite-state macro-boundaries.",
"Synthesize encodings of a cyclic tag system with bounded phase jitter.",
"For any candidate, prove finite transition certificates and formulate an induction invariant in Lean."
],
"decisive_result": "A periodic-background compiler with exact certificates for both tag productions, erasure, appending and clock restoration, followed by a Lean simulation theorem.",
"failure_signature": "No closed macro-transition set survives adversarial compositions of 10,000 random tag steps despite exhaustive periods and defect bounds; this falsifies the searched compiler family, not universality.",
"tools": [
"turing-machine-enumeration",
"large-scale-computation",
"lean4"
],
"datasets": [],
"references": [],
"difficulty": "long-shot",
"missing": "No completeness theorem says universality must use a short periodic background or local tag encoding.",
"nanopub": null
},
{
"id": "smallest-universal-turing-machine-2-3--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Decidability via bounded crossing sequences",
"kind": "proof",
"method": "Seek a negative result for eventually periodic tapes. Define crossing sequences at cell boundaries and prove that the 2-state control plus periodic tails forces either bounded crossing complexity or a pumpable translation cycle. A finite quotient of boundary state, tape color and tail phase would then decide halting and exclude conventional universality.",
"steps": [
"Enumerate local boundary-return types for each left and right tail period up to symbolic period p.",
"Conjecture a monotone measure or pumping lemma from exhaustive runs up to 10^9 steps.",
"Formalize crossing sequences and the tail-phase quotient in Lean.",
"Derive a halting decider if the bounded-crossing lemma is proved."
],
"decisive_result": "A Lean theorem that every eventually periodic input either enters a computably detected cycle or halts, with an explicit computable bound from the finite defect and periods.",
"failure_signature": "A generated family has unbounded nonrepeating crossing sequences or simulates an unbounded counter, directly refuting the proposed bounded-crossing lemma.",
"tools": [
"lean4",
"turing-machine-enumeration",
"computability-theory"
],
"datasets": [],
"references": [],
"difficulty": "long-shot",
"missing": "A viable global invariant; finite experiments cannot establish decidability.",
"nanopub": null
},
{
"id": "smallest-universal-turing-machine-2-3--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Prove Smith's tape is not eventually periodic",
"kind": "formalization",
"method": "Encode Wolfram machine 596440 exactly as published: state 1 on colors 0, 1, 2 writes (1, right, state 2), (2, left, state 1), (1, left, state 1); state 2 writes (2, left, state 1), (2, right, state 2), (0, right, state 1). Define eventually periodic as the existence of a center bound N and periods p, q such that the tape repeats with period p to the right of N and period q to the left. Define finite support as the case where both periodic parts are the blank symbol. Formalize Smith's doubling recurrence for the infinite initial condition, as described in his prize proof and in the public development at the Wolfram Institute Lean notes, and prove that this particular tape is not eventually periodic. Separately audit that public development against this repo's rules: no sorry, no extra axioms. The audit's job is to isolate the infinite-tape hypothesis, not to re-prove universality under a periodic background.",
"steps": [
"Define machine 596440 and the two tape restrictions as Lean predicates.",
"Transcribe Smith's doubling construction of the infinite initial tape.",
"Prove that tape is not eventually periodic.",
"Audit the public Lean development and list every hypothesis it still assumes, in particular the non-periodic background.",
"Reject the module if universality is stated without that hypothesis."
],
"decisive_result": "A no-sorry lemma that Smith's infinite initial condition is not eventually periodic, plus an axiom audit showing his universality theorem uses that tape.",
"failure_signature": "The doubling recurrence in fact produces an eventually periodic tape, or the public development's universality theorem does not depend on a non-periodic background. Either outcome would change the status of the objection rather than confirm it.",
"tools": [
"lean4"
],
"datasets": [
{
"name": "Wolfram (2,3) universality in Lean",
"url": "https://wolframinstitute.github.io/TuringMachine/",
"note": "Public development claiming theorems Smith.wolfram23_universal_ic and Smith.wolfram23_infinite_ic; used as the artifact to audit, not as a finished no-sorry proof."
},
{
"name": "Wolfram 2,3 Turing machine technical details",
"url": "https://www.wolframscience.com/prizes/tm23/technicaldetails.html",
"note": "Transition table of machine 596440."
}
],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.29.1.1",
"title": "Universality of Wolfram’s 2, 3 Turing Machine",
"year": 2020,
"note": "Smith 2020, Universality of Wolfram's 2, 3 Turing Machine"
}
],
"difficulty": "hard",
"missing": "The public Lean notes list open items. Closing those items under a no-sorry rule may exceed a month even though the non-periodicity lemma itself is the smaller target.",
"nanopub": null
},
{
"id": "smallest-universal-turing-machine-2-3--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Decide the machine on eventually periodic tapes",
"kind": "proof",
"method": "Try to refute conventional universality by a regularity theorem. Fix an eventually periodic initial tape with left period q, right period p and a finite center of length L. The head will destroy periodicity, so the configuration space is not the finite set of phases. The attack is to find an invariant of machine 596440 that restores finiteness: a bound on the number of non-periodic cells, or a vector of local statistics (for example counts of 1s and 2s modulo a small integer) that determines the future and takes values in a finite monoid. If every space-time diagram from an eventually periodic tape is generated by a finite automaton whose state is that invariant, then halting and reachability of any finite target pattern are decidable, which refutes universality under the conventional restriction. The invariant has to be proved for all p, q and L, not checked up to a bound. A bound-only result belongs to the search approach and is not this theorem.",
"steps": [
"Define the configuration graph of machine 596440 on eventually periodic tapes.",
"Search symbolically for a finite invariant by enumerating small moduli and local window statistics.",
"If a candidate invariant closes, prove it is preserved by all six transitions.",
"Prove that the invariant's state, plus the head state, determines the next window.",
"Conclude decidability of reachability, which is the anti-universality theorem."
],
"decisive_result": "A no-sorry theorem that reachability for machine 596440 on every eventually periodic tape is decidable.",
"failure_signature": "Every candidate finite invariant fails on an explicit tape: the non-periodic region grows without a periodic residue. That kills this proof route and leaves conventional universality open.",
"tools": [
"lean4",
"turing-machine-enumeration"
],
"datasets": [
{
"name": "Wolfram 2,3 Turing machine technical details",
"url": "https://www.wolframscience.com/prizes/tm23/technicaldetails.html",
"note": "The six transitions of machine 596440."
}
],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.29.1.1",
"title": "Universality of Wolfram’s 2, 3 Turing Machine",
"year": 2020,
"note": "Smith 2020, the non-periodic background this approach is not allowed to use"
},
{
"url": "https://doi.org/10.1016/j.tcs.2008.09.051",
"title": "The complexity of small universal Turing machines: A survey",
"year": 2009,
"note": "Woods and Neary 2009, the blank and periodic-background convention for small universal machines"
}
],
"difficulty": "long-shot",
"missing": "No known finite invariant for this machine. Finding one is open mathematics, not a missing dataset.",
"nanopub": null
},
{
"id": "smallest-universal-turing-machine-2-3--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Search a bounded periodic-background encoding family",
"kind": "computation",
"method": "Search for a positive construction inside an explicit family, so exhaustion is a real partial result. The family is: a right-infinite periodic background of period at most 32 over the alphabet {0,1,2}, a finite program region of length at most 10^4, and a block encoding that uses only the block sizes already appearing in Smith's system-0 to system-3 schedule. For each candidate, simulate machine 596440 and test whether the successive visits of the head to the left end reproduce one full step of a fixed universal cyclic tag system (two appendants, the smallest Cook system that is already universal once cyclic-tag universality is granted) for at least 100 tag steps, with the background period unchanged outside the program. The search is exhaustive over periods and over program regions drawn from that block grammar, not over all strings of length 10^4. Publish the grammar, the seed, and the list of simulated candidates.",
"steps": [
"Fix the target cyclic tag system and the block grammar before any simulation.",
"Enumerate background periods from 1 to 32 and program blocks within the grammar up to length 10^4.",
"Simulate machine 596440 from each candidate for enough steps to cover 100 tag steps or until the background is corrupted.",
"Accept a candidate only if the decoded tag step matches and the background period is intact.",
"If the grammar is exhausted with no accept, report the bound as a negative partial result."
],
"decisive_result": "Either an explicit periodic-background initial tape, of period at most 32 and program length at most 10^4, that emulates 100 steps of the chosen cyclic tag system, or a machine-readable exhaustion certificate for that grammar.",
"failure_signature": "Exhaustion of this grammar does not refute conventional universality. The approach has failed to settle the question if the grammar is exhausted, and it has failed as a search if a candidate is accepted by one simulator and rejected by an independent reimplementation.",
"tools": [
"turing-machine-enumeration"
],
"datasets": [
{
"name": "Wolfram 2,3 Turing machine technical details",
"url": "https://www.wolframscience.com/prizes/tm23/technicaldetails.html",
"note": "Machine 596440 transition table used as the simulator specification."
}
],
"references": [
{
"url": "https://doi.org/10.25088/ComplexSystems.29.1.1",
"title": "Universality of Wolfram’s 2, 3 Turing Machine",
"year": 2020,
"note": "Smith 2020, source of the block sizes the grammar is allowed to use"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
}
]
},
{
"id": "rule54-universality",
"title": "Is Rule 54 computationally universal?",
"field": "mathematics",
"tier": "A",
"question": "Is elementary cellular automaton Rule 54 capable of universal computation, as Wolfram conjectures, or can a bound on its information transmission rule it out?",
"known_state": "Open since 2002. Martínez, Adamatzky and McIntosh catalogued Rule 54's gliders and collisions and built logic gates in 2006, but no complete emulation of a universal system exists. Wolfram's search of 100 million initial conditions (wolfram2002-ch11-21) found only repetitive and nested structures.",
"acceptance_criteria": [
"A construction (glider-based or otherwise) emulating a known universal system, with the collision table verified by exhaustive simulation and formalized in Lean 4, or",
"A proof that Rule 54 cannot be universal (for example a regularity or transmission bound on all configurations)."
],
"falsifiers": [
"Partial results only (new gates without a complete emulation) leave the problem open."
],
"required_capabilities": [
"cellular-automaton-simulation",
"glider-collision-search",
"lean4"
],
"data_access": "None needed.",
"scales": [],
"frontier_problem": null,
"related_claims": [
{
"id": "wolfram2002-ch11-21",
"label": "Rule 54 shows only repetitive or nested forms in 100 million initial conditions",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": "supports",
"theorem": false
},
{
"id": "wolfram2002-ch11-19",
"label": "Class 1 and 2 cellular automata cannot be universal",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": "mixed",
"theorem": false
},
{
"id": "wolfram2002-ch11-20",
"label": "Nested-pattern and additive cellular automata cannot be universal",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": "supports",
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.chaos.2005.05.013",
"title": "Phenomenology of glider collisions in cellular automaton Rule 54 and associated logical gates",
"year": 2006,
"note": "Martínez, Adamatzky and McIntosh 2006, Phenomenology of glider collisions in cellular automaton Rule 54"
}
],
"nanopub": null,
"approaches": [
{
"id": "rule54-universality--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Automated synthesis of a cyclic-tag emulation from the complete collision graph",
"kind": "computation",
"method": "Treat universality search as program synthesis over glider collisions. First compute the complete set of Rule 54 gliders on the standard ether up to period 120 and width 80 (de Bruijn diagrams), and the full binary collision table over all phases and gaps (outgoing gliders, phase shifts, and whether the outcome depends on the gap). Then search the collision graph for the four functional components of a Cook-style CTS emulation: stationary or slow 'tape' gliders encoding 0 and 1; a periodic left-moving 'program' stream encoding appendants; an annihilation or conversion reaction that reads the leading tape symbol; and a reaction that appends or deletes appendant material conditionally. Encode timing and spacing constraints as an SMT problem (z3, integer gaps modulo the ether period) and search for satisfying assignments that emulate a 2-appendant CTS. A solution is validated by direct simulation and then formalized as in the Rule 110 approach.",
"steps": [
"Compute and publish the glider catalogue with periods, velocities and phases; compare with Martínez, Adamatzky and McIntosh 2006.",
"Compute the complete binary collision table and identify collisions that are gap-sensitive, the raw material for conditional behaviour.",
"Encode CTS-emulation components as SMT constraints over glider choices and gaps; search with increasing component sizes.",
"Validate any candidate by simulating a small universal CTS for 10^7 generations and decoding."
],
"decisive_result": "An explicit initial configuration (ether, finite seed, periodic stream) on which Rule 54 emulates arbitrary CTS, validated by simulation and then proved in Lean. Short of that, a proof that the collision graph restricted to gliders up to the bound lacks a conditional-append reaction, which pins down what a universal construction would need.",
"failure_signature": "The SMT search finds no conditional reaction among gliders up to the bound; every collision outcome is independent of the data it would need to read. That is evidence that the bound is too small or that the second (impossibility) approach is the right one.",
"tools": [
"cellular-automaton-simulation",
"glider-collision-search",
"smt-solver",
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.chaos.2005.05.013",
"title": "Phenomenology of glider collisions in cellular automaton Rule 54 and associated logical gates",
"year": 2006,
"note": "Martínez, Adamatzky and McIntosh 2006, Phenomenology of glider collisions in cellular automaton Rule 54 and associated logical gates (Chaos, Solitons & Fractals 28:100). The DOI listed in problems.json resolves to a different paper."
},
{
"url": "https://doi.org/10.25088/ComplexSystems.15.1.1",
"title": "Universality in Elementary Cellular Automata",
"year": 2004,
"note": "Cook 2004, template for the CTS-emulation architecture"
},
{
"url": "https://github.com/z3prover/z3",
"title": null,
"year": null,
"note": "Z3 SMT solver for the timing-constraint search"
}
],
"difficulty": "long-shot",
"missing": "Conditional (data-reading) reactions may not exist among small gliders; the search may need structures no one has catalogued.",
"nanopub": null
},
{
"id": "rule54-universality--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Impossibility route: a potential function on glider dynamics forcing ballistic decay",
"kind": "proof",
"method": "Try to prove that Rule 54 cannot be universal in a precise sense, for example that on every finite perturbation of the ether the long-run behaviour is decidable. Strategy: (a) prove a decomposition theorem that every configuration differing from the ether on a finite set evolves, after a transient bounded in terms of the perturbation's width, into a superposition of catalogued gliders separated by ether, using a de Bruijn-diagram argument that any non-ether, non-glider region shrinks; (b) compute the full collision algebra and look for a potential Φ (a weighted glider count, or a count weighted by velocity class) that never increases in any collision and strictly decreases in some. If Φ exists and the number of gliders produced by a collision is bounded, every finite perturbation reaches, in finitely many collisions, a state of non-interacting outgoing gliders. Then 'eventual configuration' is computable, which rules out universality for the standard definition with finite seeds. The integrable reversible RCA54 of Bobenko et al., a different automaton whose name and local rule are related, has solitons that only phase-shift on collision; checking whether ECA 54's ether-restricted dynamics inherit such a conservation law is a concrete sub-question.",
"steps": [
"Compute the collision algebra (outcome multiset for each glider pair, phase and gap) from the approach-1 catalogue.",
"Solve an integer linear program for weights making Φ non-increasing over all catalogued collisions; record strict decreases.",
"Attempt the decomposition theorem: prove by exhaustive de Bruijn analysis that non-glider defects shrink, giving a bound on transient length.",
"If both hold, formalize: decomposition plus potential implies decidability of the eventual glider content; state precisely what this excludes (universality with finite seeds on ether)."
],
"decisive_result": "A proof, ideally Lean-checked, that for every finite perturbation of the ether the multiset of eventual outgoing gliders is computable. This rules out universality from finite seeds on ether, and the result states exactly which universality notions remain open (for example, infinite periodic program streams).",
"failure_signature": "The ILP is infeasible, meaning some collision cycle increases every candidate potential (glider multiplication), or an exhaustive search finds a finite perturbation that produces unboundedly many gliders (a 'gun' or breeder). Both suggest universality is plausible and point back to approach 1.",
"tools": [
"proof",
"integer-linear-programming",
"cellular-automaton-simulation",
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.chaos.2005.05.013",
"title": "Phenomenology of glider collisions in cellular automaton Rule 54 and associated logical gates",
"year": 2006,
"note": "Martínez, Adamatzky and McIntosh 2006, Rule 54 glider collisions"
},
{
"url": "https://doi.org/10.1007/BF02097234",
"title": "On two integrable cellular automata",
"year": 1993,
"note": "Bobenko, Bordemann, Gunn and Pinkall 1993, On two integrable cellular automata (the reversible RCA54, a related but distinct system)"
}
],
"difficulty": "long-shot",
"missing": "Infinite periodic program streams (as in Cook's Rule 110 proof) escape a finite-seed impossibility argument, so even success settles only one universality notion.",
"nanopub": null
},
{
"id": "rule54-universality--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Lean-certified completeness of the Rule 54 glider catalogue up to explicit bounds",
"kind": "formalization",
"method": "Both other approaches depend on knowing all gliders, and current catalogues are lists of what has been seen, not completeness theorems. Prove in Lean 4: every pattern that is spacetime-periodic on the ether, with period p ≤ P and width w ≤ W, is one of the listed gliders up to phase and translation. The method is de Bruijn-diagram enumeration. Periodic patterns of period p correspond to cycles in a finite graph whose vertices are width-2p words, so completeness reduces to enumerating cycles in finite graphs. Extract the enumeration as a certificate (cycle list plus a proof that no other cycle passes the ether boundary conditions) and check it in Lean by kernel reduction. Pick P and W so the kernel check stays feasible (P = 24 first, then extend).",
"steps": [
"Define the de Bruijn graph for period-p spacetime patterns in Lean and prove the correspondence theorem (periodic pattern ⇔ graph cycle).",
"Generate the cycle enumeration externally for each p ≤ P with ether boundary conditions.",
"Check the certificate in Lean; derive the completeness theorem for the glider list.",
"Repeat for the binary collision table: prove that outcomes for all pairs from the complete list are as tabulated."
],
"decisive_result": "A Lean theorem 'the gliders of Rule 54 on ether with period ≤ P and width ≤ W are exactly G_1 … G_k' plus a verified collision table. This does not settle universality, but turns the search space of approaches 1 and 2 from an observation into a proved object.",
"failure_signature": "The de Bruijn graphs for useful P are too large for kernel checking (they grow as 2^(2p) vertices), capping the theorem at a P too small to include the gliders Martínez et al. used for gates.",
"tools": [
"lean4",
"de-bruijn-diagrams",
"cellular-automaton-simulation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.chaos.2005.05.013",
"title": "Phenomenology of glider collisions in cellular automaton Rule 54 and associated logical gates",
"year": 2006,
"note": "Martínez, Adamatzky and McIntosh 2006, catalogue to be certified"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "rule54-universality--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Collision-circuit synthesis for Rule 54",
"kind": "computation",
"method": "Build a complete phase-resolved catalog of Rule 54 gliders and collisions, then use SAT-based spacetime synthesis to implement a universal cyclic tag system. Each gate must restore a common ether phase and tolerate every symbol context; exact traces become Lean certificates.",
"steps": [
"Enumerate periodic-domain defects of width at most 60 and classify velocity and phase.",
"Enumerate binary and ternary collisions over all relative phases.",
"Synthesize fanout, delay, annihilation and conditional production modules.",
"Compose modules into tag steps and test at least 100,000 adversarial random steps before Lean certification."
],
"decisive_result": "A finite library of phase-restoring modules that exactly emulates a known universal cyclic tag system, plus checked traces and a compositional simulation invariant.",
"failure_signature": "The exhaustive glider basis lacks either reliable fanout or conditional production under all contexts, closing this glider-circuit architecture within the declared bounds.",
"tools": [
"cellular-automaton-simulation",
"sat-solving",
"lean4"
],
"datasets": [],
"references": [],
"difficulty": "long-shot",
"missing": "Failure of one glider architecture would not prove Rule 54 nonuniversal.",
"nanopub": null
},
{
"id": "rule54-universality--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Finite-factor obstruction to universality",
"kind": "proof",
"method": "Search for a symbolic-dynamics obstruction: a finite-state transducer or invariant cone showing every finite perturbation of each Rule 54 periodic domain decomposes into ballistic particles with ultimately periodic interactions. Infer candidate invariants from de Bruijn graphs, then prove closure and an upper bound on crossing-sequence complexity in Lean.",
"steps": [
"Construct order-k de Bruijn graphs for k through 24 and minimize future-equivalence classes.",
"Infer conserved additive charges and particle interaction normal forms.",
"Attempt a finite-defect decomposition and confluence theorem.",
"If obtained, prove finite-defect prediction decidable and state the exact restricted nonuniversality result."
],
"decisive_result": "A machine-checked theorem that finite perturbations of specified periodic backgrounds have decidable long-term behavior via a finite particle quotient.",
"failure_signature": "Future-equivalence class counts grow without stabilization and collision products generate unbounded novel defects, defeating the finite-factor premise.",
"tools": [
"symbolic-dynamics",
"large-scale-computation",
"lean4"
],
"datasets": [],
"references": [],
"difficulty": "long-shot",
"missing": "Even a finite-defect theorem may not cover arbitrary computable initial configurations.",
"nanopub": null
},
{
"id": "rule54-universality--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Exhaustive collision census of the 2006 glider catalog",
"kind": "computation",
"method": "Take every glider and compound glider named by Martínez, Adamatzky and McIntosh 2006, on the periodic ether they use, and collide every ordered pair at every separation from 1 to 256 cells. Evolve each collision for 10^5 steps or until the configuration is a disjoint union of catalogued gliders on the ether. Two independent simulators must agree. From the resulting table, search for a finite gate set that simulates one append of a cyclic tag system: a memory-carrying glider that survives a round trip, a fan-out, and a collision sequence whose net effect is that append. New logic gates that do not compose into that append are recorded and then counted as failure, matching the problem's falsifier that partial gates leave the question open.",
"steps": [
"Transcribe the 2006 glider words and ether; do not invent new names.",
"Collide all pairs at separations 1 through 256, cutoff 10^5 steps, with two simulators.",
"Classify products as ether, catalogued gliders, or a new persistent object.",
"Search the table for a cyclic-tag append realized as a finite collision sequence.",
"If found, emit the sequence as a witness; if not, emit the census and stop."
],
"decisive_result": "A collision sequence, reproduced by both simulators, whose net glider content is one step of a cyclic tag system, or a complete census up to separation 256 with no such sequence.",
"failure_signature": "The census finds gates, oscillators or a new glider but no composition that implements a tag append. That is the partial-result failure the problem already names. Disagreement between the two simulators invalidates the census.",
"tools": [
"cellular-automaton-simulation",
"glider-collision-search"
],
"datasets": [],
"references": [
{
"url": null,
"note": "Martínez, Adamatzky and McIntosh 2006, Phenomenology of glider collisions in cellular automaton Rule 54"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "rule54-universality--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Prove a defect invariant that blocks unbounded memory",
"kind": "proof",
"method": "Attempt a negative theorem that applies to every finite perturbation of the ether, not only to the 2006 gliders. Rule 54's ether is periodic. Treat gliders as defects. Search for a conserved quantity taking values in a finite monoid: a sum of local window weights modulo m, or a pairing between left-going and right-going defects, that every space-time diagram must preserve. The universality-refuting conclusion is that the set of defects reachable from a finite perturbation is a regular language, so the diagram cannot encode an arbitrary binary work string. The conservation law has to be proved from the 8-bit table for every configuration, then specialized to finite ether perturbations. Compatibility of the invariant with a glider that stores an arbitrary string is an immediate failure of this route.",
"steps": [
"Define the ether and the defect of a finite window relative to it.",
"Enumerate small modular weightings and test them against the 8-bit table.",
"Prove any surviving weighting is invariant for every configuration.",
"Prove that finite perturbations stay inside a regular defect language.",
"Conclude that no such perturbation simulates a cyclic tag system on arbitrary inputs."
],
"decisive_result": "A proof that every finite perturbation of the Rule 54 ether has space-time defect language regular, hence cannot emulate a cyclic tag system on all inputs.",
"failure_signature": "The invariant, if it exists, still permits a defect whose internal tape is an arbitrary finite word. The route has then failed even if the invariant is true.",
"tools": [
"lean4",
"cellular-automaton-simulation"
],
"datasets": [],
"references": [
{
"url": null,
"note": "Martínez, Adamatzky and McIntosh 2006, source of the ether and the known defects the invariant must reproduce"
}
],
"difficulty": "long-shot",
"missing": "No published conservation law of this strength. The swarm lacks a theorem, not a dataset.",
"nanopub": null
},
{
"id": "rule54-universality--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Lean-check the published gates and their generated monoid",
"kind": "formalization",
"method": "Formalize only the collisions Martínez, Adamatzky and McIntosh actually used to build gates, as kernel-checked lemmas of the same kind as the Rule 110 table, then stop at the algebra those gates generate. Represent each gate as a relation on a finite set of glider occupancy vectors. Compute the monoid generated by composition up to a stated vector length. Prove either that the monoid contains a copy of a cyclic-tag append, or that every element is a function of a finite set of counters that cannot store an arbitrary work string. This can succeed as a non-universality proof for this gate set while the wider Rule 54 question stays open, and it fails differently from the census: the census can find a new glider that this monoid never sees.",
"steps": [
"Transcribe the published gate collisions and check them by kernel reduction.",
"Encode each gate as a relation on occupancy vectors.",
"Close the monoid under composition up to the stated length.",
"Test whether a cyclic-tag append embeds in that monoid.",
"If it does not, prove the monoid's memory is bounded by the counters found in the closure."
],
"decisive_result": "Either a checked embedding of one cyclic-tag step in the monoid of the 2006 gates, or a proof that this monoid cannot store an arbitrary work string.",
"failure_signature": "The monoid is too large to close and no embedding is found. That is an inconclusive stop, not a non-universality proof. A simulator mismatch on a published gate also fails the approach.",
"tools": [
"lean4",
"cellular-automaton-simulation"
],
"datasets": [],
"references": [
{
"url": null,
"note": "Martínez, Adamatzky and McIntosh 2006, the gate collisions to formalize"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
}
]
},
{
"id": "rule30-center-column",
"title": "The Rule 30 center column",
"field": "mathematics",
"tier": "A",
"question": "Is the center column of Rule 30 from a single black cell non-periodic, does it have asymptotic density 1/2, and does computing its n-th cell require effort growing at least linearly in n?",
"known_state": "The three Wolfram Rule 30 Prize problems (2019), all unsolved. Non-periodicity has been verified computationally far beyond a million steps (wolfram2002-ch02-05 tests the statistical claim). Partial results exist for related sequences (surjectivity of Rule 30 on bi-infinite configurations, left-permutivity), none settles the column.",
"acceptance_criteria": [
"A proof of any of the three statements, machine-checked in Lean 4.",
"A weaker but new result: a proven lower bound on the period (if periodic) that exceeds all computational checks, or a proof that the column is not eventually periodic under an added hypothesis made explicit."
],
"falsifiers": [
"Discovery of a period (would refute problem 1) or a proven density different from 1/2."
],
"required_capabilities": [
"lean4",
"symbolic-dynamics",
"large-scale-simulation"
],
"data_access": "None needed.",
"scales": [],
"frontier_problem": "Rule30 center column periodicity",
"related_claims": [
{
"id": "wolfram2002-ch02-05",
"label": "Rule 30's center column shows no deviation from randomness over a million steps",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": "supports",
"theorem": false
},
{
"id": "wolfram2002-ch12-27",
"label": "Inverse problems for rule 30 are likely NP-complete",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "wolfram2002-ch06-14",
"label": "Class 3 randomness can arise without randomness in initial conditions",
"thinker": "Stephen Wolfram",
"status": "untested",
"evidence": "supports",
"theorem": false
}
],
"references": [
{
"url": "https://www.rule30prize.org/",
"title": null,
"year": null,
"note": "Wolfram Rule 30 Prizes, 2019"
}
],
"nanopub": null,
"approaches": [
{
"id": "rule30-center-column--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Lean: left-permutivity, Jen's two-column theorem and conditional extensions",
"kind": "formalization",
"method": "Formalize the only rigorous progress on Prize problem 1 and push it as far as it goes. Rule 30 is x_i^{t+1} = x_{i−1}^t XOR (x_i^t OR x_{i+1}^t). It is left-permutive: x_{i−1}^t = x_i^{t+1} XOR (x_i^t OR x_{i+1}^t), so two adjacent columns determine everything to their left. Jen (1990, announced 1986) proved that no two columns of the single-seed pattern can both be eventually periodic. Prove in Lean: (a) the reconstruction lemma; (b) if columns i and i+1 are eventually periodic with period p, every column to the left is eventually periodic with period p; (c) Jen's theorem, from the fact that the left boundary of the light cone moves left one cell per step, contradicting periodicity of the reconstructed region. Then attempt new conditional theorems with explicit hypotheses, for example: if the center column is eventually periodic with period p from time t₀, then column +1 is not eventually periodic and its subword complexity grows at least linearly; or, the periodic-diagonal region on the left (Wolfram's observation) cannot reach the center column before time f(p).",
"steps": [
"Define Rule 30 and the single-seed evolution on ℤ × ℕ; prove the light-cone and left-permutivity lemmas.",
"Prove the reconstruction and periodicity-propagation lemmas; derive Jen's two-column theorem.",
"Formulate and attempt the conditional single-column statements; record which hypotheses each needs.",
"Report as in the Thompson Lean reports, with axioms checked."
],
"decisive_result": "A Lean module proving Jen's theorem (no two columns both eventually periodic) with only standard axioms, plus at least one new conditional theorem about the center column whose hypothesis is stated explicitly. A proof of non-periodicity itself would settle Prize problem 1.",
"failure_signature": "All conditional extensions reduce back to assuming information about a second column; the left-permutivity method then provably cannot reach a single column without new input.",
"tools": [
"lean4",
"mathlib",
"symbolic-dynamics"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/0167-2789(90)90169-p",
"title": "Aperiodicity in one-dimensional cellular automata",
"year": 1990,
"note": "Jen 1990, Aperiodicity in one-dimensional cellular automata, Physica D 45:3"
},
{
"url": "https://writings.stephenwolfram.com/2019/10/announcing-the-rule-30-prizes/",
"title": null,
"year": null,
"note": "Wolfram 2019, prize announcement; summarizes Jen's result and the left-permutivity reconstruction"
},
{
"url": "https://www.rule30prize.org/",
"title": null,
"year": null,
"note": "Rule 30 Prize problems"
}
],
"difficulty": "hard",
"missing": "Jen's theorem is tractable; a genuinely new single-column result needs a mathematical idea no one has yet found.",
"nanopub": null
},
{
"id": "rule30-center-column--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Extend the column to 10^10–10^11 bits with a pre-registered density and period test",
"kind": "computation",
"method": "Extend the known prefix beyond the published billion bits and pre-register what the extension can and cannot show. Computing n center bits needs about n²/2 cell updates, but the light cone can be trimmed: cells at distance d from the center at time t only matter if d ≤ n − t. With bit-sliced 64-wide words on GPUs this gives roughly 10^19 to 10^20 word operations for n = 10^10, which is feasible on a modest cluster. Pre-register: (a) exclusion of every eventual period p ≤ n/3 with transient ≤ n/3, using a suffix-array or Z-function scan; (b) a density test, the running deviation D(n) = #1s − n/2 compared with the law-of-iterated-logarithm envelope √(2n log log n)/2 expected for an i.i.d. fair sequence, with pass and fail bands fixed in advance; (c) block-frequency and linear-complexity profiles (Berlekamp–Massey) at block lengths 1 to 20. The result is computational evidence, recorded as data_kind 'computational', not a proof.",
"steps": [
"Validate the simulator against the published billion-bit dataset (SHA-256 match).",
"Write result.json with hypotheses, bands and planned outputs before running beyond 10^9.",
"Run to n = 10^10 (stretch 10^11) with checkpointing; publish the prefix checksum and a fetch script, not the data.",
"Run the period-exclusion scan, the LIL density test and Berlekamp–Massey linear complexity; record verdicts."
],
"decisive_result": "A certified statement that no eventual period with transient and period each ≤ n/3 exists in the first n = 10^10 bits, and density deviations inside or outside the pre-registered LIL band. A detected period would refute problem 1 outright.",
"failure_signature": "Compute or memory limits stop the run below 10^10, so it adds little beyond the published billion bits; or the trimming is shown to be wrong because the result disagrees with the reference prefix.",
"tools": [
"large-scale-computation",
"gpu",
"data-test"
],
"datasets": [
{
"name": "A Billion Bits of the Center Column of the Rule 30 Cellular Automaton (Wolfram Data Repository)",
"url": "https://datarepository.wolframcloud.com/resources/A-Billion-Bits-of-the-Center-Column-of-the-Rule-30-Cellular-Automaton",
"note": "Reference prefix to validate the simulator and as the starting baseline"
}
],
"references": [
{
"url": "https://writings.stephenwolfram.com/2019/10/announcing-the-rule-30-prizes/",
"title": null,
"year": null,
"note": "States that the column is known non-periodic for the first billion steps"
}
],
"difficulty": "tractable",
"missing": "GPU time on the order of 10^4 GPU-hours for 10^10 bits; less if a better light-cone trimming is found.",
"nanopub": null
},
{
"id": "rule30-center-column--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Restricted-model lower bounds: the column is not generated by small automata",
"kind": "mixed",
"method": "Attack weak forms of problems 1 and 3 that are provable. Problem 3 asks for a lower bound on the effort to compute bit n, and unrestricted lower bounds are out of reach, but lower bounds against restricted models are finite questions. (a) For each base k in {2, 3, 4} and state count s ≤ 12, prove that no deterministic finite automaton with output (DFAO) reading n in base k reproduces the first N center bits. Encode as SAT (variables for transition and output tables; constraints from each prefix bit) and extract unsatisfiability certificates, which are DRAT proofs checkable by a verified checker. (b) Use Walnut to test whether the prefix is consistent with low-complexity morphic generation. (c) Prove in Lean, from (a) and the definitions, the theorem 'the center column is not k-automatic with ≤ s states'. That theorem implies, for example, that no constant-memory streaming algorithm over base-k digits of n computes the column with that much memory.",
"steps": [
"Generate SAT instances for (k, s) pairs over a prefix of N = 10^6 bits (smaller if unsatisfiability is reached earlier).",
"Solve with a DRAT-producing solver; check proofs with a verified checker.",
"Formalize the DFAO definition in Lean and state the non-automaticity theorems; import the certificate check by kernel-verified decoding of the minimal refuting prefix.",
"Record the largest (k, s) reached and the shortest refuting prefix for each."
],
"decisive_result": "Lean-checked theorems 'the Rule 30 center column is not generated by any base-k DFAO with at most s states' for stated (k, s). These are new and weak; their value is a precise, citable lower bound in a restricted model, and any satisfiable instance that extends to 10^9 bits would be a striking structural discovery.",
"failure_signature": "SAT instances become intractable at small s (for example s ≤ 6), giving results too weak to matter; or proofs need native_decide-level computation that Lean cannot do in kernel.",
"tools": [
"sat-solver",
"lean4",
"automata-theory",
"walnut"
],
"datasets": [
{
"name": "A Billion Bits of the Center Column of the Rule 30 Cellular Automaton",
"url": "https://datarepository.wolframcloud.com/resources/A-Billion-Bits-of-the-Center-Column-of-the-Rule-30-Cellular-Automaton",
"note": "Prefix used to generate SAT constraints"
}
],
"references": [
{
"url": "https://github.com/Walnut-Theorem-Prover/Walnut",
"title": null,
"year": null,
"note": "Walnut, decision procedure for automatic sequences"
},
{
"url": "https://www.rule30prize.org/",
"title": null,
"year": null,
"note": "Prize problem 3 (computational irreducibility of the n-th cell)"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "rule30-center-column--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "SAT-certified lower bounds on center-column periods",
"kind": "computation",
"method": "Encode the triangular Rule 30 light cone and the assertion that the center column is eventually p-periodic after preperiod q as SAT. Use incremental solving for all p up to a declared P and q up to Q, produce DRAT certificates for unsatisfiable cases, and verify certificates independently; formalize the reduction from a certificate to the excluded finite period claim.",
"steps": [
"Generate bit-blasted Rule 30 constraints with shared subformulas.",
"Exclude every pair p <= 100000 and q <= 100000 using incremental SAT and symmetry reductions.",
"Check all UNSAT certificates with an independent verifier.",
"Formalize the constraint semantics and certificate implication in Lean."
],
"decisive_result": "A Lean-checked theorem excluding every eventual period and preperiod in the stated rectangle, exceeding previous certified bounds.",
"failure_signature": "A satisfying assignment reproduces from the single-cell seed and yields a genuine period; otherwise solver scaling before a novel bound defeats this approach.",
"tools": [
"sat-solving",
"large-scale-computation",
"lean4"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Finite lower bounds do not prove nonperiodicity.",
"nanopub": null
},
{
"id": "rule30-center-column--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Algebraic dependency and entropy bounds",
"kind": "proof",
"method": "Exploit left-permutivity to express center bits as Boolean polynomials of boundary variables, then seek a lemma that any eventual center period imposes repeated polynomial identities of increasing degree. Compute algebraic normal forms to identify a monotone leading monomial and formalize the resulting degree or influence lower bound.",
"steps": [
"Compute exact algebraic normal forms for center bits through n = 4096.",
"Conjecture and prove a recurrence for a distinguished monomial or discrete derivative.",
"Show an eventual period would force equality of incompatible derivatives.",
"Formalize the recurrence and contradiction in Lean."
],
"decisive_result": "A Lean theorem proving non-eventual-periodicity, or a new theorem that center-bit algebraic degree or influence grows unboundedly.",
"failure_signature": "The proposed distinguished monomial cancels infinitely often or the derivative recurrence admits periodic center sequences, invalidating the invariant.",
"tools": [
"lean4",
"symbolic-computation",
"cellular-automaton-simulation"
],
"datasets": [],
"references": [],
"difficulty": "long-shot",
"missing": "A currently unknown invariant connecting global left-permutivity to the single seeded center column.",
"nanopub": null
},
{
"id": "rule30-center-column--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Certify a period lower bound past the published checks",
"kind": "mixed",
"method": "Prove the elementary step in Lean: the Rule 30 cell is left XOR (center OR right). Then certify a long center-column prefix by blocked replay. An external bit-packed simulator emits a row hash every K steps; Lean replays each block from the certified start row and checks the hash and the center bit. The theorem is that if the center column is periodic with period p, then p exceeds N, where N is the length of the certified prefix with no period. N must be strictly larger than the longest prefix the Rule 30 prize page reports as already computed on the day the run starts. Read that horizon from the prize page first and write it into the pre-registration. A certified prefix that does not beat the published horizon is not a result under the acceptance criteria.",
"steps": [
"Record the computational horizon stated on the prize page before running.",
"Prove the Rule 30 step function in Lean.",
"Emit block hashes from a bit-packed simulator for a prefix longer than that horizon.",
"Check every block in Lean.",
"Prove that absence of a repeat in the certified prefix rules out every period at most N."
],
"decisive_result": "A no-sorry theorem that any period of the center column exceeds the published computational horizon recorded at the start of the run.",
"failure_signature": "A repeat is found, which refutes non-periodicity if it is a true period, or the certified N does not exceed the published horizon, in which case the approach has not met its own bar.",
"tools": [
"lean4",
"large-scale-simulation"
],
"datasets": [
{
"name": "Wolfram Rule 30 Prizes",
"url": "https://www.rule30prize.org/",
"note": "Source of the three problem statements and of the computational horizon the certified prefix must beat."
}
],
"references": [],
"difficulty": "hard",
"missing": "Lean kernel time. If the published horizon is far beyond what blocked replay can certify in a month, this approach cannot finish as stated.",
"nanopub": null
},
{
"id": "rule30-center-column--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Prove density one half from left permutivity",
"kind": "proof",
"method": "The local rule f(l, c, r) = l XOR (c OR r) is left-permutive: for every fixed center and right, the two choices of the left neighbor give opposite outputs. Prove that permutivity, then the standard consequences that can be formalized: the global map on bi-infinite configurations is surjective, and the fair Bernoulli product measure is invariant. The prize question is not about almost every configuration. It is about the single orbit of the configuration with one black cell. The remaining lemma is equidistribution of that orbit's center cell: the visit frequency of black at the center tends to 1/2. State equidistribution as a separate theorem whose hypotheses, if any extra ergodicity assumption is used, are written explicitly. An extra assumption means the prize problem is not solved.",
"steps": [
"Prove left permutivity from the XOR-OR formula.",
"Prove surjectivity on bi-infinite configurations.",
"Prove invariance of the fair Bernoulli measure.",
"Attempt equidistribution of the single-seed center cell with no added assumption.",
"If an ergodicity hypothesis is required, stop and label the prize statement unproved."
],
"decisive_result": "A no-sorry proof that the center-cell frequency along the single-seed orbit tends to 1/2, with no hypothesis beyond the rule and the initial cell.",
"failure_signature": "The measure lemmas go through but equidistribution of this orbit needs an assumption that is itself open. The approach has then failed on the prize statement even if the lemmas are worth keeping.",
"tools": [
"lean4",
"symbolic-dynamics"
],
"datasets": [],
"references": [
{
"url": null,
"note": "Wolfram Rule 30 Prizes, 2019, problem 2: each color occurs equally often on average. Page: https://www.rule30prize.org/"
}
],
"difficulty": "long-shot",
"missing": "A proof idea that upgrades invariance of Bernoulli measure to equidistribution of one explicit orbit. That is the open step.",
"nanopub": null
},
{
"id": "rule30-center-column--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Show the center column is not 2-automatic",
"kind": "proof",
"method": "Attack the linear-effort prize by way of automatic sequences. A shortcut that computed the n-th center cell from the binary digits of n in o(n) time, uniformly, would make the sequence 2-automatic or at least would give a finite 2-kernel. Compute the 2-kernel of the center column: the subsequences u(2^k n + r) for residues r. Prove in Lean that the step function commutes with the kernel construction in the usual way for a cellular automaton. Then exhibit more distinct kernel sequences than any finite automaton of a given size can hold, using a certified prefix from the first approach as the witness for the small kernels, and a theorem that a new distinct subsequence appears at arbitrarily large scales. Refuting all linear recurrences over GF(2) up to a fixed order is a weaker lemma and should be stated separately; it does not by itself give an Ω(n) lower bound.",
"steps": [
"Define the 2-kernel of the center column.",
"Prove the Rule 30 step constrains kernel sequences.",
"Certify, from a checked prefix, that the kernel already has more than M distinct sequences.",
"Prove distinctness persists at arbitrary scale, or isolate the missing lemma.",
"Separately refute linear recurrences up to an order fixed by the certified prefix, and do not call that the prize theorem."
],
"decisive_result": "A proof that the center column is not 2-automatic, together with a reduction showing that any uniform o(n) digit-based algorithm would imply automaticity. That is a form of the linear-effort lower bound, not yet a general Ω(n) time lower bound against all algorithms.",
"failure_signature": "The kernel collapses to a finite set, or a new distinct subsequence cannot be forced past the certified prefix. Recurrence-refutation alone is a failure to reach the prize statement.",
"tools": [
"lean4",
"large-scale-simulation"
],
"datasets": [
{
"name": "Wolfram Rule 30 Prizes",
"url": "https://www.rule30prize.org/",
"note": "Problem 3 asks for at least linear effort in n for the n-th center cell."
}
],
"references": [],
"difficulty": "long-shot",
"missing": "A general time lower bound against arbitrary algorithms is beyond a kernel argument. This approach only targets digit-based and automatic shortcuts.",
"nanopub": null
}
]
},
{
"id": "turing-capable-network-topologies",
"title": "Classify the network topologies that admit diffusion-driven instability",
"field": "both",
"tier": "A",
"question": "For reaction networks of n species with m diffusing, which interaction sign-patterns (topologies) admit a Turing instability for some parameters, and is it true that for n >= 3 almost all topologies with at least one feedback loop do?",
"known_state": "Turing (1952) settled n = 2 (activator-inhibitor or substrate-depletion sign patterns, with the inhibitor diffusing faster). Marcon et al. (2016) and Scholes et al. (2019) enumerated 2- and 3-node topologies numerically and found many that pattern without differential diffusivity, but there is no proof characterizing the Turing-capable set for general n, and the genericity conjecture is numerical only. turing-ddi has formal status partial in the repo.",
"acceptance_criteria": [
"A theorem, machine-checked in Lean 4 or at least rigorously proved, giving necessary and sufficient sign-pattern conditions for the existence of parameters producing a Turing instability for n = 3, and",
"An exhaustive, reproducible enumeration for n = 3 and n = 4 against which the theorem is checked."
],
"falsifiers": [
"A 3-node topology that the enumeration finds Turing-capable but the theorem excludes, or vice versa."
],
"required_capabilities": [
"linear-algebra",
"sign-pattern-matrices",
"enumeration",
"lean4"
],
"data_access": "None needed; verification against published topology atlases (Marcon 2016, Scholes 2019) which are open access.",
"scales": [
"tissue",
"developmental"
],
"frontier_problem": null,
"related_claims": [
{
"id": "turing-ddi",
"label": "Turing diffusion-driven instability",
"thinker": "Alan Turing",
"status": "partial",
"evidence": "refutes",
"theorem": false
},
{
"id": "turing-wavelength",
"label": "Turing wavelength selection",
"thinker": "Alan Turing",
"status": "proved",
"evidence": null,
"theorem": true
},
{
"id": "turing-morphogen-hypothesis",
"label": "Reaction-diffusion of morphogens accounts for biological pattern",
"thinker": "Alan Turing",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1098/rstb.1952.0012",
"title": "The chemical basis of morphogenesis",
"year": 1952,
"note": "Turing 1952, The chemical basis of morphogenesis"
},
{
"url": "https://doi.org/10.7554/eLife.14022",
"title": "High-throughput mathematical analysis identifies Turing networks for patterning with equally diffusing signals",
"year": 2016,
"note": "Marcon et al. 2016, High-throughput mathematical analysis identifies Turing networks for patterning with equally diffusing signals"
},
{
"url": "https://doi.org/10.1016/j.cels.2019.07.007",
"title": "A Comprehensive Network Atlas Reveals That Turing Patterns Are Common but Not Robust",
"year": 2019,
"note": "Scholes et al. 2019, A comprehensive network atlas reveals that Turing patterns are common but not robust"
}
],
"nanopub": null,
"approaches": [
{
"id": "turing-capable-network-topologies--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Sign-pattern theorem via additive D-stability and sign stability",
"kind": "proof",
"method": "Reformulate. With linearization J at a stable steady state and diagonal diffusion D ≥ 0, a Turing instability is J stable with J − k²D unstable for some k > 0. Absorbing k² into D, the pattern P is Turing-capable (all species may diffuse) iff some realization J of P is stable but not additively D-stable, that is, J − D is unstable for some nonnegative diagonal D (Cross 1978). Lemma A: sign-stable patterns, characterized combinatorially by Quirk–Ruppert and Jeffries–Klee–van den Driessche (nonpositive diagonal, no positive 2-cycles, no cycles of length ≥ 3, nonzero determinant, a colour test), stay sign-stable when negative diagonal is added, hence are Turing-incapable. Conjecture B for n = 3: P is Turing-capable iff P is potentially stable and not sign-stable. Prove B by case analysis over which Jeffries condition fails (a 3-cycle, a positive self-loop, a mutualistic 2-cycle, or the colour test). For each case give an explicit parametric family (J(ε), D) that is stable, satisfying the 3×3 Routh–Hurwitz conditions a₁ > 0, a₃ > 0, a₁a₂ > a₃, with det(J − D) of the wrong sign or Routh–Hurwitz violated for J − D. For m < n diffusing species, D is supported on the diffusing set S, and the analogous criterion involves principal submatrices of J off S (Satnoianu et al. 2000).",
"steps": [
"Prove in Lean: for 3×3 real matrices, define stability by the Routh–Hurwitz inequalities on the characteristic polynomial coefficients, and prove that the Turing condition is equivalent to 'not additively D-stable'.",
"Prove Lemma A (sign-stable implies Turing-incapable) from the Jeffries conditions.",
"For each failure mode of sign stability, construct witness families and prove the inequalities with nlinarith or polyrith.",
"Extend to restricted diffusion sets S ⊂ {1,2,3} and state the theorem for each m.",
"Check the theorem against approach 2's enumeration (any disagreement is a counterexample to B)."
],
"decisive_result": "A Lean-checked theorem: for n = 3 and each diffusing set S, the Turing-capable sign patterns are exactly those satisfying an explicit combinatorial condition (conjecturally 'potentially stable and not S-restricted sign-stable'), with zero disagreements against the exhaustive enumeration.",
"failure_signature": "The enumeration finds a potentially stable, non-sign-stable pattern that is Turing-incapable (every stable realization is additively D-stable without being sign-stable), so Conjecture B is false and the characterization needs a finer invariant.",
"tools": [
"lean4",
"mathlib",
"sign-pattern-matrices",
"linear-algebra"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rstb.1952.0012",
"title": "The chemical basis of morphogenesis",
"year": 1952,
"note": "Turing 1952, The chemical basis of morphogenesis"
},
{
"url": "https://doi.org/10.1016/0024-3795(78)90021-6",
"title": "Three types of matrix stability",
"year": 1978,
"note": "Cross 1978, Three types of matrix stability (D-stability, additive D-stability)"
},
{
"url": "https://doi.org/10.4153/CJM-1977-035-3",
"title": "When is a Matrix Sign Stable?",
"year": 1977,
"note": "Jeffries, Klee and van den Driessche 1977, When is a matrix sign stable?"
},
{
"url": "https://doi.org/10.2307/2295838",
"title": "Qualitative Economics and the Stability of Equilibrium",
"year": 1965,
"note": "Quirk and Ruppert 1965, Qualitative economics and the stability of equilibrium"
},
{
"url": "https://doi.org/10.1007/s002850000056",
"title": "Turing instabilities in general systems",
"year": 2000,
"note": "Satnoianu, Menzinger and Maini 2000, Turing instabilities in general systems"
}
],
"difficulty": "hard",
"missing": "Mathlib lacks a ready Routh–Hurwitz theorem connecting coefficient inequalities to root location; for n = 3 it must be proved, or stability defined by the coefficient conditions with that equivalence proved separately.",
"nanopub": null
},
{
"id": "turing-capable-network-topologies--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Exact exhaustive enumeration for n = 3 and n = 4 with certificates",
"kind": "computation",
"method": "Decide Turing-capability for every sign pattern exactly, not by sampling. For n = 3 there are 3^9 = 19,683 patterns, about 3,400 up to relabelling. For each pattern and diffusing set S, the question ∃J consistent with P, ∃D ≥ 0 supported on S, ∃k: J stable and J − k²D unstable is a sentence in the existential theory of the reals, decidable by cylindrical algebraic decomposition or nlsat. Pipeline: (1) random-search for a numerical witness (log-uniform entries, D spanning six orders of magnitude); (2) if found, rationalize and verify exactly (rational arithmetic, Routh–Hurwitz), and emit the witness as a Lean `norm_num` lemma; (3) otherwise run Redlog CAD or z3 nlsat on the exact sentence and keep the 'unsat' result, or prove incapability from sign stability (approach 1, Lemma A). For n = 4, 3^16 ≈ 4.3 × 10^7 patterns, about 1.8 × 10^6 up to relabelling. Numerical witnesses should settle most capable patterns; the unresolved remainder goes to nlsat with a time cap, and capped cases are reported as open. Compare with the Marcon et al. 2016 and Scholes et al. 2019 atlases pattern by pattern.",
"steps": [
"Canonicalize patterns under node relabelling (respecting S) and generate the work list.",
"Run witness search with exact rational verification; export capable witnesses as Lean lemmas.",
"Run CAD or nlsat on unresolved patterns; record certificates or timeouts.",
"Cross-check with the published atlases; publish the classification table with SHA-256 and reproduction scripts."
],
"decisive_result": "A complete n = 3 table (every pattern × S labelled capable with an exact rational witness, or incapable with a CAD/nlsat certificate or sign-stability proof), and an n = 4 table with the fraction left undecided stated. Any capable pattern that a published atlas lists as non-Turing, or the reverse, is reported.",
"failure_signature": "CAD or nlsat times out on a substantial set of n = 3 patterns, so incapability cannot be certified and the table is incomplete even at n = 3.",
"tools": [
"enumeration",
"computer-algebra",
"smt-solver",
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.7554/eLife.14022",
"title": "High-throughput mathematical analysis identifies Turing networks for patterning with equally diffusing signals",
"year": 2016,
"note": "Marcon et al. 2016, atlas of 2- and 3-node Turing networks"
},
{
"url": "https://doi.org/10.1016/j.cels.2019.07.007",
"title": "A Comprehensive Network Atlas Reveals That Turing Patterns Are Common but Not Robust",
"year": 2019,
"note": "Scholes et al. 2019, comprehensive network atlas"
},
{
"url": "https://doi.org/10.1103/PhysRevX.8.021071",
"title": "Key Features of Turing Systems are Determined Purely by Network Topology",
"year": 2018,
"note": "Diego, Marcon, Müller and Sharpe 2018, Key features of Turing systems are determined purely by network topology"
},
{
"url": "https://www.redlog.eu/",
"title": null,
"year": null,
"note": "Redlog, real quantifier elimination"
},
{
"url": "https://github.com/z3prover/z3",
"title": null,
"year": null,
"note": "Z3 with the nlsat procedure for nonlinear real arithmetic"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "turing-capable-network-topologies--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Genericity for n ≥ 3 via counting non-sign-stable patterns and a 3-node embedding lemma",
"kind": "proof",
"method": "Prove the genericity conjecture without first characterizing every n. Embedding lemma: if a principal subpattern on 3 nodes is Turing-capable (with a stable realization), and the remaining nodes have negative self-loops, then the full pattern is Turing-capable. Proof sketch: realize the 3-node block with a witness, give other nodes strong self-damping −M, and scale off-block couplings by ε. For small ε and large M, the eigenvalues of J and of J − k²D are perturbations of the block eigenvalues plus values near −M, so stability and the instability persist by continuity of eigenvalues (Gershgorin plus a Rouché-type argument). Counting lemma: among n-node patterns with all diagonal entries negative, the fraction with no 3-node principal subpattern from the n = 3 capable list tends to 0 as n → ∞. This holds because the n = 3 list is nonempty and the C(n,3) triples are only weakly dependent, so a second-moment (Janson) bound applies. Together these give: almost all n-node patterns with negative self-loops are Turing-capable. Weaker hypotheses on the diagonal (for example, at least one self-loop) need a refined embedding.",
"steps": [
"State and prove the embedding lemma (eigenvalue continuity under block-diagonal dominance) in Lean, or rigorously on paper if Mathlib's spectral continuity is insufficient.",
"Take the n = 3 capable list from approach 2 and prove the counting lemma via Janson's inequality.",
"Combine into the genericity theorem; state precisely the class of patterns (diagonal sign assumptions) it covers.",
"Check numerically against approach 2's n = 4 table: predicted capable fraction versus measured."
],
"decisive_result": "A theorem: the fraction of n-node sign patterns (in the stated class) that are Turing-capable tends to 1 as n → ∞, with an explicit rate, checked against the n = 4 enumeration.",
"failure_signature": "The embedding lemma fails because coupling to the other nodes destroys stability or instability in ways small ε cannot control for some sign configurations, or the n = 4 enumeration shows a capable fraction inconsistent with the bound.",
"tools": [
"proof",
"lean4",
"combinatorics",
"enumeration"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.cels.2019.07.007",
"title": "A Comprehensive Network Atlas Reveals That Turing Patterns Are Common but Not Robust",
"year": 2019,
"note": "Scholes et al. 2019, numerical evidence that Turing topologies are common"
},
{
"url": "https://doi.org/10.1007/s002850000056",
"title": "Turing instabilities in general systems",
"year": 2000,
"note": "Satnoianu, Menzinger and Maini 2000, Turing instability in general n-species systems"
}
],
"difficulty": "hard",
"missing": "Eigenvalue-perturbation results in Mathlib are thin; the Lean version may need to be replaced by a rigorous paper proof.",
"nanopub": null
},
{
"id": "turing-capable-network-topologies--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Quantifier-eliminate all three-node sign patterns",
"kind": "mixed",
"method": "Represent a three-species Jacobian with prescribed zero/sign entries and diagonal diffusion with one to three positive diffusivities. Turing capability is an existential semialgebraic formula: stability at wave number zero plus instability for some q > 0. Use cylindrical algebraic decomposition per sign pattern, then translate compact certificates into determinant, trace and Routh-Hurwitz lemmas checked in Lean.",
"steps": [
"Canonicalize all 3-node signed directed graphs under permutation and global rescaling.",
"Write exact Routh-Hurwitz and diffusion-shift inequalities.",
"Run exact quantifier elimination and extract parameter witnesses or infeasibility certificates.",
"Prove the classification theorem and check every graph against numerical witnesses."
],
"decisive_result": "Necessary and sufficient conditions for every canonical 3-node sign pattern, with exact rational witnesses for capable classes and checked algebraic contradictions for incapable classes.",
"failure_signature": "Quantifier elimination cannot finish after decomposition, or numerical capability disagrees with an exact certificate, exposing an encoding error.",
"tools": [
"symbolic-computation",
"enumeration",
"lean4",
"linear-algebra"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Efficient proof-certificate translation from quantifier elimination into Lean.",
"nanopub": null
},
{
"id": "turing-capable-network-topologies--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Asymptotic capability fraction for random sign graphs",
"kind": "computation",
"method": "Define an explicit random topology ensemble with independent directed edges, signs and m diffusers. For n = 3 through 8, estimate existence of Turing parameters by global optimization followed by interval verification, stratified by feedback-cycle signatures. Fit finite-size scaling and seek constructive sufficient conditions based on an embedded capable motif plus weak couplings.",
"steps": [
"Pre-register graph ensemble, parameter box and interval verification criterion.",
"Enumerate n <= 5 exactly and sample at least 100000 graphs for each larger n and m.",
"Verify candidate instabilities with interval eigenvalue bounds.",
"Prove an embedding lemma that preserves instability under sufficiently small added couplings."
],
"decisive_result": "A verified lower bound on the capable fraction with 99% binomial intervals and a theorem showing a stated motif family remains capable for all n.",
"failure_signature": "Capability fraction remains bounded away from one after conditioning on feedback, or supposed motif extensions lose instability under every admissible coupling.",
"tools": [
"large-scale-computation",
"interval-arithmetic",
"lean4"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Sampling cannot by itself settle the almost-all conjecture.",
"nanopub": null
},
{
"id": "turing-capable-network-topologies--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Formalize equal-diffusion impossibility and the n=2 list",
"kind": "formalization",
"method": "Two theorems, both classical, both absent as checked lemmas. First, for every dimension n, if the diffusion matrix is a scalar multiple of the identity, every eigenvalue of J minus q squared D is an eigenvalue of J shifted by a non-positive real, so Hurwitz stability of J implies stability at every wave number. Equal diffusion cannot create a Turing instability. This does not contradict Marcon et al., who found patterning by mechanisms other than that linear instability; the theorem's statement must say diffusion-driven linear instability, not patterning. Second, for n = 2, derive the necessary and sufficient sign patterns: the inhibitor diffuses faster, and the Jacobian signs are the activator-inhibitor pattern or the substrate-depletion pattern. Use the Routh–Hurwitz criteria on the quadratic characteristic polynomial. Hypotheses are the signs and which species diffuse.",
"steps": [
"Define the Jacobian, the diagonal diffusion matrix and the dispersion relation.",
"Prove the equal-diffusion theorem for general n.",
"Specialize to n = 2 and compute the Hurwitz inequalities.",
"Case-split the sign patterns and prove the two classical patterns are exactly the Turing-capable ones.",
"Audit axioms."
],
"decisive_result": "No-sorry theorems: equal diffusivities forbid a Turing instability in every dimension, and for two species the Turing-capable sign patterns are exactly the two classical ones.",
"failure_signature": "A concrete 2-by-2 sign pattern outside those two classes, with numerical rates, that is stable without diffusion and unstable at some wave number. That would falsify the classification.",
"tools": [
"lean4",
"linear-algebra"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rstb.1952.0012",
"title": "The chemical basis of morphogenesis",
"year": 1952,
"note": "Turing 1952, The chemical basis of morphogenesis"
},
{
"url": "https://doi.org/10.7554/eLife.14022",
"title": "High-throughput mathematical analysis identifies Turing networks for patterning with equally diffusing signals",
"year": 2016,
"note": "Marcon et al. 2016, patterning with equal diffusion, which this theorem does not claim to forbid"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "turing-capable-network-topologies--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Exact n=3 sign-pattern census by interval bounds",
"kind": "computation",
"method": "Enumerate all 3^9 = 19683 sign patterns in {-1, 0, +1}^{3×3}. Quotient by simultaneous permutation of the three species only after the raw census, so the quotient cannot hide a disagreement. A pattern is Turing-capable if there exist positive rate magnitudes and a non-negative diffusion vector, not a scalar multiple of the identity, such that J is Hurwitz and J minus q squared D is not Hurwitz for some q > 0. Hurwitz for 3×3 is the three Routh–Hurwitz inequalities on the characteristic polynomial. Search by log-uniform samples of the nonzero magnitudes in [10^{-2}, 10^{2}], 200 draws per pattern, and a log grid of q. Every pattern not decided by a sample goes to interval branch-and-bound on the log-magnitudes. Capable means a concrete witness box. Incapable means the parameter domain is covered by boxes that violate the conditions. A pattern still unresolved below a minimum box width is a failure of the census, not a silent negative. Compare witnesses with the Turing-instability labels in Marcon et al. 2016 and Scholes et al. 2019, not with their broader patterning labels.",
"steps": [
"Enumerate the 19683 sign patterns and the diffusion supports.",
"Apply the Routh–Hurwitz test to samples and to the q grid.",
"Branch-and-bound every undecided pattern with interval arithmetic.",
"Emit a witness or a covering certificate for each pattern.",
"Match the capable set against the Turing labels in the two atlases and list every disagreement."
],
"decisive_result": "A complete partition of the 19683 patterns into capable and incapable, with a witness or a covering certificate for each, and an empty disagreement list against the published Turing labels or an explicit list of disagreements.",
"failure_signature": "More than 1 percent of patterns remain unresolved, or a published Turing-capable network is certified incapable, or the converse. Either is a failed census or a failed atlas entry, and the approach must say which.",
"tools": [
"enumeration",
"sign-pattern-matrices",
"linear-algebra"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.7554/eLife.14022",
"title": "High-throughput mathematical analysis identifies Turing networks for patterning with equally diffusing signals",
"year": 2016,
"note": "Marcon et al. 2016, atlas used only for its Turing-instability labels"
},
{
"url": "https://doi.org/10.1016/j.cels.2019.07.007",
"title": "A Comprehensive Network Atlas Reveals That Turing Patterns Are Common but Not Robust",
"year": 2019,
"note": "Scholes et al. 2019, network atlas of Turing patterns"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "turing-capable-network-topologies--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Monte Carlo the n=4 genericity conjecture",
"kind": "computation",
"method": "The genericity claim is that for n at least 3, almost all topologies with a feedback loop are Turing-capable. For n = 4 an exact census of 3^16 patterns is not the target. Draw 2000 sign patterns uniformly among those whose interaction digraph contains at least one directed cycle, assign diffusion to a non-empty random subset of species, and test 100 log-uniform rate draws with the same Routh–Hurwitz and wave-number sweep as the n = 3 census. Estimate the capable fraction and a Clopper–Pearson 95 percent interval. Almost-all is pre-registered as a fraction above 0.9. A cycle is required in the sample by construction, so the estimate is conditional on that event. This sample can refute the conjecture and cannot prove it.",
"steps": [
"Sample 2000 four-species sign patterns conditioned on having a directed cycle.",
"Sample diffusion supports that are not scalar.",
"Test 100 rate draws per pattern with the Hurwitz criteria.",
"Compute the capable fraction and the Clopper–Pearson interval.",
"Declare the conjecture refuted for n = 4 if the interval lies below 0.9, and unrefuted otherwise."
],
"decisive_result": "A Clopper–Pearson 95 percent interval for the Turing-capable fraction among 4-node cyclic sign patterns. The conjecture is refuted for this n if the interval lies entirely below 0.9.",
"failure_signature": "The interval covers both sides of 0.9, so the sample does not decide the conjecture. That is a failed test, not evidence for the conjecture. Patterns found capable only because diffusion was accidentally scalar are a bug, not a result.",
"tools": [
"enumeration",
"sign-pattern-matrices"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.cels.2019.07.007",
"title": "A Comprehensive Network Atlas Reveals That Turing Patterns Are Common but Not Robust",
"year": 2019,
"note": "Scholes et al. 2019, the numerical genericity claim this sample tests at n = 4"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
}
]
},
{
"id": "hamiltons-rule-exact-scope",
"title": "The exact scope of Hamilton's rule",
"field": "both",
"tier": "A",
"question": "Under precisely which population-genetic models is Hamilton's rule rb > c (i) a theorem with predictive content, (ii) an identity with no content beyond the Price equation, and (iii) false?",
"known_state": "Nowak, Tarnita and Wilson (2010) argued the rule is almost never a theorem and when general is tautological; Gardner, West and Wild (2011) replied that the regression form is exact and general. Birch and Okasha (2015) separate the readings. The dispute is partly about theorem statements, which makes it formalizable; no formalization exists. The repo records hamilton-rule as proved in the regression form and hamilton-haplodiploidy as refuted.",
"acceptance_criteria": [
"A Lean 4 development stating the general regression form, the additive-fitness form and a frequency-dependent counterexample as separate theorems, each proved or refuted, with the model assumptions as explicit hypotheses.",
"A short statement of which disputed claims in the 2010-2011 exchange correspond to which formal theorem."
],
"falsifiers": [
"A formal model in which the regression form fails to predict the direction of allele-frequency change would refute Gardner et al.'s generality claim."
],
"required_capabilities": [
"lean4",
"population-genetics",
"price-equation"
],
"data_access": "None needed for the formal part; empirical follow-up can use pedigreed cooperative-breeder datasets.",
"scales": [
"evolutionary",
"collective"
],
"frontier_problem": null,
"related_claims": [
{
"id": "hamilton-rule",
"label": "Hamilton's rule",
"thinker": "W. D. Hamilton",
"status": "proved",
"evidence": "inconclusive",
"theorem": true
},
{
"id": "hamilton-haplodiploidy",
"label": "Haplodiploidy hypothesis for eusociality",
"thinker": "W. D. Hamilton",
"status": "refuted",
"evidence": null,
"theorem": false
},
{
"id": "hamilton-lmc",
"label": "Local mate competition sex ratio",
"thinker": "W. D. Hamilton",
"status": "proved",
"evidence": "inconclusive",
"theorem": true
}
],
"references": [
{
"url": "https://doi.org/10.1016/0022-5193(64)90038-4",
"title": "The genetical evolution of social behaviour. I",
"year": 1964,
"note": "Hamilton 1964, The genetical evolution of social behaviour I"
},
{
"url": "https://doi.org/10.1038/nature09205",
"title": "The evolution of eusociality",
"year": 2010,
"note": "Nowak, Tarnita and Wilson 2010, The evolution of eusociality"
},
{
"url": "https://doi.org/10.1111/j.1420-9101.2011.02236.x",
"title": "The genetical theory of kin selection",
"year": 2011,
"note": "Gardner, West and Wild 2011, The genetical theory of kin selection"
},
{
"url": "https://doi.org/10.1093/biosci/biu196",
"title": "Kin Selection and Its Critics",
"year": 2014,
"note": "Birch and Okasha 2015, Kin selection and its critics"
}
],
"nanopub": null,
"approaches": [
{
"id": "hamiltons-rule-exact-scope--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Lean: Price equation and the regression Hamilton's rule as an exact finite identity",
"kind": "formalization",
"method": "Formalize the finite-population statements both sides of the 2010–2011 dispute rely on. Population: a Finset of individuals with breeding value g_i, social partners' breeding value g'_i and fitness w_i (real, mean w̄ > 0). Theorem 1 (Price, no transmission bias): w̄ Δḡ = Cov(w, g). Theorem 2 (least-squares decomposition): with c := −β_{wg·g'} and b := β_{wg'·g} the partial regression coefficients from the 2×2 normal equations (assuming the regressors' covariance matrix is nonsingular), Cov(w, g) = Var(g)(r b − c) with r := Cov(g', g)/Var(g). Corollary (regression HR): if Var(g) > 0, then Δḡ > 0 ⇔ r b > c. Theorem 3 (content condition): if fitness is exactly additive, w_i = α − c₀ g_i + b₀ g'_i, then the regression b, c equal b₀, c₀ in every population state, so the rule predicts change from parameters fixed in advance. Theorem 4: if fitness has an interaction term d g_i g'_i, the regression b and c depend on the population state, so b and c measured in one generation do not predict the next. These separate reading (ii), an identity, from reading (i), a theorem with content.",
"steps": [
"Define finite-population mean, covariance and partial regression coefficients over Finset sums in Lean.",
"Prove Theorems 1 and 2 and the corollary with field_simp, ring and positivity.",
"Prove Theorem 3 (parameter recovery under additivity) and Theorem 4 (state dependence under synergy), giving explicit rational states for the latter.",
"Write the mapping document: each disputed sentence from Nowak et al. 2010, Gardner et al. 2011 and Birch and Okasha 2015 assigned to Theorems 1–4 or to approach 2's counterexamples."
],
"decisive_result": "A Lean module with Theorems 1–4 compiled with only standard axioms, and a mapping table assigning each disputed claim to a formal statement. This settles which claims are identities and which have content under which hypotheses.",
"failure_signature": "None expected for Theorems 1–3, which are classical. The approach fails as a resolution if the mapping shows the disputants mean incompatible definitions of 'Hamilton's rule' that no single formal statement captures; the mapping would then document the equivocation instead.",
"tools": [
"lean4",
"mathlib",
"population-genetics",
"price-equation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/227520a0",
"title": "Selection and Covariance",
"year": 1970,
"note": "Price 1970, Selection and covariance"
},
{
"url": "https://doi.org/10.1111/j.1558-5646.1992.tb02045.x",
"title": "A GENERAL MODEL FOR KIN SELECTION",
"year": 1992,
"note": "Queller 1992, A general model for kin selection (regression form)"
},
{
"url": "https://doi.org/10.1038/nature09205",
"title": "The evolution of eusociality",
"year": 2010,
"note": "Nowak, Tarnita and Wilson 2010"
},
{
"url": "https://doi.org/10.1111/j.1420-9101.2011.02236.x",
"title": "The genetical theory of kin selection",
"year": 2011,
"note": "Gardner, West and Wild 2011"
},
{
"url": "https://doi.org/10.1093/biosci/biu196",
"title": "Kin Selection and Its Critics",
"year": 2014,
"note": "Birch and Okasha 2015 (online 2014), Kin selection and its critics"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hamiltons-rule-exact-scope--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Formal counterexamples: synergistic games where parameter-based rb > c mispredicts",
"kind": "proof",
"method": "Settle reading (iii), when the rule is false, by explicit, Lean-checked counterexamples in standard population-genetic models. Model: a two-player game with payoffs to a focal individual: cooperate against cooperate R = b − c + d, cooperate against defect −c, defect against cooperate b, defect against defect 0. Here d is the synergy term, and assortment is given by relatedness r through the probability that partners share type. Under weak selection, with linear fitness and the probability of a cooperator partner given focal type p_C = r + (1 − r)p, the expected change in cooperator frequency has sign (r b − c) + d·φ(r, p), with φ an explicit polynomial. Choose rational (b, c, d, r, p) with r b > c but Δp < 0, and the reverse, and prove both in Lean by norm_num. Also prove the general form, the generalized Hamilton's rule with the synergy correction (Queller's form), as a theorem, so the counterexample marks where the simple rule's scope ends. Second family: a frequency-dependent threshold game where the sign of Δp flips with p at fixed (b, c, r), showing that no fixed parameter pair predicts the direction everywhere.",
"steps": [
"Define the game, assortment and expected allele-frequency change in Lean.",
"Derive the closed form for Δp and prove the synergy-corrected rule.",
"Prove two explicit counterexample instances and the frequency-flip instance with norm_num.",
"Add these as the (iii) entries of the approach-1 mapping table."
],
"decisive_result": "Lean theorems exhibiting model states in which the parameter-based rule r b > c and the actual direction of selection disagree, alongside a theorem stating the corrected rule. This is the formal refutation the problem's falsifier asks about for the parameter reading, while approach 1 shows the regression reading is never refuted.",
"failure_signature": "None for the mathematics. The approach fails as a contribution if reviewers judge the model class non-standard; mitigate by using the exact models of van Veelen et al. 2017 and Lehmann and Keller 2006.",
"tools": [
"lean4",
"mathlib",
"evolutionary-game-theory"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.jtbi.2016.08.019",
"title": "Hamilton's rule",
"year": 2017,
"note": "van Veelen, Allen, Hoffman, Simon and Veller 2017, Hamilton's rule"
},
{
"url": "https://doi.org/10.1111/j.1420-9101.2006.01119.x",
"title": "The evolution of cooperation and altruism – a general framework and a classification of models",
"year": 2006,
"note": "Lehmann and Keller 2006, classification of models of cooperation and altruism"
},
{
"url": "https://doi.org/10.1111/j.1558-5646.1992.tb02045.x",
"title": "A GENERAL MODEL FOR KIN SELECTION",
"year": 1992,
"note": "Queller 1992, general model for kin selection"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hamiltons-rule-exact-scope--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Computational phase map of when parameter-based Hamilton's rule predicts selection",
"kind": "computation",
"method": "Map the boundary between readings (i) and (iii) across model space, not just at isolated counterexamples. Model classes: Wright–Fisher island model (n demes ∈ {2, 5, 20}, migration m ∈ [0.001, 0.5]); Moran process on cycles, regular graphs (degree 3–10) and random graphs with N ∈ {20, 50, 200}; two-player games with synergy d ∈ [−b, b]; and multiplicative versus additive fitness, at selection strengths δ from 10^-4 to 1. In each cell compute exactly (small N, linear algebra on the Markov chain) or by simulation with 10^6 replicates (larger N) the fixation probability of a single cooperator relative to neutral 1/N. Compare three predictors: parameter HR with model b, c and pedigree or identity-by-descent r; the weak-selection inclusive-fitness condition (Taylor–Frank); and regression HR, which holds by approach 1. Output the fraction of model space where parameter HR gives the right sign, as a function of δ and d.",
"steps": [
"Implement exact Markov-chain fixation for small graphs and validated Monte Carlo for larger ones.",
"Pre-register the predictors and the agreement metric (sign agreement and, where nonzero, rank correlation of effect sizes).",
"Sweep the grid; publish the phase maps with code and seeds.",
"Identify the analytic boundary (for example agreement whenever d = 0 and δ → 0) and feed it back to approach 1 as a theorem to prove."
],
"decisive_result": "Phase maps showing, for each model class, the region where parameter-based r b > c correctly predicts the direction of selection. The pre-stated expectation is 100% agreement at d = 0, δ → 0, with measured degradation as |d| and δ grow; a region of disagreement at d = 0 under weak selection would itself be notable.",
"failure_signature": "Monte Carlo noise at small fixation-probability differences makes signs undecidable in large parts of the grid (Wilson intervals overlapping 1/N), so the map is mostly 'unresolved'.",
"tools": [
"large-scale-computation",
"markov-chains",
"population-genetics"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1111/j.1420-9101.2006.01119.x",
"title": "The evolution of cooperation and altruism – a general framework and a classification of models",
"year": 2006,
"note": "Lehmann and Keller 2006, model classification used to define the grid"
},
{
"url": "https://doi.org/10.1038/nature09205",
"title": "The evolution of eusociality",
"year": 2010,
"note": "Nowak, Tarnita and Wilson 2010, the models they claim fall outside inclusive fitness"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hamiltons-rule-exact-scope--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Lean taxonomy of Hamilton-rule statements",
"kind": "formalization",
"method": "Formalize finite populations as weighted probability spaces and allele-frequency change as a covariance identity. Separate three statements: the Price-equation regression identity, an additive causal fitness theorem with exogenous coefficients, and a frequency-dependent predictive rule. Prove the first two under explicit hypotheses and encode a minimal counterexample to the third.",
"steps": [
"Define reproductive value, fitness, genotype and social-partner genotype on a finite probability space.",
"Prove the Price covariance identity and least-squares decomposition including zero-variance cases.",
"Derive rb > c iff positive allele-frequency change under additive linear fitness and transmission assumptions.",
"Exhaustively search two- and three-player games for a rational frequency-dependent counterexample and check it in Lean."
],
"decisive_result": "A no-sorry Lean module whose theorem boundaries exactly identify when rb > c is an identity, a predictive theorem, or false, including a checked finite countermodel.",
"failure_signature": "The proposed countermodel disappears when relatedness and costs are defined consistently, or the predictive theorem needs assumptions equivalent to its conclusion.",
"tools": [
"lean4",
"population-genetics",
"finite-enumeration"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hamiltons-rule-exact-scope--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Exhaustive finite-population boundary map",
"kind": "computation",
"method": "Enumerate finite haploid models with population size 2 through 8, binary action, mutation, assortment and all rational payoff tables on a coarse grid. Compute exact one-generation allele change, regression r, b and c, and causal intervention effects. Classify disagreements among identity, additive and causal readings and mine minimal assumption sets with decision trees.",
"steps": [
"Predefine each Hamilton-rule estimator and degeneracy handling.",
"Enumerate transition kernels using exact rational arithmetic.",
"Record minimal models where each reading predicts the wrong sign.",
"Generalize recurring boundaries into conjectured lemmas for Lean."
],
"decisive_result": "A complete exact atlas for the declared model class, with minimal counterexamples and zero unexplained disagreements between implementation and formal definitions.",
"failure_signature": "Classification depends primarily on arbitrary regression conventions rather than model structure, preventing a stable scope taxonomy.",
"tools": [
"large-scale-computation",
"population-genetics",
"lean4"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hamiltons-rule-exact-scope--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Three Lean theorems for the three readings of rb > c",
"kind": "formalization",
"method": "Formalize the Birch–Okasha separation as three theorems with disjoint hypotheses. Price's identity first: the change in mean breeding value equals the covariance of fitness and breeding value divided by mean fitness, plus the expected fitness-weighted transmission term. Under fair meiosis the transmission term is zero, so the sign of allele-frequency change is the sign of that covariance. Regression form: define r as the least-squares regression of partner genotype on actor genotype, and b, c as the partial regressions of fitness on partner and actor genotype. Prove that r b - c equals the covariance ratio by the normal equations, so the regression form predicts the sign of frequency change whenever the transmission term vanishes. This is an identity, and the theorem should say so. Additive form: fitness is exactly 1 - c x + b y with b, c constant and y the partner phenotype. Prove the same sign prediction with genealogical relatedness in place of the regression. Counterexample theorem: with fitness 1 - c x + b x y, b and c frozen at one frequency do not preserve the sign at another frequency. State the 2010–2011 correspondence in the module docstring: Nowak, Tarnita and Wilson attack dynamic sufficiency of inclusive-fitness maximization; Gardner, West and Wild defend the regression identity; those are different theorems.",
"steps": [
"Prove Price's identity and the fair-meiosis corollary.",
"Prove the regression form from the normal equations, labeled as an identity.",
"Prove the additive constant-b, constant-c form with genealogical relatedness.",
"Prove a frequency-dependent counterexample to the frozen additive predictor.",
"Map each theorem onto the 2010 and 2011 claims in the module documentation."
],
"decisive_result": "Three no-sorry theorems with explicit hypotheses, plus a short map from those theorems onto the 2010–2011 dispute.",
"failure_signature": "A model inside the regression theorem's hypotheses where the regression form has the wrong sign of allele-frequency change. That would refute the identity and the formalization together.",
"tools": [
"lean4",
"population-genetics",
"price-equation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/227520a0",
"title": "Selection and Covariance",
"year": 1970,
"note": "Price 1970, Selection and covariance"
},
{
"url": "https://doi.org/10.1016/0022-5193(64)90038-4",
"title": "The genetical evolution of social behaviour. I",
"year": 1964,
"note": "Hamilton 1964, The genetical evolution of social behaviour I"
},
{
"url": "https://doi.org/10.1038/nature09205",
"title": "The evolution of eusociality",
"year": 2010,
"note": "Nowak, Tarnita and Wilson 2010, The evolution of eusociality"
},
{
"url": "https://doi.org/10.1111/j.1420-9101.2011.02236.x",
"title": "The genetical theory of kin selection",
"year": 2011,
"note": "Gardner, West and Wild 2011, The genetical theory of kin selection"
},
{
"url": "https://doi.org/10.1093/biosci/biu196",
"title": "Kin Selection and Its Critics",
"year": 2014,
"note": "Birch and Okasha 2015, Kin selection and its critics"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "hamiltons-rule-exact-scope--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Exact Moran chain where frozen rb > c has the wrong sign",
"kind": "computation",
"method": "Give a finite witness that does not rely on Monte Carlo error. Population of N = 6 haploid individuals, reproduction a Moran birth-death update, interaction in three fixed pairs so genealogical relatedness inside a pair is 1 and between pairs is 0. Phenotype x in {0,1}. Fitness w = 1 - c x + b x y with b = 5 and c = 1, where y is the partner phenotype. The state is the integer count of cooperators, 0 through 6, and the chain is written in closed form. The additive predictor freezes b and c at frequency 1/2 by an ordinary least-squares regression on the exact stationary-conditional fitnesses at states with three cooperators, then is applied at every other state. The regression predictor is refit at each state. Compare signs with the exact expected frequency change. Pre-register success as: the refit regression matches the sign of expected change at every non-absorbing state where the absolute expected change exceeds 0, and the frozen predictor disagrees at least at one endpoint state.",
"steps": [
"Write the 7-state Moran transition matrix for the paired population and the synergistic payoff.",
"Compute exact expected frequency change at each state.",
"Fit the frozen additive regression at the three-cooperator states.",
"Refit the regression at every state.",
"Compare signs and emit the disagreement table."
],
"decisive_result": "An exact table, with rational arithmetic, where the refit regression matches every nonzero expected frequency change and the additive coefficients frozen at frequency 1/2 disagree in sign at least once.",
"failure_signature": "The frozen additive predictor agrees in sign at every state, or the refit regression disagrees. The second outcome would contradict the regression identity and fail the formalization too.",
"tools": [
"population-genetics"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1111/j.1420-9101.2011.02236.x",
"title": "The genetical theory of kin selection",
"year": 2011,
"note": "Gardner, West and Wild 2011, the regression form this chain is built to separate from the additive form"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hamiltons-rule-exact-scope--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Separate the regression identity from ESS maximization",
"kind": "computation",
"method": "Nowak, Tarnita and Wilson's target is dynamic sufficiency of inclusive-fitness maximization, not the regression identity. Use the same Moran chain as the sign witness. Compute its exact stationary distribution by solving the linear balance equations on the transient states with mutation probability u = 0.01 each birth, so the chain is irreducible. Define the inclusive-fitness ESS as the strategy that maximizes r b_marg - c with genealogical r from the pair structure and b_marg the marginal benefit at that frequency. Pre-register the comparison: the stationary mean frequency is not within 0.05 of the ESS frequency, while the one-step regression identity still holds at every state. That package shows an identity with predictive content for one-step change and no content for the long-run optimum. It fails differently from the sign witness, which never looks at the stationary distribution.",
"steps": [
"Add mutation u = 0.01 and solve the stationary distribution exactly.",
"Compute the inclusive-fitness ESS frequency from the marginal benefit.",
"Check the one-step regression identity at each state.",
"Compare stationary mean frequency to the ESS with tolerance 0.05.",
"Report the pair of outcomes as the scope statement for this model."
],
"decisive_result": "Exact stationary mean at least 0.05 away from the inclusive-fitness ESS, together with the regression identity holding at every state of the same chain.",
"failure_signature": "The stationary mean sits within 0.05 of the ESS. This model then does not witness the gap Nowak et al. claim, and another payoff would be a new approach rather than a rescue of this one.",
"tools": [
"population-genetics"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nature09205",
"title": "The evolution of eusociality",
"year": 2010,
"note": "Nowak, Tarnita and Wilson 2010, dynamic sufficiency is the claim under test"
},
{
"url": "https://doi.org/10.1093/biosci/biu196",
"title": "Kin Selection and Its Critics",
"year": 2014,
"note": "Birch and Okasha 2015, the separation between the identity and the maximization reading"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
}
]
},
{
"id": "phyllotaxis-attractor-classification",
"title": "Rigorous classification of attractors in dynamical phyllotaxis",
"field": "both",
"tier": "A",
"question": "In the Douady-Couder class of models (new primordium placed at the point of least repulsion from existing ones on a growing cylinder or disc), what is the complete set of attractors and their basins as the growth parameter varies, and does the golden-angle Fibonacci branch have full-measure basin for small growth parameter?",
"known_state": "Douady and Couder (1992) showed numerically and experimentally that the Fibonacci branch is selected; Levitov (1991) explained the energetics; Atela, Golé and Hotton (2002) proved convergence to Fibonacci lattices for a cylindrical model under restrictions. Pennybacker, Shipman and Newell (2015) review the gap: no rigorous classification covering Lucas and bijugate branches, basin boundaries, or the disc geometry. Thompson's phyllotaxis claims (ch14) sit on this.",
"acceptance_criteria": [
"A theorem characterizing the attractors for the cylindrical Douady-Couder map over a stated parameter range, including the Lucas branch, with proof (Lean 4 or rigorous computer-assisted interval proof).",
"Numerical basin maps matching the theorem."
],
"falsifiers": [
"Numerical discovery of an attractor outside the classified set."
],
"required_capabilities": [
"dynamical-systems",
"interval-arithmetic",
"lean4",
"simulation"
],
"data_access": "None needed; sunflower and cone parastichy counts in the repo (thompson1917-ch14-05/06 tests) can check predictions.",
"scales": [
"organ",
"developmental"
],
"frontier_problem": null,
"related_claims": [
{
"id": "thompson1917-ch14-01",
"label": "Spirals of sunflower, thistle and cone are deformations of helices on a stem",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "thompson1917-ch14-05",
"label": "Parastichy numbers change when the structure varies greatly in size",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": "supports",
"theorem": false
},
{
"id": "thompson1917-ch14-06",
"label": "Right- and left-handed parastichy directions occur about equally often",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": "supports",
"theorem": false
},
{
"id": "thompson1917-ch14-09",
"label": "Which parastichies are conspicuous depends on leaf proportions, not on divergence",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "thompson1917-ch14-02",
"label": "Sunflower spirals intersect isogonally, rarely orthogonally",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1103/PhysRevLett.68.2098",
"title": "Phyllotaxis as a physical self-organized growth process",
"year": 1992,
"note": "Douady and Couder 1992, Phyllotaxis as a physical self-organized growth process"
},
{
"url": "https://doi.org/10.1103/PhysRevLett.66.224",
"title": "Phyllotaxis of flux lattices in layered superconductors",
"year": 1991,
"note": "Levitov 1991, Phyllotaxis of flux lattices in layered superconductors"
},
{
"url": "https://doi.org/10.1016/j.physd.2015.05.003",
"title": "Phyllotaxis: Some progress, but a story far from over",
"year": 2015,
"note": "Pennybacker, Shipman and Newell 2015, Phyllotaxis: some progress, but a story far from over"
}
],
"nanopub": null,
"approaches": [
{
"id": "phyllotaxis-attractor-classification--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Computer-assisted proof of attractors via interval Newton and Conley–Morse graphs",
"kind": "proof",
"method": "Use the Atela–Golé–Hotton reduction: in the cylindrical hard-disk version of the Douady–Couder model (each new primordium at the lowest available position at distance ≥ D from existing ones on a cylinder of circumference 1), the dynamics reduce to a finite-dimensional map on 'fronts', the chains of most recent primordia. Rhombic lattices (Fibonacci, Lucas, bijugate and other parastichy pairs) are fixed points of this map up to translation. For the smooth-repulsion version (energy Σ exp(−|x − x_j|/λ) or a power law), the step map is piecewise smooth. Plan: (a) for each candidate branch (m, n), enclose the fixed point over parameter boxes in D (or Γ) with the interval Newton or Krawczyk operator in Arb/FLINT or CAPD, and enclose the Jacobian spectrum inside the unit disk to certify local attraction; (b) for global classification, build a Conley–Morse graph over a box grid of the reduced state space for each parameter box (the Arai et al. 2009 database method), certifying that every recurrent set is one of the enclosed fixed points; (c) the Lucas and bijugate branches enter as additional Morse sets with their own parameter ranges.",
"steps": [
"Implement the front map exactly (hard-disk model) and the smooth-repulsion map; reproduce Douady–Couder bifurcation diagrams numerically.",
"Certify fixed points and stability with interval Newton or Krawczyk across D ∈ [D_min, 0.5] in boxes, adaptively refined.",
"Compute Conley–Morse graphs per parameter box; check that the Morse sets match the certified fixed points.",
"Publish the certified bifurcation diagram and code; describe precisely the parameter range covered."
],
"decisive_result": "A computer-assisted theorem: for all D in a stated range, the only attractors of the cylindrical front map are the certified lattice fixed points, which include the Fibonacci and Lucas branches with explicit ranges, and every other initial front converges to one of them. Basin boundaries are bounded by the Morse decomposition.",
"failure_signature": "Conley–Morse graphs at feasible resolutions contain Morse sets that are not single fixed points (unresolved recurrent sets) in significant parameter ranges, or the discontinuities of the hard-disk map break the interval Newton enclosures near bifurcations.",
"tools": [
"interval-arithmetic",
"dynamical-systems",
"large-scale-computation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1007/s00332-002-0513-1",
"title": "A Dynamical System for Plant Pattern Formation: A Rigorous Analysis",
"year": 2003,
"note": "Atela, Golé and Hotton 2002, A dynamical system for plant pattern formation: a rigorous analysis (J Nonlinear Sci)"
},
{
"url": "https://doi.org/10.1103/PhysRevLett.68.2098",
"title": "Phyllotaxis as a physical self-organized growth process",
"year": 1992,
"note": "Douady and Couder 1992, Phyllotaxis as a physical self-organized growth process"
},
{
"url": "https://doi.org/10.1006/jtbi.1996.0024",
"title": "Phyllotaxis as a Dynamical Self Organizing Process Part I: The Spiral Modes Resulting from Time-Periodic Iterations",
"year": 1996,
"note": "Douady and Couder 1996, Phyllotaxis as a dynamical self organizing process, Part I"
},
{
"url": "https://doi.org/10.1137/080734935",
"title": "A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems",
"year": 2009,
"note": "Arai et al. 2009, A database schema for the analysis of global dynamics of multiparameter systems"
},
{
"url": "https://arblib.org/",
"title": null,
"year": null,
"note": "Arb (now part of FLINT), ball arithmetic for interval enclosures"
},
{
"url": "http://capd.ii.uj.edu.pl/",
"title": null,
"year": null,
"note": "CAPD library for computer-assisted proofs in dynamics"
}
],
"difficulty": "hard",
"missing": "The disc geometry (non-cylindrical, radially varying spacing) is out of scope for this reduction; the result would cover only the cylinder.",
"nanopub": null
},
{
"id": "phyllotaxis-attractor-classification--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Lean: lattice parastichy theorems from continued fractions",
"kind": "formalization",
"method": "Formalize the static half of the theory, which every dynamical claim refers to and which is classical. For the lattice of points (kα mod 1, k·h), k ∈ ℤ, on a cylinder, with divergence α and rise h, prove: (a) the visible opposed parastichy pairs, the two shortest lattice vectors forming a basis and pointing in opposite directions, have numbers (q_j, q_{j+1}) given by consecutive denominators of convergents or intermediate fractions of α; (b) as h decreases, the visible pair changes exactly at the h-values where three vectors have equal length (triangular lattice points), and the sequence of pairs is the convergent sequence; (c) for α = the golden angle fraction (3 − √5)/2, all visible pairs are consecutive Fibonacci numbers, for α = 1/(3 + 1/φ) (the Lucas angle) consecutive Lucas numbers, and bijugate lattices give 2× Fibonacci pairs. Mathlib has generalized continued fractions (GenContFract) and Fibonacci (Nat.fib), so most groundwork exists.",
"steps": [
"Define the cylindrical lattice, visible parastichy pairs and the shortest-vector basis in Lean.",
"Prove the convergent characterization using Mathlib's continued-fraction API and best-approximation lemmas.",
"Specialize to golden, Lucas and bijugate cases; prove the Fibonacci and Lucas statements.",
"Write a report linking each theorem to Thompson's ch14 claims in the repo."
],
"decisive_result": "A Lean module proving that the visible opposed parastichy numbers of the golden-angle lattice are consecutive Fibonacci numbers for every rise h, with the Lucas and bijugate analogues. It does not classify dynamical attractors, but certifies the lattice side that approach 1's fixed points are identified by.",
"failure_signature": "Mathlib's best-approximation lemmas for continued fractions are not in the form needed, and proving them from scratch exceeds the time budget.",
"tools": [
"lean4",
"mathlib",
"number-theory"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.physd.2015.05.003",
"title": "Phyllotaxis: Some progress, but a story far from over",
"year": 2015,
"note": "Pennybacker, Shipman and Newell 2015, review of phyllotaxis theory and open gaps"
},
{
"url": "https://doi.org/10.1209/0295-5075/14/6/006",
"title": "Energetic Approach to Phyllotaxis",
"year": 1991,
"note": "Levitov 1991, Energetic approach to phyllotaxis (Europhysics Letters). The DOI 10.1103/PhysRevLett.66.224 in problems.json is Levitov's flux-lattice PRL paper."
},
{
"url": "https://github.com/leanprover-community/mathlib4",
"title": null,
"year": null,
"note": "Mathlib: GenContFract and Nat.fib"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "phyllotaxis-attractor-classification--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Simulated basin measures tested against citizen-science sunflower counts",
"kind": "data-test",
"method": "Turn the model into a quantitative prediction about real plants. Simulate the Douady–Couder disc model and the cylindrical model over a slowly decreasing growth parameter Γ (the ontogenetic path from seedling to capitulum), starting from initial fronts sampled from a stated distribution (cotyledon arrangement: decussate or two-fold at stated rates, plus Gaussian positional noise σ ∈ {0.01, 0.03, 0.1}). Record the terminal parastichy pair class (Fibonacci, Lucas, double Fibonacci, other) and the basin measures with Wilson intervals from 10^5 runs per setting. Pre-register predicted class frequencies at each σ and compare with the Swinton et al. 2016 sunflower data and the repo's thompson1917-ch14-05/06 parastichy count tests, using a multinomial likelihood-ratio test. The pre-registered rule: the model supports the data if some σ in the grid gives p > 0.05 for the class distribution; it is refuted if all σ give p < 0.001 or if the model cannot produce the observed non-Fibonacci classes at all.",
"steps": [
"Implement disc and cylinder simulators with an inhibition-field placement rule; validate against Douady and Couder 1996 diagrams.",
"Write result.json with predictions per σ and the decision rule before downloading the Swinton data.",
"Run 10^5 simulations per setting; estimate class frequencies.",
"Fetch the sunflower data, classify heads by published parastichy counts, run the multinomial test."
],
"decisive_result": "A pre-registered verdict on whether the Douady–Couder model, with one noise parameter, reproduces the observed frequencies of Fibonacci, Lucas and other classes in sunflowers. If it does, the basin measures are what the rigorous classification must explain.",
"failure_signature": "Model frequencies cannot match the observed share of non-Fibonacci heads for any σ, or the data lack enough class detail to distinguish Lucas from other non-Fibonacci heads (then inconclusive, underpowered).",
"tools": [
"simulation",
"data-test",
"large-scale-computation"
],
"datasets": [
{
"name": "Swinton et al. 2016 sunflower parastichy counts (supplementary data to the Royal Society Open Science article)",
"url": "https://doi.org/10.1098/rsos.160091",
"note": "Parastichy counts and class labels for citizen-science sunflower heads"
}
],
"references": [
{
"url": "https://doi.org/10.1098/rsos.160091",
"title": "Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment",
"year": 2016,
"note": "Swinton, Ochu et al. 2016, Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment"
},
{
"url": "https://doi.org/10.1006/jtbi.1996.0024",
"title": "Phyllotaxis as a Dynamical Self Organizing Process Part I: The Spiral Modes Resulting from Time-Periodic Iterations",
"year": 1996,
"note": "Douady and Couder 1996, spiral modes resulting from time-periodic iterations"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "phyllotaxis-attractor-classification--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Interval-certified cylinder attractors",
"kind": "proof",
"method": "Fix a smooth finite-range repulsion kernel and reduce the cylindrical placement process to a finite-memory map after proving an exponential tail bound. Use interval Newton and cone conditions to certify fixed or periodic lattice branches, then cover the state space with boxes whose images enter certified basins or declared unresolved sets.",
"steps": [
"Specify kernel, growth parameter interval and finite-memory truncation error.",
"Continue Fibonacci, Lucas and bijugate branches with interval arithmetic.",
"Certify hyperbolicity and trapping neighborhoods.",
"Run a validated box-cover basin computation and formally check interval certificates."
],
"decisive_result": "For a nontrivial growth interval, a rigorous list of all attractors and basin-volume bounds whose uncovered state-space measure is below 10^-6.",
"failure_signature": "Persistent positive-measure unresolved boxes or a certified additional attractor outside the proposed branches disproves completeness.",
"tools": [
"interval-arithmetic",
"dynamical-systems",
"lean4",
"simulation"
],
"datasets": [],
"references": [],
"difficulty": "long-shot",
"missing": "Validated numerics infrastructure for the model's discontinuous least-repulsion placement rule.",
"nanopub": null
},
{
"id": "phyllotaxis-attractor-classification--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Empirical basin frequencies from randomized primordia",
"kind": "simulation",
"method": "Pre-register a computational experiment over growth parameters and perturbation amplitudes. Sample 100000 initial states per stratum using a declared uniform phase measure, simulate until convergence diagnostics stabilize, classify parastichy pairs, and estimate basin probabilities with simultaneous multinomial confidence intervals. Compare cylinder and disc geometries.",
"steps": [
"Fix repulsion kernel, numerical precision, initial-state measure and attractor classifier.",
"Validate convergence classifications against high-precision reruns.",
"Estimate basin probabilities on an adaptive parameter grid.",
"Search rare trajectories for unknown attractors and hand them to interval certification."
],
"decisive_result": "A reproducible basin atlas with 99% lower confidence bound above 0.999 for the Fibonacci basin in any claimed small-growth interval, or a measured non-Fibonacci mass exceeding 0.001.",
"failure_signature": "Classifier instability under precision doubling or discovery of persistent unclassified trajectories above 0.1% prevents the full-measure inference.",
"tools": [
"large-scale-computation",
"simulation",
"interval-arithmetic"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": "Simulation cannot prove full measure.",
"nanopub": null
},
{
"id": "phyllotaxis-attractor-classification--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Interval-certify cylindrical Fibonacci and Lucas branches",
"kind": "computation",
"method": "Implement the discrete cylindrical map exactly as defined by Atela, Golé and Hotton 2003, whose fixed points are cylindrical lattices and whose bifurcation parameter is the growth parameter of their diagram. Do not substitute a different potential. On a uniform grid of that parameter across the interval their paper analyzes, certify attracting cycles by interval Newton on the return map: a cycle is certified when an interval box maps into itself and the interval derivative excludes magnitude 1 or greater. Classify certified limits by parastichy pair: consecutive Fibonacci numbers, consecutive Lucas numbers, or the bijugate doubles. A trajectory whose interval enclosure stays away from every classified cycle is an unclassified attractor. The golden-angle branch is the divergence 2π/φ^2 with φ = (1 + sqrt(5))/2. Full Lebesgue measure of its basin is not claimed by a grid; the grid certifies which cycles exist.",
"steps": [
"Encode the Atela–Golé–Hotton cylindrical map from the paper, not a proxy.",
"Sweep their growth parameter on a stated grid.",
"Certify attracting cycles by interval Newton.",
"Label each cycle as Fibonacci, Lucas, bijugate, or unclassified.",
"Emit the certified partition of the parameter interval."
],
"decisive_result": "A certified partition of the paper's parameter interval into basins of Fibonacci, Lucas and bijugate cycles, with every grid point accounted for.",
"failure_signature": "An interval-certified attractor that is none of those families. That is the falsifier the problem states: an attractor outside the classified set.",
"tools": [
"interval-arithmetic",
"dynamical-systems",
"simulation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1007/s00332-002-0513-1",
"title": "A Dynamical System for Plant Pattern Formation: A Rigorous Analysis",
"year": 2003,
"note": "Atela, Golé and Hotton 2003, A dynamical system for plant pattern formation"
},
{
"url": "https://doi.org/10.1103/PhysRevLett.68.2098",
"title": "Phyllotaxis as a physical self-organized growth process",
"year": 1992,
"note": "Douady and Couder 1992, the physical model this map discretizes"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "phyllotaxis-attractor-classification--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Formalize the Atela–Golé–Hotton fixed-point theorem",
"kind": "formalization",
"method": "Turn the 2003 existence-and-stability argument into Lean lemmas, restricted to the hypotheses printed in that paper. The first theorem is the characterization of fixed points as cylindrical lattices whose divergence angle is a noble number, with the Fibonacci and Lucas branches identified by their continued-fraction endings. The second is stability of those fixed points on the branches where the paper proves stability, with the same restrictions. Do not extend the theorem to the disc, to bijugate branches the paper does not cover, or to a full-measure basin. Those are different claims. A proof that uses an unformalized estimate must keep that estimate as a named hypothesis. The module is a success only if the named hypotheses are exactly the paper's.",
"steps": [
"Define the cylindrical lattice and the divergence angle.",
"Prove the fixed-point characterization and the continued-fraction labeling.",
"Formalize the stability estimates under the paper's restrictions.",
"Leave any unformalized estimate as a named hypothesis and list it.",
"Audit axioms."
],
"decisive_result": "A no-sorry fixed-point theorem and a stability theorem whose hypotheses match Atela, Golé and Hotton 2003, including the Lucas branch where their proof includes it.",
"failure_signature": "A numerically certified attractor from the interval sweep lies outside the theorem's fixed-point list inside the theorem's parameter range. The formalization and the sweep then disagree.",
"tools": [
"lean4",
"dynamical-systems"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1007/s00332-002-0513-1",
"title": "A Dynamical System for Plant Pattern Formation: A Rigorous Analysis",
"year": 2003,
"note": "Atela, Golé and Hotton 2003, the theorem being formalized"
},
{
"url": "https://doi.org/10.1016/j.physd.2015.05.003",
"title": "Phyllotaxis: Some progress, but a story far from over",
"year": 2015,
"note": "Pennybacker, Shipman and Newell 2015, the review that marks the remaining gaps"
}
],
"difficulty": "long-shot",
"missing": "The paper is a long real-analysis argument. A month is enough to formalize definitions and perhaps the fixed-point labeling, and unlikely to finish every stability estimate.",
"nanopub": null
},
{
"id": "phyllotaxis-attractor-classification--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Monte Carlo basin volumes on the disc",
"kind": "simulation",
"method": "The disc geometry is the gap the cylindrical theorem does not close. Implement the Douady–Couder disc rule: each new primordium is placed at the angular minimum of the repulsive field of the existing primordia on a growing disc, with the growth parameter in the same role as in their 1992 simulations. Sweep 200 parameter values. At each, launch 500 random initial angles and place 300 primordia. Classify the last 50 divergence angles by distance to the golden angle, the Lucas angle, and the bijugate angles. Estimate basin volume as the fraction of seeds in each class, with a bootstrap percentile interval of 1000 resamples. An unclassified attractor has angular error above 10^{-3} radians from every listed noble angle over those 50 primordia and estimated measure above 0.01. This atlas can falsify a proposed complete list and cannot prove full measure.",
"steps": [
"Implement the disc placement rule from Douady and Couder 1992.",
"Sweep 200 growth parameters and 500 seeds, 300 primordia each.",
"Classify terminal divergence angles against the noble list.",
"Bootstrap the basin-volume estimates.",
"Flag any class with measure above 0.01 that is not on the list."
],
"decisive_result": "A basin-volume table with bootstrap intervals. The classified list fails if any unclassified class has estimated measure above 0.01.",
"failure_signature": "Discovery of such an unclassified attractor. Failure of a different kind is a simulator bug: two implementations disagree on the terminal angle of a shared seed by more than 10^{-3} radians.",
"tools": [
"simulation",
"dynamical-systems"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1103/PhysRevLett.68.2098",
"title": "Phyllotaxis as a physical self-organized growth process",
"year": 1992,
"note": "Douady and Couder 1992, Phyllotaxis as a physical self-organized growth process"
},
{
"url": "https://doi.org/10.1103/PhysRevLett.66.224",
"title": "Phyllotaxis of flux lattices in layered superconductors",
"year": 1991,
"note": "Levitov 1991, Energetic approach to phyllotaxis"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
}
]
},
{
"id": "lewontins-paradox-of-variation",
"title": "Lewontin's paradox of variation",
"field": "biology",
"tier": "B",
"question": "Why does nucleotide diversity scale so weakly with census population size across species, and can linked selection plus demographic history account quantitatively for the gap predicted by neutral theory?",
"known_state": "Kimura's neutral theory predicts heterozygosity proportional to 4Nu; observed diversity spans about three orders of magnitude while census sizes span more than twelve (Lewontin 1974). Leffler et al. (2012) compiled diversity across 167 species; Corbett-Detig et al. (2015) argued linked selection explains part of it; Buffalo (2021) estimated the diversity-census slope with phylogenetic correction and found linked selection insufficient. The quantitative residual is unexplained.",
"acceptance_criteria": [
"A pre-registered test combining a current polymorphism compilation with census-size and range-size estimates (GBIF occurrence extents from the organisms layer) and recombination-rate or genome-size covariates, with explicit quantitative predictions from each hypothesis (neutral, linked selection, boom-bust demography) stated before analysis.",
"A verdict on whether linked selection models fitted to measured recombination landscapes close the gap."
],
"falsifiers": [
"If linked-selection models with measured parameters predict the observed slope, the paradox is resolved; if the residual stays at orders of magnitude, demographic explanations are required."
],
"required_capabilities": [
"population-genetics",
"phylogenetic-comparative-methods",
"gbif",
"data-test"
],
"data_access": "Leffler 2012 and Buffalo 2021 supplementary tables (open); GBIF via the organisms layer; recombination maps per species where published.",
"scales": [
"evolutionary",
"ecological"
],
"frontier_problem": null,
"related_claims": [
{
"id": "kimura-heterozygosity",
"label": "Neutral equilibrium heterozygosity (Kimura-Crow)",
"thinker": "Motoo Kimura",
"status": "proved",
"evidence": "refutes",
"theorem": true
},
{
"id": "kimura-neutral-theory",
"label": "Neutral theory: most molecular substitutions are neutral",
"thinker": "Motoo Kimura",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "kimura-substitution-rate",
"label": "Neutral substitution rate equals the mutation rate",
"thinker": "Motoo Kimura",
"status": "proved",
"evidence": null,
"theorem": true
}
],
"references": [
{
"url": "https://doi.org/10.1038/217624a0",
"title": "Evolutionary Rate at the Molecular Level",
"year": 1968,
"note": "Kimura 1968, Evolutionary rate at the molecular level"
},
{
"url": "https://doi.org/10.1371/journal.pbio.1001388",
"title": "Revisiting an Old Riddle: What Determines Genetic Diversity Levels within Species?",
"year": 2012,
"note": "Leffler et al. 2012, Revisiting an old riddle: what determines genetic diversity levels within species?"
},
{
"url": "https://doi.org/10.1371/journal.pbio.1002112",
"title": "Natural Selection Constrains Neutral Diversity across A Wide Range of Species",
"year": 2015,
"note": "Corbett-Detig, Hartl and Sackton 2015, Natural selection constrains neutral diversity across a wide range of species"
},
{
"url": "https://doi.org/10.7554/eLife.67509",
"title": "Quantifying the relationship between genetic diversity and population size suggests natural selection cannot explain Lewontin’s Paradox",
"year": 2021,
"note": "Buffalo 2021, Quantifying the relationship between genetic diversity and population size suggests natural selection cannot explain Lewontin's paradox"
}
],
"nanopub": null,
"approaches": [
{
"id": "lewontins-paradox-of-variation--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Pre-registered phylogenetic test of three quantitative predictions for the π–N slope",
"kind": "data-test",
"method": "Extend Buffalo's (2021) compilation with independent census proxies from the organisms layer and state, before analysis, the slope each hypothesis predicts. Response: log₁₀ π at synonymous or neutral sites. Census proxy: log₁₀(range area × density), where range area is the convex hull or alpha hull of cleaned GBIF occurrences and density is predicted from body mass by Damuth's allometry (D ∝ M^−0.75), as Buffalo did, but with GBIF-derived ranges to test robustness. Predictions: (H0, neutral) slope of log π on log N_c ≈ 1 if N_e ∝ N_c and μ is constant; (H1, linked selection) slope reduced by the factor predicted from each species' map length and coding density using a Corbett-Detig-style model of background selection plus sweeps, computed per species before looking at π; (H2, demographic) π tracks the harmonic mean of population size, so slope ≈ 1 against harmonic-mean N but much less against current N_c. Fit PGLS with Pagel's λ (phylolm) on an Open Tree of Life synthetic tree with time calibration. Decision rule: H1 closes the gap if the observed slope lies within the 95% interval of the H1-predicted slope; H0 is refuted if the slope interval excludes 0.5.",
"steps": [
"Write result.json with predicted slopes and intervals for H0, H1 and H2 before computing observed slopes.",
"Resolve species through the organisms layer; fetch GBIF occurrences with a cleaning pipeline (coordinate checks, outlier removal) and compute ranges.",
"Compute per-species predicted linked-selection reduction from published map lengths and genome annotations; record n for which inputs exist.",
"Fit PGLS; compare slopes; report residual orders of magnitude unexplained."
],
"decisive_result": "Observed PGLS slope with 95% CI compared against the pre-registered H0 and H1 intervals: the gap is closed if the observed slope falls in the H1 interval, and the residual is quantified in orders of magnitude otherwise. The minimum sample is 100 species with π and range; Buffalo's set is larger, so recombination-map availability limits only the H1 subset.",
"failure_signature": "Census proxies from GBIF ranges and Damuth density have measurement error so large (or so correlated with body mass) that slope intervals overlap all three predictions: inconclusive, underpowered.",
"tools": [
"gbif",
"organisms-layer",
"phylogenetic-comparative-methods",
"data-test"
],
"datasets": [
{
"name": "Buffalo 2021 data and code",
"url": "https://github.com/vsbuffalo/paradox_variation",
"note": "Species diversity estimates, census proxies and the analysis code to extend"
},
{
"name": "GBIF occurrence API",
"url": "https://api.gbif.org/v1/occurrence/search?limit=1",
"note": "Occurrence coordinates for range estimation"
},
{
"name": "Open Tree of Life",
"url": "https://opentreeoflife.github.io/",
"note": "Synthetic phylogeny for PGLS"
}
],
"references": [
{
"url": "https://doi.org/10.7554/eLife.67509",
"title": "Quantifying the relationship between genetic diversity and population size suggests natural selection cannot explain Lewontin’s Paradox",
"year": 2021,
"note": "Buffalo 2021, Quantifying the relationship between genetic diversity and population size"
},
{
"url": "https://doi.org/10.1371/journal.pbio.1002112",
"title": "Natural Selection Constrains Neutral Diversity across A Wide Range of Species",
"year": 2015,
"note": "Corbett-Detig, Hartl and Sackton 2015, linked selection across 40 species"
},
{
"url": "https://doi.org/10.1371/journal.pbio.1001388",
"title": "Revisiting an Old Riddle: What Determines Genetic Diversity Levels within Species?",
"year": 2012,
"note": "Leffler et al. 2012, diversity across 167 species"
},
{
"url": "https://doi.org/10.1038/290699a0",
"title": "Population density and body size in mammals",
"year": 1981,
"note": "Damuth 1981, Population density and body size in mammals"
}
],
"difficulty": "hard",
"missing": "Recombination maps exist for a minority of species, so H1 can be tested only on a subset; census-size estimation remains crude.",
"nanopub": null
},
{
"id": "lewontins-paradox-of-variation--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Decompose π into N_e and μ using pedigree mutation rates",
"kind": "data-test",
"method": "Part of the paradox may sit in μ, not N_e: the drift-barrier hypothesis predicts that per-generation mutation rate falls as N_e rises, which flattens π = 4N_eμ against N_c. Pedigree-based germline mutation rates now exist for dozens of vertebrates (Bergeron et al. 2023). For every species with both a pedigree μ and a published π (Romiguier et al. 2014, Leffler et al. 2012, genome-scale estimates in NCBI-linked studies), compute N̂_e = π/(4μ) and fit log N̂_e on log N_c (PGLS). Pre-registered predictions: if the paradox is entirely in N_e, then log μ is uncorrelated with log N_c and the slope of log N̂_e equals the slope of log π; if drift-barrier compression matters, log μ declines with log N_c and the N̂_e slope is steeper than the π slope by the μ slope. The quantity of interest is the fraction of the gap between neutral slope 1 and the observed π slope that μ accounts for.",
"steps": [
"Pre-register the decomposition and predicted signs; list admissible μ estimates (trio-based, with stated callable-genome correction).",
"Assemble the matched set (species with both μ and π); resolve taxa via the organisms layer.",
"Fit PGLS for log π, log μ and log N̂_e on log N_c; propagate μ uncertainty by parametric bootstrap.",
"Report the share of the paradox attributable to μ with its 95% interval."
],
"decisive_result": "The estimated slope of log μ on log N_c and the implied share of the π–N_c gap explained by mutation-rate variation, with 95% CI. If the share's interval excludes 0, μ is a quantified part of the answer; if it is centred near 0 with a tight interval, the paradox is entirely in N_e.",
"failure_signature": "Fewer than about 30 species with both pedigree μ and comparable π, giving intervals spanning 0 to 100%: inconclusive, underpowered.",
"tools": [
"phylogenetic-comparative-methods",
"organisms-layer",
"data-test"
],
"datasets": [
{
"name": "Bergeron et al. 2023 germline mutation rates (supplementary tables)",
"url": "https://doi.org/10.1038/s41586-023-05752-y",
"note": "Pedigree-based per-generation mutation rates across vertebrates"
},
{
"name": "Buffalo 2021 compilation",
"url": "https://github.com/vsbuffalo/paradox_variation",
"note": "π estimates and census proxies to match against"
}
],
"references": [
{
"url": "https://doi.org/10.1038/s41586-023-05752-y",
"title": "Evolution of the germline mutation rate across vertebrates",
"year": 2023,
"note": "Bergeron et al. 2023, Evolution of the germline mutation rate across vertebrates"
},
{
"url": "https://doi.org/10.1038/nrg.2016.104",
"title": "Genetic drift, selection and the evolution of the mutation rate",
"year": 2016,
"note": "Lynch et al. 2016, Genetic drift, selection and the evolution of the mutation rate"
},
{
"url": "https://doi.org/10.1038/nature13685",
"title": "Comparative population genomics in animals uncovers the determinants of genetic diversity",
"year": 2014,
"note": "Romiguier et al. 2014, Comparative population genomics in animals uncovers the determinants of genetic diversity"
},
{
"url": "https://doi.org/10.1038/nrg.2016.58",
"title": "Determinants of genetic diversity",
"year": 2016,
"note": "Ellegren and Galtier 2016, Determinants of genetic diversity"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "lewontins-paradox-of-variation--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Simulation bound on how much linked selection can flatten π across N",
"kind": "simulation",
"method": "Settle whether linked selection can close the gap by computing the most it could do under generous but empirically bounded parameters. Use SLiM forward simulation with tree-sequence recording plus msprime recapitation, and stdpopsim genome models and DFEs where available, across a ladder of census sizes N from 10^3 to 10^7 simulated directly. Extrapolate to 10^12 with the analytic background-selection and sweep formulas (B = exp(−Σ u/(s(1+r/s)²)) plus a recurrent-sweep coalescent rate), calibrated against the simulations in the overlap range. Upper-bound parameters: the deleterious mutation rate at the top of published estimates, a DFE fraction of strongly selected new mutations at the top of estimates, and adaptive substitution rate α from McDonald–Kreitman ranges. Output: the curve π(N) and the maximum achievable flattening (slope of log π on log N over 10^4–10^12). Compare with the observed slope from approach 1 or from Buffalo 2021.",
"steps": [
"Choose 3 representative genome architectures (vertebrate, insect, plant) from stdpopsim; fix parameter ranges from literature before simulating.",
"Run SLiM simulations over the N ladder with 20 replicates each; compute π at neutral sites.",
"Calibrate the analytic B and sweep model to simulations; extrapolate to large N.",
"Compute the maximum flattening and compare with the observed slope."
],
"decisive_result": "The minimum π–N slope achievable by linked selection under upper-bound parameters. If that minimum exceeds the observed slope plus its CI, linked selection cannot explain the paradox, as Buffalo argued, and demography or μ must; if it falls below, the bound shows which parameter regime is needed.",
"failure_signature": "Extrapolation from 10^7 to 10^12 is unreliable: the analytic model misfits simulations at the top of the ladder, or sweep interference (genetic draft saturation) makes predictions model-dependent across plausible formulas.",
"tools": [
"simulation",
"population-genetics",
"large-scale-computation"
],
"datasets": [],
"references": [
{
"url": "https://github.com/MesserLab/SLiM",
"title": null,
"year": null,
"note": "SLiM forward population-genetic simulator"
},
{
"url": "https://tskit.dev/msprime/docs/stable/",
"title": null,
"year": null,
"note": "msprime coalescent simulator for recapitation"
},
{
"url": "https://popsim-consortium.github.io/stdpopsim-docs/stable/",
"title": null,
"year": null,
"note": "stdpopsim species models and DFEs"
},
{
"url": "https://doi.org/10.7554/eLife.76065",
"title": "Broad-scale variation in human genetic diversity levels is predicted by purifying selection on coding and non-coding elements",
"year": 2023,
"note": "Murphy, Elyashiv, Amster and Sella 2023, purifying selection predicts human diversity landscapes (calibration of B maps)"
},
{
"url": "https://doi.org/10.1093/molbev/msy092",
"title": "The Neutral Theory in Light of Natural Selection",
"year": 2018,
"note": "Kern and Hahn 2018, The neutral theory in light of natural selection"
}
],
"difficulty": "hard",
"missing": "Direct simulation at N ≥ 10^8 is infeasible, so the bound rests on extrapolation.",
"nanopub": null
},
{
"id": "lewontins-paradox-of-variation--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Phylogenetic decomposition of diversity constraints",
"kind": "data-test",
"method": "Pre-register a phylogenetic hierarchical model for autosomal synonymous diversity across at least 150 animal species. Predict log diversity from log census proxy, recombination-map length, functional genomic density, body size, range area from GBIF, and generation time, with clade random slopes and measurement error. Compare neutral, linked-selection and boom-bust models by leave-one-clade-out expected log predictive density.",
"steps": [
"Freeze taxon inclusion, one diversity estimate per species and GBIF coverage rules before fitting.",
"Require at least 30 species with measured recombination maps and 150 total with range proxies.",
"Fit matched neutral, background-selection and recurrent-sweep models with a fixed phylogeny.",
"Report slope residuals, variance explained and held-out calibration."
],
"decisive_result": "Linked selection closes the gap only if its model improves leave-one-clade-out ELPD by at least 10 and leaves an absolute census-diversity slope residual below 0.05; otherwise the residual is quantitatively unresolved.",
"failure_signature": "Fewer than 30 independent recombination landscapes or posterior predictions dominated by range-size measurement error make the mechanism comparison underpowered.",
"tools": [
"gbif",
"phylogenetic-comparative-methods",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Comparable recombination maps and defensible census sizes are sparse outside model organisms.",
"nanopub": null
},
{
"id": "lewontins-paradox-of-variation--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Genome-window test of linked-selection predictions",
"kind": "data-test",
"method": "Within species, use genome windows to avoid uncertain census sizes. For at least 20 species with polymorphism, recombination and annotation data, preregister the reduction in neutral diversity near conserved or selected sites predicted by background-selection maps, then aggregate the measured maximum genome-wide reduction into a cross-species upper bound on how much linked selection can compress diversity.",
"steps": [
"Select species and neutral-site filters without inspecting diversity-recombination associations.",
"Fit a common monotone recombination-diversity model with species-specific intercepts.",
"Cross-validate chromosome-held-out predictions and estimate diversity absent linked selection.",
"Regress corrected diversity on GBIF range and life-history proxies."
],
"decisive_result": "If linked selection explains at least 80% of within-genome neutral-diversity variance and corrected cross-species diversity spans under two orders of magnitude, it is sufficient for the tested taxa; a corrected span above three orders refutes sufficiency.",
"failure_signature": "Neutral-site definitions or recombination maps are incomparable enough that chromosome-held-out prediction R2 is below 0.1 in more than half the species.",
"tools": [
"large-scale-computation",
"gbif",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Harmonized population-genomic callsets and recombination maps.",
"nanopub": null
},
{
"id": "lewontins-paradox-of-variation--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Pre-register the diversity-census slope with a range covariate",
"kind": "data-test",
"method": "Pre-register before downloading tables. Response is log10 of nucleotide diversity π from the Leffler et al. 2012 compilation, with the Buffalo 2021 estimates used only as a second, separately reported column where the species overlap. Predictor is log10 of census size as published in those supplements. Covariate is log10 of GBIF range area: occurrences for the species with coordinate uncertainty below 10 km, reduced to presence on a 0.1 degree grid, range area the axis-aligned bounding box, species with fewer than 20 occupied cells dropped. Primary estimator is a nested taxonomic random-effects model, species within genus within family within order, fit by REML, not a single time-calibrated tree of all life. Neutral prediction is slope 1 of log10 π on log10 N. Secondary estimator, mammals only, is PGLS with Pagel's lambda on 100 trees from the Upham, Esselstyn and Jetz credible set. Linked selection is not fit in this approach. The verdict is about the slope: if the 95 percent profile interval includes 1, neutral scaling of diversity with census size is not rejected and the paradox sits in the intercept.",
"steps": [
"Write the model, the GBIF filter and the drop rule before opening the supplements.",
"Join Leffler diversity and census size to GBIF bounding-box area.",
"Fit the nested random-effects model by REML and record the profile interval for the slope.",
"Repeat PGLS on the mammal overlap across 100 Upham trees.",
"Call the estimators mixed if one interval contains 1 and the other excludes it."
],
"decisive_result": "A 95 percent interval for the slope of log10 π on log10 census N. Neutral scaling is rejected only if that interval excludes 1 in the primary model.",
"failure_signature": "Fewer than 30 species survive the GBIF cell filter, in which case the verdict is inconclusive and underpowered. A slope interval that includes both 0 and 1 is also indecisive and is reported as such.",
"tools": [
"gbif",
"phylogenetic-comparative-methods",
"data-test"
],
"datasets": [
{
"name": "Leffler et al. 2012 diversity compilation",
"url": "https://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.1001388",
"note": "Species-level nucleotide diversity and the census sizes used in that paper."
},
{
"name": "Buffalo 2021 diversity-census reanalysis",
"url": "https://elifesciences.org/articles/67509",
"note": "Second diversity column, reported separately, not pooled with Leffler in the primary fit."
},
{
"name": "GBIF occurrence API",
"url": "https://api.gbif.org/v1/",
"note": "Occurrences used only to build range bounding boxes under the pre-registered filter."
},
{
"name": "Upham mammal phylogenies",
"url": "https://vertlife.org/data/mammals/",
"note": "One hundred node-dated trees for the mammal-only PGLS."
}
],
"references": [
{
"url": "https://doi.org/10.1371/journal.pbio.1001388",
"title": "Revisiting an Old Riddle: What Determines Genetic Diversity Levels within Species?",
"year": 2012,
"note": "Leffler et al. 2012, Revisiting an old riddle: what determines genetic diversity levels within species?"
},
{
"url": "https://doi.org/10.7554/eLife.67509",
"title": "Quantifying the relationship between genetic diversity and population size suggests natural selection cannot explain Lewontin’s Paradox",
"year": 2021,
"note": "Buffalo 2021, Quantifying the relationship between genetic diversity and population size"
},
{
"url": "https://doi.org/10.1371/journal.pbio.3000494",
"title": "Inferring the mammal tree: Species-level sets of phylogenies for questions in ecology, evolution, and conservation",
"year": 2019,
"note": "Upham, Esselstyn and Jetz 2019, the mammal trees"
},
{
"url": "https://doi.org/10.1038/217624a0",
"title": "Evolutionary Rate at the Molecular Level",
"year": 1968,
"note": "Kimura 1968, the neutral prediction that heterozygosity scales with N"
}
],
"difficulty": "hard",
"missing": "Census sizes are only as good as the supplements. GBIF bounding boxes are a range proxy, not a census, and the model treats them as a covariate rather than as N.",
"nanopub": null
},
{
"id": "lewontins-paradox-of-variation--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Test whether published recombination maps close the residual",
"kind": "data-test",
"method": "Pre-register a second test aimed at linked selection, using only species that have both a diversity estimate in the Leffler or Buffalo tables and a public genetic map with at least 10^3 markers in centimorgans. Homo sapiens enters through the 1000 Genomes resource; every other species enters only with a map meeting the same marker floor. For each such species compute map length per megabase. Fit log10 π = a + b log10 N + c (map length per megabase) in the same nested taxonomic model as the slope test. Linked selection closes the gap only if, after this covariate, the residual slope on log10 N has a 95 percent interval containing 0 and the root-mean-square residual of log10 π is below 0.5. Both conditions are required. If fewer than 8 species qualify, the verdict is inconclusive, underpowered, and is not filled in by dropping the marker floor.",
"steps": [
"Freeze the marker floor and the two numerical conditions before listing species.",
"Admit a species only when both a diversity row and a qualifying public map exist.",
"Fit the nested model with map length per megabase as covariate.",
"Record the residual slope interval and the residual RMSE.",
"Stop as underpowered if fewer than 8 species qualify."
],
"decisive_result": "Residual slope interval containing 0 and residual RMSE of log10 π below 0.5, on at least 8 species with real maps. Otherwise linked selection, as measured here, does not close the gap.",
"failure_signature": "The residual slope interval excludes 0, or the RMSE stays at or above 0.5. Dropping the marker floor after seeing the fit is a failed pre-registration, not a result.",
"tools": [
"data-test",
"population-genetics"
],
"datasets": [
{
"name": "1000 Genomes Project",
"url": "https://www.internationalgenome.org/",
"note": "Human genetic map and diversity resources. Other species enter only with a comparable public map."
},
{
"name": "Leffler et al. 2012 diversity compilation",
"url": "https://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.1001388",
"note": "Diversity values joined to maps."
}
],
"references": [
{
"url": "https://doi.org/10.1371/journal.pbio.1002112",
"title": "Natural Selection Constrains Neutral Diversity across A Wide Range of Species",
"year": 2015,
"note": "Corbett-Detig, Hartl and Sackton 2015, linked selection as the hypothesis under test"
},
{
"url": "https://doi.org/10.7554/eLife.67509",
"title": "Quantifying the relationship between genetic diversity and population size suggests natural selection cannot explain Lewontin’s Paradox",
"year": 2021,
"note": "Buffalo 2021, the claim that linked selection does not close the gap"
}
],
"difficulty": "hard",
"missing": "Public genetic maps for most of the Leffler species. The test is designed to return underpowered rather than to invent maps.",
"nanopub": null
},
{
"id": "lewontins-paradox-of-variation--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Forward simulations of sweeps, background selection and bottlenecks",
"kind": "computation",
"method": "Ask whether a pre-specified demographic-plus-selection process can reproduce the empirical slope, using the slope from the first approach as a fixed target and not refitting it. Three regimes, parameters frozen first: neutral Wright–Fisher with N_e equal to census N; recurrent hitchhiking plus background selection with a stated deleterious mutation rate and a beneficial rate scaled per megabase; and a bottleneck regime with a contraction of factor 100 lasting a stated fraction of each run. For each regime, simulate a grid of census sizes spanning the empirical range, 20 replicates per size, and estimate the slope of mean log10 π on log10 N. A regime accounts for the paradox only if its slope's simulation interval covers the empirical slope and its diversity range covers the empirical diversity range within a factor of two at both ends. Code and seeds are part of the result.",
"steps": [
"Freeze mutation rates, selection coefficients and the bottleneck schedule before fitting.",
"Simulate the three regimes across the census-size grid, 20 replicates.",
"Estimate each regime's slope of log10 π on log10 N.",
"Compare slope and endpoint diversity with the empirical target.",
"Reject a regime that misses either the slope or the endpoint range."
],
"decisive_result": "A regime whose simulated slope interval covers the empirical slope and whose diversity range matches both empirical endpoints within a factor of two. If none does, demography-plus-linked-selection of this form does not resolve the paradox.",
"failure_signature": "All three regimes miss the slope or the endpoint range. Tuning the rates after seeing the empirical slope is a failed pre-registration.",
"tools": [
"population-genetics"
],
"datasets": [
{
"name": "Leffler et al. 2012 diversity compilation",
"url": "https://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.1001388",
"note": "Empirical slope and diversity range used only as the pre-registered target."
}
],
"references": [
{
"url": "https://doi.org/10.1371/journal.pbio.1002112",
"title": "Natural Selection Constrains Neutral Diversity across A Wide Range of Species",
"year": 2015,
"note": "Corbett-Detig, Hartl and Sackton 2015, the linked-selection regime"
},
{
"url": "https://doi.org/10.7554/eLife.67509",
"title": "Quantifying the relationship between genetic diversity and population size suggests natural selection cannot explain Lewontin’s Paradox",
"year": 2021,
"note": "Buffalo 2021, the empirical target the simulations must hit"
}
],
"difficulty": "hard",
"missing": "Species-specific recombination maps are not used here; the regimes are deliberately generic. A species-matched map is the other approach.",
"nanopub": null
}
]
},
{
"id": "metabolic-scaling-exponent",
"title": "Is there a universal metabolic scaling exponent?",
"field": "biology",
"tier": "B",
"question": "Does basal metabolic rate scale with body mass with a single exponent across taxa, and if so is it 3/4 (fractal supply networks) or 2/3 (surface-to-volume), or does the exponent vary systematically with taxon, temperature and body size?",
"known_state": "Kleiber (1932) reported 3/4; West, Brown and Enquist (1997) derived 3/4 from space-filling branching networks; White and Seymour (2003) found 2/3 for mammals once body temperature and digestive state are controlled; Kozlowski and Konarzewski (2004) disputed the WBE derivation. Current syntheses find curvature and taxon dependence. Thompson's surface-volume claims (ch02-01, ch03-07) encode the 2/3 alternative.",
"acceptance_criteria": [
"A pre-registered phylogenetically controlled analysis across the organism taxa with AnAge/PanTHERIA body mass and metabolic data, with stated predictions: a single exponent with 95% interval excluding one of 2/3 and 3/4, or a model-comparison verdict that a curved or taxon-specific model beats any single power law.",
"A Lean 4 formalization of the WBE derivation stating its assumptions as hypotheses, so the disputed step is explicit."
],
"falsifiers": [
"If the pooled exponent interval contains both 2/3 and 3/4 after controls, the universality question is empirically underdetermined with present data."
],
"required_capabilities": [
"phylogenetic-comparative-methods",
"allometry",
"data-test",
"lean4"
],
"data_access": "AnAge (open), PanTHERIA (open), White and Seymour 2003 compilation; taxon resolution via test-data/organisms.",
"scales": [
"organism",
"evolutionary"
],
"frontier_problem": "Mammalian body size control",
"related_claims": [
{
"id": "thompson1917-ch03-07",
"label": "Weight tends to vary as the cube of linear dimensions during growth",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": "mixed",
"theorem": true
},
{
"id": "thompson1917-ch02-01",
"label": "Changes in the length-weight constant k reveal changes of form during growth",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": "supports",
"theorem": true
},
{
"id": "thompson1917-ch03-10",
"label": "Cold-blooded animals approach a maximal size only asymptotically",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": "mixed",
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.3733/hilg.v06n11p315",
"title": "Body size and metabolism",
"year": 1932,
"note": "Kleiber 1932, Body size and metabolism"
},
{
"url": "https://doi.org/10.1126/science.276.5309.122",
"title": "A General Model for the Origin of Allometric Scaling Laws in Biology",
"year": 1997,
"note": "West, Brown and Enquist 1997, A general model for the origin of allometric scaling laws in biology"
},
{
"url": "https://doi.org/10.1073/pnas.0436428100",
"title": "Mammalian basal metabolic rate is proportional to body mass\n 2/3",
"year": 2003,
"note": "White and Seymour 2003, Mammalian basal metabolic rate is proportional to body mass 2/3"
},
{
"url": "https://doi.org/10.1111/j.0269-8463.2004.00830.x",
"title": "Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant?",
"year": 2004,
"note": "Kozlowski and Konarzewski 2004, Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant?"
}
],
"nanopub": null,
"approaches": [
{
"id": "metabolic-scaling-exponent--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Pre-registered PGLS model comparison: single power law vs curvature vs clade slopes",
"kind": "data-test",
"method": "Assemble mammal BMR and mass from AnAge (metabolic rate, body mass, body temperature fields) and PanTHERIA (BasalMetRate, AdultBodyMass), deduplicated by species via the organisms layer. Exclude records not meeting basal criteria where flagged, and require n ≥ 300 species with temperature for the temperature-corrected analysis. Phylogeny: Upham et al. 2019 node-dated mammal trees, with analyses integrated over 100 posterior trees. Models: (M1) log B = a + b log M; (M2) M1 + Boltzmann temperature term −E/(kT_b) (Gillooly form) or T_b as covariate; (M3) quadratic in log M (Kolokotrones curvature); (M4) order-level random slopes. Fit by PGLS with Pagel's λ, compare by AICc and by leave-one-order-out predictive log-likelihood. Pre-registered decision: 'single exponent' if M1 or M2 wins and the 95% CI of b excludes one of 2/3 or 3/4 (reported as such); 'no universal exponent' if M3 or M4 improves predictive log-likelihood by > 10 units summed over orders; 'underdetermined' if the CI contains both.",
"steps": [
"Write result.json with models, decision rule and minimum n before downloading.",
"Download AnAge (dataset.zip) and PanTHERIA; record SHA-256; merge and resolve taxa.",
"Prune Upham trees to the matched species; fit M1–M4 over 100 trees.",
"Report b with CI, curvature coefficient, order-slope variance and the verdict."
],
"decisive_result": "A verdict among 'exponent b with CI excluding 2/3 (or 3/4)', 'curved or clade-specific model wins by > 10 log-likelihood units out of sample', or 'underdetermined', with effect sizes. Mammals alone give n of several hundred species, enough to separate 2/3 from 3/4 if the relation is linear (White and Seymour's CI was about ±0.03).",
"failure_signature": "Strong dependence of b on inclusion criteria (basal versus resting, captive versus wild) that the pre-registered criteria cannot resolve; the verdict then rests on data curation rather than biology.",
"tools": [
"anage",
"phylogenetic-comparative-methods",
"organisms-layer",
"data-test"
],
"datasets": [
{
"name": "AnAge database of animal ageing and longevity",
"url": "https://genomics.senescence.info/species/dataset.zip",
"note": "Body mass, metabolic rate and body temperature fields"
},
{
"name": "PanTHERIA (Ecological Archives E090-184)",
"url": "https://esapubs.org/archive/ecol/E090/184/",
"note": "Mammal basal metabolic rate and adult body mass"
},
{
"name": "VertLife mammal phylogenies (Upham et al. 2019)",
"url": "https://vertlife.org/",
"note": "Posterior distribution of node-dated mammal trees"
}
],
"references": [
{
"url": "https://doi.org/10.1073/pnas.0436428100",
"title": "Mammalian basal metabolic rate is proportional to body mass\n 2/3",
"year": 2003,
"note": "White and Seymour 2003, mammalian BMR ∝ M^(2/3)"
},
{
"url": "https://doi.org/10.1038/nature08920",
"title": "Curvature in metabolic scaling",
"year": 2010,
"note": "Kolokotrones et al. 2010, Curvature in metabolic scaling"
},
{
"url": "https://doi.org/10.1371/journal.pbio.3000494",
"title": "Inferring the mammal tree: Species-level sets of phylogenies for questions in ecology, evolution, and conservation",
"year": 2019,
"note": "Upham, Esselstyn and Jetz 2019, mammal tree"
},
{
"url": "https://cran.r-project.org/package=phylolm",
"title": null,
"year": null,
"note": "phylolm, phylogenetic regression"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "metabolic-scaling-exponent--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Lean formalization of the WBE network derivation with the finite-size exponent",
"kind": "formalization",
"method": "Make the disputed step explicit. Model: a branching network with N levels, branching ratio n, radius ratio β and length ratio γ per level, invariant terminal units (capillaries) and total blood volume V_b ∝ M. Define V_b(N) = Σ_{k=0}^{N} n^k π r_k² l_k with r_k = β^k r₀, l_k = γ^k l₀, so V_b = π r₀² l₀ Σ (nβ²γ)^k. Theorem 1 (exact finite form): with terminal size fixed, N_cap = n^N and V_b = V_cap · ((nβ²γ)^{−(N+1)} − 1)/((nβ²γ)^{−1} − 1) for nβ²γ < 1. Theorem 2 (asymptotic exponent): under area preservation β = n^{−1/2} and space filling γ = n^{−1/3}, and V_b ∝ M, the ratio log N_cap / log M → 3/4 as N → ∞, but for finite N the effective exponent is strictly less than 3/4 with an explicit correction term, matching Savage et al. 2008's finite-size analysis. Theorem 3: if area-preserving branching holds only below a transition level and Poiseuille scaling (β = n^{−1/3}) above it, the limiting exponent is different (explicit formula). The hypotheses V_b ∝ M and invariant terminal units are exactly where Kozłowski and Konarzewski object, and they appear as named hypotheses.",
"steps": [
"Define the network sums in Lean (Finset.sum over levels, geometric series).",
"Prove Theorem 1 by geometric-series lemmas, and Theorem 2 by limits (Filter.Tendsto) on logarithms.",
"Prove Theorem 3 for the two-regime network.",
"Write a report with each hypothesis named and linked to Thompson's ch03-07 and ch02-01 claims; state which are empirical."
],
"decisive_result": "A Lean module proving that WBE's 3/4 is the N → ∞ limit under four named hypotheses, with the exact finite-N exponent, compiled with only standard axioms. Each published objection then corresponds to dropping a named hypothesis.",
"failure_signature": "The formalization shows that the published derivation needs a hypothesis not listed (for example a specific relation between N and M) to get 3/4, which is a substantive finding about WBE rather than a failure, or the limit proofs in Mathlib take longer than budgeted.",
"tools": [
"lean4",
"mathlib",
"allometry"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1126/science.276.5309.122",
"title": "A General Model for the Origin of Allometric Scaling Laws in Biology",
"year": 1997,
"note": "West, Brown and Enquist 1997, the derivation to formalize"
},
{
"url": "https://doi.org/10.1111/j.0269-8463.2004.00830.x",
"title": "Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant?",
"year": 2004,
"note": "Kozłowski and Konarzewski 2004, the objection to encode as a dropped hypothesis"
},
{
"url": "https://doi.org/10.1371/journal.pcbi.1000171",
"title": "Sizing Up Allometric Scaling Theory",
"year": 2008,
"note": "Savage et al. 2008, Sizing up allometric scaling theory (finite-size corrections)"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "metabolic-scaling-exponent--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Cross-domain heterogeneity test: one exponent across prokaryotes, protists and animals?",
"kind": "data-test",
"method": "Universality claims cover all life, but approach 1 tests mammals. DeLong et al. 2010 compiled metabolic rates for prokaryotes, protists and metazoans and reported superlinear, approximately linear and sublinear exponents respectively. Re-analyse with a pre-registered hierarchical model: log B = a_g + b_g log M + temperature correction, with group g ∈ {prokaryotes, protists, ectothermic metazoans, endotherms}, b_g ~ Normal(b̄, τ²). Pre-registered decision: universality is refuted if τ's 95% posterior interval excludes 0.05 and at least two b_g intervals are disjoint; supported if τ < 0.03 and b̄'s interval contains 3/4 or 2/3. Add AnAge ectotherm entries with temperature to test whether temperature correction collapses ectotherm and endotherm slopes, as metabolic theory predicts.",
"steps": [
"Pre-register the model, priors and decision thresholds.",
"Retrieve DeLong et al. 2010 supplementary data; add AnAge ectotherm records with temperature.",
"Fit the hierarchical model (Stan or PyMC) with measurement-error terms on M where reported.",
"Report τ, group slopes and the verdict; check sensitivity to the temperature correction."
],
"decisive_result": "The posterior for between-group slope heterogeneity τ, and whether group slopes overlap. Disjoint slopes across domains refute a single universal exponent regardless of the mammal result.",
"failure_signature": "Group samples (especially prokaryotes and protists) too small or too heterogeneous in measurement protocol for slopes to be compared, leaving τ's interval wide: inconclusive, underpowered.",
"tools": [
"anage",
"bayesian-hierarchical-models",
"data-test"
],
"datasets": [
{
"name": "DeLong et al. 2010 supplementary data",
"url": "https://doi.org/10.1073/pnas.1007783107",
"note": "Metabolic rate and mass across prokaryotes, protists and metazoans"
},
{
"name": "AnAge",
"url": "https://genomics.senescence.info/species/dataset.zip",
"note": "Ectotherm metabolic rates with temperatures"
}
],
"references": [
{
"url": "https://doi.org/10.1073/pnas.1007783107",
"title": "Shifts in metabolic scaling, production, and efficiency across major evolutionary transitions of life",
"year": 2010,
"note": "DeLong, Okie, Moses, Sibly and Brown 2010, Shifts in metabolic scaling across major evolutionary transitions of life"
},
{
"url": "https://doi.org/10.1126/science.276.5309.122",
"title": "A General Model for the Origin of Allometric Scaling Laws in Biology",
"year": 1997,
"note": "West, Brown and Enquist 1997"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "metabolic-scaling-exponent--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Phylogenetic model comparison across endotherms",
"kind": "data-test",
"method": "Pre-register adult, postabsorptive, thermoneutral basal-metabolism inclusion rules and fit mammal and bird data with phylogenetic generalized least squares. Compare fixed slopes 2/3 and 3/4, an estimated common slope, clade-specific slopes and a quadratic log-mass model using leave-one-family-out prediction.",
"steps": [
"Resolve taxa against the organisms layer and deduplicate measurements by source.",
"Require at least 300 species and 20 families, recording temperature and measurement protocol.",
"Fit all five models with body-temperature and study random effects.",
"Use family-held-out ELPD and calibrated residuals for the preregistered verdict."
],
"decisive_result": "Universality is supported only if one common-slope model beats every curved or clade-varying model by ELPD at least 10 and its 95% interval width is below 0.04; otherwise report the best heterogeneity structure.",
"failure_signature": "Protocol covariates are missing for over 30% of records or 2/3 and 3/4 remain inside the common-slope interval, yielding an inconclusive universal exponent.",
"tools": [
"phylogenetic-comparative-methods",
"anage",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "metabolic-scaling-exponent--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Formal audit of the WBE exponent derivation",
"kind": "formalization",
"method": "Encode an idealized branching network with branch ratio, length ratio, radius ratio, terminal-unit invariance and space-filling assumptions. Prove each exponent implication separately in Lean, including where area preservation or impedance minimization enters, and construct countermodels when assumptions are weakened.",
"steps": [
"Represent finite self-similar trees and derive total serviced volume and network volume.",
"Prove logarithmic exponent lemmas with positivity and nondegeneracy hypotheses.",
"Derive 3/4 only from a named minimal assumption set.",
"Use exact finite trees to exhibit assumption-compatible alternatives producing 2/3 or other exponents."
],
"decisive_result": "A no-sorry theorem stating the minimal explicit hypotheses implying exponent 3/4, plus checked countermodels showing which omitted hypotheses destroy the conclusion.",
"failure_signature": "The biological assumptions cannot be stated without directly assuming the desired scaling relation, showing that the derivation has no independent formal content.",
"tools": [
"lean4",
"symbolic-computation",
"allometry"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Formalization settles logical validity, not whether organisms satisfy the assumptions.",
"nanopub": null
},
{
"id": "metabolic-scaling-exponent--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "PGLS of PanTHERIA basal rate against body mass",
"kind": "data-test",
"method": "Use PanTHERIA 1.0, Wilson and Reeder 2005 file, columns 18-1_BasalMetRate_mLO2hr and the paired mass 5-2_BasalMetRateMass_g. The Ecological Archives metadata says metabolic rate was accepted only with a mass from the same individuals, measured resting, post-absorptive, in the thermoneutral zone. Drop the missing-value code -999. Primary regression is log10(BMR) on log10(paired mass), PGLS with Pagel's lambda, repeated on 100 trees from the Upham credible set. A species-typical mass sensitivity uses 5-1_AdultBodyMass_g and is reported separately. Model comparison, pre-registered: a single slope, order-specific slopes, and a quadratic term in log mass. The curved or order-specific model wins if its AIC beats the single power law by more than 10 on at least 95 of the 100 trees. The single-exponent claim is decided by the 95 percent interval of the single slope: it must exclude either 2/3 or 3/4. If the interval contains both, the question is underdetermined, which is the problem's own falsifier. Geographic mean temperature in PanTHERIA is not body temperature and is not used as a thermoregulatory control.",
"steps": [
"Pre-register the columns, the missing code, the AIC gap and the two target exponents.",
"Drop -999 and match species to the Upham trees.",
"Fit Pagel's lambda PGLS on 100 trees for the paired-mass regression.",
"Compare single-slope, order-specific and quadratic models by AIC.",
"Repeat with adult body mass as a labeled sensitivity and do not pool the two masses."
],
"decisive_result": "On at least 95 of 100 trees, either a single-slope 95 percent interval that excludes 2/3 or excludes 3/4, or an order-specific or quadratic model that beats the power law by AIC of more than 10.",
"failure_signature": "The single-slope interval contains both 2/3 and 3/4 and no alternative model beats the power law by the AIC gap. The universality question is then underdetermined on PanTHERIA.",
"tools": [
"phylogenetic-comparative-methods",
"allometry",
"data-test"
],
"datasets": [
{
"name": "PanTHERIA 1.0 Ecological Archives E090-184",
"url": "https://esapubs.org/archive/ecol/E090/184/metadata.htm",
"note": "Basal metabolic rate in mL O2/hr and the paired body mass; missing code -999; Wilson and Reeder 2005 file has 5416 species records."
},
{
"name": "Upham mammal phylogenies",
"url": "https://vertlife.org/data/mammals/",
"note": "One hundred trees for Pagel's lambda PGLS."
}
],
"references": [
{
"url": "https://doi.org/10.1890/08-1494.1",
"title": "PanTHERIA: a species‐level database of life history, ecology, and geography of extant and recently extinct mammals",
"year": 2009,
"note": "Jones et al. 2009, PanTHERIA"
},
{
"url": "https://doi.org/10.1371/journal.pbio.3000494",
"title": "Inferring the mammal tree: Species-level sets of phylogenies for questions in ecology, evolution, and conservation",
"year": 2019,
"note": "Upham, Esselstyn and Jetz 2019, mammal phylogenies"
},
{
"url": "https://doi.org/10.1126/science.276.5309.122",
"title": "A General Model for the Origin of Allometric Scaling Laws in Biology",
"year": 1997,
"note": "West, Brown and Enquist 1997, the 3/4 prediction"
},
{
"url": "https://doi.org/10.1073/pnas.0436428100",
"title": "Mammalian basal metabolic rate is proportional to body mass\n 2/3",
"year": 2003,
"note": "White and Seymour 2003, the 2/3 claim after physiological controls"
},
{
"url": "https://doi.org/10.3733/hilg.v06n11p315",
"title": "Body size and metabolism",
"year": 1932,
"note": "Kleiber 1932, Body size and metabolism"
}
],
"difficulty": "tractable",
"missing": "A species-level body-temperature table joined to this taxonomy. PanTHERIA's geographic temperature is not a substitute, so the White and Seymour control is not replicated here.",
"nanopub": null
},
{
"id": "metabolic-scaling-exponent--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Lean statement of the West–Brown–Enquist exponent",
"kind": "formalization",
"method": "Formalize the scaling argument under explicit hypotheses, and make the step Kozlowski and Konarzewski dispute a separate lemma. Hypotheses: a self-similar branching network with fixed branching ratio; a space-filling condition relating the linear scale factor to that ratio; a terminal unit whose size does not depend on body mass; and a flow or impedance condition that fixes how cross-section scales across a branch. From those, prove that metabolic rate scales as mass to the 3/4. Then prove a second lemma that names the alternative service-volume assumption used in the critique and shows whether it is consistent with invariant terminal units. The module must not hide either assumption inside a definition. Empirical truth of the hypotheses is out of scope, as in the Thompson Lean standard.",
"steps": [
"Define branching ratio, scale factor, terminal-unit size and service volume.",
"Prove the 3/4 exponent from the stated hypotheses.",
"State the critiqued service-volume assumption as a second definition.",
"Prove consistency or inconsistency with invariant terminal units.",
"Audit axioms and write the scope note that the hypotheses are not data."
],
"decisive_result": "A no-sorry derivation of the 3/4 exponent under named hypotheses, and a second lemma that settles consistency of the critiqued alternative with invariant capillaries.",
"failure_signature": "The 3/4 exponent is obtained only by assuming the conclusion, or the two lemmas use incompatible definitions of service volume without saying so.",
"tools": [
"lean4",
"allometry"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1126/science.276.5309.122",
"title": "A General Model for the Origin of Allometric Scaling Laws in Biology",
"year": 1997,
"note": "West, Brown and Enquist 1997, A general model for the origin of allometric scaling laws in biology"
},
{
"url": "https://doi.org/10.1111/j.0269-8463.2004.00830.x",
"title": "Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant?",
"year": 2004,
"note": "Kozlowski and Konarzewski 2004, the critique of the derivation"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "metabolic-scaling-exponent--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Finite-generation networks and the approach to 3/4",
"kind": "computation",
"method": "The Lean theorem is an infinite-generation statement. Build the finite network the hypotheses describe: branching ratio 2, generations G from 2 to 24, area-preserving branch radii, terminal units of fixed size, service volume the sum of the terminal service volumes. Define metabolic rate as total terminal flow. Regress log rate on log service volume across the generation sequence and report the local exponent between successive generations. The numerical claim matching the theorem is that the local exponent is within 0.01 of 3/4 by some G at most 24. Also build the alternative network from the critique's service-volume assumption and report its exponent. This checks the algebra of both lemmas and says nothing about real mammals.",
"steps": [
"Implement both finite networks from the Lean hypotheses.",
"Compute total flow and service volume at each generation from 2 to 24.",
"Estimate the local log-log exponent between successive generations.",
"Test whether the primary network is within 0.01 of 3/4 by generation 24.",
"Report the alternative network's exponent beside it."
],
"decisive_result": "The primary network's local exponent lies within 0.01 of 3/4 at some generation at most 24, and the alternative network's exponent is reported from the same code.",
"failure_signature": "The primary exponent stays more than 0.01 away from 3/4 through generation 24. The finite network then does not realize the theorem, and one of the two must be wrong.",
"tools": [
"allometry"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1126/science.276.5309.122",
"title": "A General Model for the Origin of Allometric Scaling Laws in Biology",
"year": 1997,
"note": "West, Brown and Enquist 1997, the network being generated"
},
{
"url": "https://doi.org/10.1111/j.0269-8463.2004.00830.x",
"title": "Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant?",
"year": 2004,
"note": "Kozlowski and Konarzewski 2004, the alternative service-volume assumption"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
}
]
},
{
"id": "species-area-exponent",
"title": "Why does the species-area exponent cluster where it does?",
"field": "biology",
"tier": "B",
"question": "Is the island species-area relationship a power law with a near-universal exponent z around 0.25, and if so what sets that value; or is the power law an approximation to a non-power form (as the maximum entropy theory of ecology predicts) whose apparent exponent depends on scale?",
"known_state": "Darwin recorded that islands hold fewer species than equal continental areas (ch12-04). Drakare, Lennon and Hillebrand (2006) meta-analysed 794 species-area curves and found z depends on sampling design, habitat and body size; Harte et al. (2009) derived a universal non-power curve from METE; Triantis, Guilhaumon and Whittaker (2012) compared 20 functional forms across 601 island datasets and found the power law best overall but with substantial variation in z. Whether a single mechanism sets z is unresolved.",
"acceptance_criteria": [
"A pre-registered test on GBIF occurrence counts for a defined set of island archipelagos (from the organisms layer), fitting the power law, METE and at least two alternatives with identical sampling corrections, stating in advance the decision rule for 'universal exponent' versus 'scale-dependent form'."
],
"falsifiers": [
"METE predictions failing systematically at the island scale would refute the universal non-power form; z intervals that do not overlap across archipelagos refute universality."
],
"required_capabilities": [
"gbif",
"macroecology",
"model-comparison",
"data-test"
],
"data_access": "GBIF (open), island area from Triantis et al. 2012 supplementary data (open).",
"scales": [
"ecological"
],
"frontier_problem": null,
"related_claims": [
{
"id": "darwin1859-ch12-04",
"label": "Oceanic islands have fewer species than equal continental areas",
"thinker": "Charles Darwin",
"status": "proved",
"evidence": "supports",
"theorem": true
},
{
"id": "darwin1859-ch12-05",
"label": "Oceanic islands have a very high proportion of endemic species",
"thinker": "Charles Darwin",
"status": "proved",
"evidence": "supports",
"theorem": true
},
{
"id": "darwin1859-ch11-02",
"label": "Barriers to migration correspond closely to differences between regional biotas",
"thinker": "Charles Darwin",
"status": "untested",
"evidence": "supports",
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1111/j.1461-0248.2005.00848.x",
"title": "The imprint of the geographical, evolutionary and ecological context on species–area relationships",
"year": 2005,
"note": "Drakare, Lennon and Hillebrand 2006, The imprint of the geographical, evolutionary and ecological context on species-area relationships"
},
{
"url": "https://doi.org/10.1111/j.1461-0248.2009.01328.x",
"title": "Biodiversity scales from plots to biomes with a universal species–area curve",
"year": 2009,
"note": "Harte et al. 2009, Biodiversity scales from plots to biomes with a universal species-area curve"
},
{
"url": "https://doi.org/10.1111/j.1365-2699.2011.02652.x",
"title": "The island species–area relationship: biology and statistics",
"year": 2011,
"note": "Triantis, Guilhaumon and Whittaker 2012, The island species-area relationship: biology and statistics"
}
],
"nanopub": null,
"approaches": [
{
"id": "species-area-exponent--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Pre-registered GBIF island SAR test with coverage-standardized richness",
"kind": "data-test",
"method": "Archipelagos: Canary Islands, Galápagos, Hawaiian main islands, Azores, Balearic Islands, Cape Verde, chosen before looking at data because they have island areas in GIFT and dense GBIF sampling. Taxa: vascular plants, land birds and land snails, analysed separately. For each island × taxon, estimate richness from GBIF occurrences by coverage-based rarefaction to a common sample coverage of 0.95 (iNEXT), and also use GIFT checklists for plants as a sampling-independent check. Fit with the sars package, using identical richness estimates for every model: power (S = cA^z), logarithmic, Weibull-3 and the METE-predicted island curve (Harte et al. 2009 form, anchored at the largest island with its total abundance proxy). Pre-registered rules: 'universal power law' if the power model has the lowest AICc in ≥ 70% of archipelago × taxon sets and all z 95% CIs overlap a common interval of width ≤ 0.1; 'scale-dependent form' if METE or a curved model wins in ≥ 50% and local slope declines with area as METE predicts; otherwise 'neither'.",
"steps": [
"Write result.json with archipelagos, taxa, coverage target, models and decision rules.",
"Resolve island polygons and areas from GIFT; download GBIF occurrences per island polygon with a fetch script and checksums.",
"Compute coverage-standardized richness (and GIFT checklist richness for plants).",
"Fit all models with sars; compute AICc ranks and z intervals; apply the decision rule."
],
"decisive_result": "For 18 archipelago × taxon sets (6 × 3), the share where the power law wins and whether z intervals overlap, giving a pre-registered verdict among 'universal power law', 'scale-dependent form' or 'neither'.",
"failure_signature": "GBIF coverage is too low on small islands for coverage standardization to reach 0.95 in most sets, and plant results disagree with GIFT checklists by more than the CI widths: inconclusive, data quality.",
"tools": [
"gbif",
"organisms-layer",
"macroecology",
"model-comparison",
"data-test"
],
"datasets": [
{
"name": "GBIF occurrence API",
"url": "https://api.gbif.org/v1/occurrence/search?limit=1",
"note": "Occurrence records per island polygon"
},
{
"name": "GIFT – Global Inventory of Floras and Traits",
"url": "https://gift.uni-goettingen.de/",
"note": "Island geometries and areas, plant checklists"
}
],
"references": [
{
"url": "https://doi.org/10.1111/j.1365-2699.2011.02652.x",
"title": "The island species–area relationship: biology and statistics",
"year": 2011,
"note": "Triantis, Guilhaumon and Whittaker 2012, island SAR across 601 datasets"
},
{
"url": "https://doi.org/10.1111/j.1461-0248.2009.01328.x",
"title": "Biodiversity scales from plots to biomes with a universal species–area curve",
"year": 2009,
"note": "Harte et al. 2009, universal species–area curve (METE)"
},
{
"url": "https://doi.org/10.1111/ecog.04271",
"title": "sars: an R package for fitting, evaluating and comparing species–area relationship models",
"year": 2019,
"note": "Matthews et al. 2019, sars R package"
},
{
"url": null,
"note": "Hsieh, Ma and Chao 2016, iNEXT"
},
{
"url": null,
"note": "Weigelt et al. 2020, GIFT"
}
],
"difficulty": "hard",
"missing": "GBIF sampling bias across islands is severe for snails; METE needs abundance totals that occurrence data only approximate.",
"nanopub": null
},
{
"id": "species-area-exponent--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Sampling-only null: do lognormal SADs and random placement reproduce z?",
"kind": "computation",
"method": "Before asking what mechanism sets z ≈ 0.25, establish what z distribution arises with no mechanism beyond the species-abundance distribution and random placement. Preston (1962) and later May argued that a canonical lognormal yields z ≈ 0.26 for isolates. Compute this exactly over the range of conditions in real datasets. For each of a grid of total individuals J (10^3–10^9), species pools S₀ and SAD shapes (lognormal with σ ∈ [1, 3], log-series with α ∈ [5, 500], and Preston's canonical constraint), place individuals on islands proportional to area (Coleman random placement, expected richness E[S(a)] = Σ_i 1 − (1 − a)^{n_i}). Fit z over island-area spans matching each dataset in the Triantis et al. compilation (area range, number of islands), producing the null distribution of z per dataset. Compare the observed z distribution with the null by quantile regression and a two-sample Anderson–Darling test.",
"steps": [
"Pre-register the SAD families, parameter grid and comparison statistics.",
"Compute expected richness curves analytically (Coleman formula) and stochastically for small J.",
"Match null conditions to each published dataset's area span and island count; generate null z.",
"Compare observed and null z distributions; report the variance in z the null explains."
],
"decisive_result": "The fraction of the between-dataset variance in z explained by SAD shape and area span alone. If the sampling null reproduces the observed z distribution (Anderson–Darling p > 0.05 under a plausible SAD), z ≈ 0.25 needs no ecological mechanism; if observed z is systematically higher (for example on isolated archipelagos), the excess quantifies the mechanism's contribution.",
"failure_signature": "The null z depends so strongly on unobservable SAD parameters that a plausible SAD can be found to fit any observed z, so the null has no discriminating power.",
"tools": [
"large-scale-computation",
"macroecology"
],
"datasets": [
{
"name": "Triantis et al. 2012 island SAR compilation (supplementary data)",
"url": "https://doi.org/10.1111/j.1365-2699.2011.02652.x",
"note": "Area spans, island counts and fitted z for 601 island datasets"
}
],
"references": [
{
"url": "https://doi.org/10.2307/1931976",
"title": "The Canonical Distribution of Commonness and Rarity: Part I",
"year": 1962,
"note": "Preston 1962, The canonical distribution of commonness and rarity"
},
{
"url": "https://doi.org/10.1111/j.1365-2699.2011.02652.x",
"title": "The island species–area relationship: biology and statistics",
"year": 2011,
"note": "Triantis, Guilhaumon and Whittaker 2012"
},
{
"url": "https://doi.org/10.1111/j.1461-0248.2005.00848.x",
"title": "The imprint of the geographical, evolutionary and ecological context on species–area relationships",
"year": 2005,
"note": "Drakare, Lennon and Hillebrand 2006, 794 SAR curves"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "species-area-exponent--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Random-effects meta-analysis of z heterogeneity plus a METE scaling-collapse test",
"kind": "data-test",
"method": "Universality means the true between-dataset variance τ² of z is near zero after sampling error. Using the Triantis et al. 2012 and Drakare et al. 2006 compilations, estimate each dataset's z with its standard error (refitting from the raw S–A data where supplied, otherwise from reported fits) and fit a random-effects meta-regression: z_j = μ + x_j'β + u_j + e_j, with u_j ~ N(0, τ²), e_j ~ N(0, se_j²), and covariates for isolation, island type, taxon and body size. Pre-registered: universality refuted if τ's 95% CI excludes 0.03, a z standard deviation one eighth of the canonical 0.25. Independently, test METE's specific prediction (Harte et al. 2009) that the local slope z at a scale is a universal decreasing function of log(N/S) at that scale: for datasets with abundance N, compute local slopes and test the collapse onto the METE curve by its residual variance against a free-curve alternative.",
"steps": [
"Pre-register τ threshold, covariates and the METE collapse statistic.",
"Extract per-dataset z and SE; flag datasets with abundance data for the METE test.",
"Fit the random-effects meta-regression (REML) and report τ², I² and covariate effects.",
"Run the METE collapse test on datasets with abundances; report the residual variance ratio."
],
"decisive_result": "τ with CI (universality supported if the CI upper bound is < 0.03, refuted if the lower bound is > 0.03), the share of heterogeneity explained by covariates, and whether local slopes collapse onto the METE curve, the decisive test between 'universal exponent' and 'universal non-power form'.",
"failure_signature": "Too few datasets report abundances for the METE test (fewer than 20), and z standard errors are unavailable for most compiled curves, making τ uninterpretable.",
"tools": [
"meta-analysis",
"macroecology",
"data-test"
],
"datasets": [
{
"name": "Triantis et al. 2012 compilation",
"url": "https://doi.org/10.1111/j.1365-2699.2011.02652.x",
"note": "Island SAR datasets and fits"
},
{
"name": "Drakare et al. 2006 compilation",
"url": "https://doi.org/10.1111/j.1461-0248.2005.00848.x",
"note": "794 species–area curves with design covariates"
}
],
"references": [
{
"url": "https://doi.org/10.1111/j.1461-0248.2009.01328.x",
"title": "Biodiversity scales from plots to biomes with a universal species–area curve",
"year": 2009,
"note": "Harte et al. 2009, METE universal SAR"
},
{
"url": "https://doi.org/10.1111/j.1461-0248.2005.00848.x",
"title": "The imprint of the geographical, evolutionary and ecological context on species–area relationships",
"year": 2005,
"note": "Drakare, Lennon and Hillebrand 2006"
}
],
"difficulty": "hard",
"missing": "Raw S–A data for many compiled curves are not public, so standard errors may have to be reconstructed from the reported number of islands and R².",
"nanopub": null
},
{
"id": "species-area-exponent--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Coverage-corrected island model tournament",
"kind": "data-test",
"method": "Select at least 30 oceanic archipelagos and construct standardized GBIF checklists with equal-coverage rarefaction. Fit power law, log-series, Lomolino and METE forms using the same observation model, area covariates and spatially blocked cross-validation. Estimate archipelago-specific z with partial pooling.",
"steps": [
"Freeze island polygons, taxonomic groups, dates and minimum 1000 occurrences per archipelago.",
"Estimate completeness from singleton and doubleton frequencies and rarefy to 80% coverage.",
"Fit all candidate forms with identical detection offsets.",
"Hold out whole islands and compare predictive log density and z heterogeneity."
],
"decisive_result": "A universal power law requires ELPD at least 10 above every alternative and between-archipelago SD(z) below 0.03 around a mean within 0.03 of 0.25; nonoverlapping z intervals or superior non-power prediction refutes it.",
"failure_signature": "Coverage estimates below 0.5 for more than one third of archipelagos or severe geographic sampling clustering makes the test inconclusive.",
"tools": [
"gbif",
"macroecology",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Reliable island polygons and absence-aware inventories for poorly sampled archipelagos.",
"nanopub": null
},
{
"id": "species-area-exponent--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Scale-transition test within nested surveys",
"kind": "data-test",
"method": "Use nested quadrat and island inventories to test whether apparent z changes predictably with sampled area. Pre-register local-slope estimators d log S/d log A for power-law and METE curves, and compare them on datasets with at least six nested area scales and 50 species.",
"steps": [
"Assemble at least 50 independent nested sampling series without pooling nonnested sites.",
"Estimate local slopes with a monotone spline whose smoothing is fixed by simulation.",
"Compare constant-slope and METE-predicted slope curves by leave-one-scale-out prediction.",
"Meta-analyze effects with biome and taxon moderators."
],
"decisive_result": "Scale dependence is supported if at least 70% of series show directional local-slope change and the non-power model improves total ELPD by at least 20; otherwise a constant exponent remains adequate.",
"failure_signature": "Fewer than 30 series retain six scales after harmonization or richness estimates are dominated by unequal effort.",
"tools": [
"macroecology",
"data-test",
"large-scale-computation"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "A harmonized public collection of genuinely nested surveys.",
"nanopub": null
},
{
"id": "species-area-exponent--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Power law against METE on GIFT island floras",
"kind": "data-test",
"method": "Pre-register on GIFT island regions: complete vascular-plant checklists, native species, region area as returned by the GIFT API. Exclude incomplete lists. Fit three models with the same data and the same log-likelihood: the power law log S = log c + z log A by a negative-binomial GLM with log area as offset predictor; the logarithmic curve S = c + z log A under the same likelihood family; and the METE species-area curve implemented by meteR's meteSAR, following Harte et al. 2009, where expected richness is the sum over the MaxEnt species-abundance distribution of the probability a species is present. Universal exponent, pre-registered: the pooled z has a 95 percent interval of width less than 0.1, and every island group with at least 10 islands has a z interval overlapping that pooled interval. Groups are the island-group identifiers GIFT returns; if that field is absent, stop, do not invent clusters. METE wins only if its AIC beats the power law by more than 10 and its mean absolute residual is smaller.",
"steps": [
"Freeze the likelihood, the width criterion and the AIC gap before querying GIFT.",
"Download complete native vascular-plant island checklists and areas.",
"Fit the power law, the logarithmic curve and meteSAR.",
"Compute pooled and per-group z intervals.",
"Apply the universal-exponent rule and the METE rule separately."
],
"decisive_result": "Either a pooled z interval narrower than 0.1 that every qualifying island group overlaps, or a rejection of that universality, plus an AIC comparison of METE against the power law with gap 10.",
"failure_signature": "Group identifiers are missing, or fewer than five groups have 10 islands. The test is then inconclusive. A METE win on residual and a loss on AIC, or the converse, is a split result and does not count as a win.",
"tools": [
"gbif",
"macroecology",
"model-comparison",
"data-test"
],
"datasets": [
{
"name": "GIFT, Global Inventory of Floras and Traits",
"url": "https://gift.uni-goettingen.de/",
"note": "Island checklists, native status and region area."
},
{
"name": "GIFT R package documentation",
"url": "https://biogeomacro.github.io/GIFT/",
"note": "API access to checklists, including the island geographic type."
},
{
"name": "meteR",
"url": "https://cran.r-project.org/package=meteR",
"note": "meteSAR implementation of the Harte species-area curve."
}
],
"references": [
{
"url": "https://doi.org/10.1111/j.1461-0248.2009.01328.x",
"title": "Biodiversity scales from plots to biomes with a universal species–area curve",
"year": 2009,
"note": "Harte et al. 2009, the METE species-area curve"
},
{
"url": null,
"note": "Weigelt et al. 2019, GIFT"
},
{
"url": "https://doi.org/10.1111/2041-210X.14213",
"title": "GIFT\n —An R package to access the Global Inventory of Floras and Traits",
"year": 2023,
"note": "Denelle, Weigelt and Kreft, the GIFT R package"
},
{
"url": "https://doi.org/10.1111/j.1461-0248.2005.00848.x",
"title": "The imprint of the geographical, evolutionary and ecological context on species–area relationships",
"year": 2005,
"note": "Drakare, Lennon and Hillebrand 2006, context dependence of z"
},
{
"url": "https://doi.org/10.1111/j.1365-2699.2011.02652.x",
"title": "The island species–area relationship: biology and statistics",
"year": 2011,
"note": "Triantis, Guilhaumon and Whittaker 2012, comparison of island species-area forms"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "species-area-exponent--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "A logseries null for the island exponent",
"kind": "computation",
"method": "Test the sampling explanation of z without fitting z on the same islands used to build the null. Fit Fisher's logseries to the single largest GIFT island, or to one pre-registered mainland GIFT region if the largest island is itself in the test set. Freeze those parameters. Place individuals independently onto the other islands with probability proportional to island area, thinning the same abundance vector. Estimate z by the same negative-binomial GLM as the data test, 1000 replicates. The empirical pooled z is a sampling consequence if it lies inside the central 95 percent of this null. The null uses the empirical area distribution and does not use the empirical species lists of the test islands.",
"steps": [
"Choose the calibration region and freeze it before estimating z on the remainder.",
"Fit the logseries on the calibration region only.",
"Thin individuals onto the other islands proportional to area, 1000 times.",
"Estimate z with the same GLM as the data test.",
"Locate the empirical z in the null distribution."
],
"decisive_result": "The empirical pooled z either lies inside the central 95 percent of the logseries-placement null or it does not. Inside means the cluster of z is reproduced by this sampling model.",
"failure_signature": "The empirical z falls outside the null interval. A biological mechanism is then still required, and this approach has not explained the exponent. Using the test islands to fit the logseries is a failed pre-registration.",
"tools": [
"macroecology",
"model-comparison"
],
"datasets": [
{
"name": "GIFT, Global Inventory of Floras and Traits",
"url": "https://gift.uni-goettingen.de/",
"note": "Calibration-region abundances and the area distribution of the remaining islands."
}
],
"references": [
{
"url": "https://doi.org/10.1111/j.1461-0248.2009.01328.x",
"title": "Biodiversity scales from plots to biomes with a universal species–area curve",
"year": 2009,
"note": "Harte et al. 2009, the MaxEnt logseries this null approximates in the calibration region"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "species-area-exponent--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Prove the METE slope is not a constant power",
"kind": "formalization",
"method": "Formalize the Harte 2009 claim that the MaxEnt species-area curve is not a power law, so a local exponent near 0.25 is an accident of scale. Define expected richness S(A) as S0 times the sum over abundances of the MaxEnt species-abundance probabilities times the probability a species of that abundance is present in a nested cell of area A. Define the local exponent as the derivative of log S with respect to log A. Prove it is not constant on (0, A0) under the Harte constraints, and prove the published limiting statement that the local exponent tends to 0, not to 1/4, as mean abundance per species goes to infinity. This is a theorem about the model. It does not say islands obey the model.",
"steps": [
"Encode the MaxEnt abundance distribution and the spatial presence probability from Harte et al. 2009.",
"Define S(A) and the local exponent.",
"Prove the local exponent is non-constant for finite mean abundance.",
"Prove the infinite-abundance limit is 0.",
"Audit axioms."
],
"decisive_result": "A no-sorry proof that the METE local exponent is not constant and tends to 0 as mean abundance per species tends to infinity.",
"failure_signature": "Under the stated constraints the local exponent is in fact constant. The formalization would then have found that METE does not say what the approach claims it says.",
"tools": [
"lean4",
"macroecology"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1111/j.1461-0248.2009.01328.x",
"title": "Biodiversity scales from plots to biomes with a universal species–area curve",
"year": 2009,
"note": "Harte et al. 2009, Biodiversity scales from plots to biomes with a universal species-area curve"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
}
]
},
{
"id": "clade-trajectory-universality",
"title": "Are clade diversity trajectories universally hat-shaped?",
"field": "biology",
"tier": "B",
"question": "Do most fossil clades rise to a diversity peak and decline symmetrically, as Darwin and later Foote and Quental and Marshall proposed, or is the apparent hat shape a product of sampling and the way clades are defined?",
"known_state": "Darwin claimed groups rise gradually to a maximum then decline (ch10-06; repo verdict mixed). Foote (2007) found symmetric waxing and waning of marine invertebrate genera; Liow and Stenseth (2007) found similar shapes in mammals; Quental and Marshall (2013) linked decline to Red Queen dynamics. Critics attribute symmetry to the 'pull of the recent', paraphyletic clade definitions and sampling heterogeneity. No standardized, subsampling-corrected test across all orders and families exists.",
"acceptance_criteria": [
"A pre-registered test over all PBDB orders and families meeting stated occurrence thresholds, using shareholder-quorum subsampling and monophyly filters, with a stated statistic for hat-shapedness and a null from randomized origination-extinction sequences.",
"A verdict with effect size: fraction of clades significantly hat-shaped versus the null."
],
"falsifiers": [
"If the fraction of hat-shaped clades is indistinguishable from the random-walk null after subsampling, the pattern is an artifact."
],
"required_capabilities": [
"pbdb",
"paleobiology-statistics",
"subsampling",
"data-test"
],
"data_access": "Paleobiology Database API (open), already wired into test-data/organisms.",
"scales": [
"evolutionary"
],
"frontier_problem": null,
"related_claims": [
{
"id": "darwin1859-ch10-06",
"label": "Groups rise gradually in species number to a maximum, then gradually decline",
"thinker": "Charles Darwin",
"status": "untested",
"evidence": "mixed",
"theorem": false
},
{
"id": "darwin1859-ch10-07",
"label": "Species and groups disappear gradually, place by place, not by global catastrophes",
"thinker": "Charles Darwin",
"status": "untested",
"evidence": "mixed",
"theorem": false
},
{
"id": "darwin1859-ch10-02",
"label": "Species of different genera and classes change at different rates",
"thinker": "Charles Darwin",
"status": "untested",
"evidence": "supports",
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1666/06084.1",
"title": "Symmetric waxing and waning of marine invertebrate genera",
"year": 2007,
"note": "Foote 2007, Symmetric waxing and waning of marine invertebrate genera"
},
{
"url": "https://doi.org/10.1098/rspb.2007.1006",
"title": "The rise and fall of species: implications for macroevolutionary and macroecological studies",
"year": 2007,
"note": "Liow and Stenseth 2007, The rise and fall of species: implications for macroevolutionary and macroecological studies"
},
{
"url": "https://doi.org/10.1126/science.1239431",
"title": "How the Red Queen Drives Terrestrial Mammals to Extinction",
"year": 2013,
"note": "Quental and Marshall 2013, How the Red Queen drives terrestrial mammals to extinction"
}
],
"nanopub": null,
"approaches": [
{
"id": "clade-trajectory-universality--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Pre-registered PBDB test: hat-shapedness of subsampled trajectories versus an MBL null",
"kind": "data-test",
"method": "Download PBDB occurrences at genus resolution for all orders and families, assign occurrences to stages (ICS 2023), and estimate per-stage richness by shareholder quorum subsampling at quorum 0.7 (divDyn). Include clades with ≥ 6 stages of duration and ≥ 50 occurrences. Hat-shapedness statistics: (a) center of gravity CG ∈ [0, 1] of the subsampled diversity curve (Gould et al. 1987); (b) a unimodality score, the share of total variation explained by the best single-peaked isotonic fit; (c) a symmetry score |CG − 0.5|. Null: the Raup et al. (1973) MBL process, a homogeneous birth–death process with per-clade origination and extinction rates estimated from the clade's own boundary-crosser counts, conditioned on clade duration and terminal diversity, passed through the same sampling (per-stage sampling probabilities from three-timer analysis) and the same SQS procedure. Pre-registered verdict: hat-shape is a real pattern if the fraction of clades with unimodality above the null's 95th percentile exceeds 5% by at least 10 percentage points (binomial test p < 0.01).",
"steps": [
"Write result.json with inclusion thresholds, statistics, null and decision rule before downloading.",
"Fetch PBDB occurrences via the API with a scripted query; record checksums.",
"Compute SQS curves per clade; compute CG, unimodality and symmetry.",
"Simulate 1,000 null clades per real clade; compute empirical percentiles; apply the decision rule; report effect sizes separately for marine invertebrates and terrestrial vertebrates."
],
"decisive_result": "The fraction of clades significantly hat-shaped against a sampling-matched birth–death null, with its binomial CI. Reaching that fraction is the effect size the problem asks for. Several hundred qualifying families are expected, enough to detect a 10-point excess.",
"failure_signature": "The null's percentiles are dominated by the conditioning on duration and terminal diversity (conditioned random walks are themselves hat-shaped), leaving no power to detect an excess; or SQS fails for many clades with low occurrence counts.",
"tools": [
"pbdb",
"organisms-layer",
"subsampling",
"paleobiology-statistics",
"data-test"
],
"datasets": [
{
"name": "Paleobiology Database API",
"url": "https://paleobiodb.org/data1.2/",
"note": "Occurrences with taxonomy and stratigraphy"
}
],
"references": [
{
"url": "https://doi.org/10.1666/06084.1",
"title": "Symmetric waxing and waning of marine invertebrate genera",
"year": 2007,
"note": "Foote 2007, Symmetric waxing and waning of marine invertebrate genera"
},
{
"url": "https://doi.org/10.1098/rspb.2007.1006",
"title": "The rise and fall of species: implications for macroevolutionary and macroecological studies",
"year": 2007,
"note": "Liow and Stenseth 2007, The rise and fall of species. The DOI 10.1098/rspb.2007.0712 in problems.json resolves to a different paper."
},
{
"url": "https://doi.org/10.1126/science.236.4807.1437",
"title": "Asymmetry of Lineages and the Direction of Evolutionary Time",
"year": 1987,
"note": "Gould, Gilinsky and German 1987, Asymmetry of lineages (center of gravity)"
},
{
"url": "https://doi.org/10.1086/627905",
"title": "Stochastic Models of Phylogeny and the Evolution of Diversity",
"year": 1973,
"note": "Raup, Gould, Schopf and Simberloff 1973, MBL stochastic phylogeny model"
},
{
"url": null,
"note": "Alroy 2010, shareholder quorum subsampling"
},
{
"url": "https://doi.org/10.1111/2041-210X.13161",
"title": "The\n r\n package divDyn for quantifying diversity dynamics using fossil sampling data",
"year": 2019,
"note": "Kocsis et al. 2019, divDyn R package"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "clade-trajectory-universality--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Model-based rates: does net diversification decline through clade life?",
"kind": "data-test",
"method": "Hat-shapedness from diversity curves is indirect; the mechanistic claim (Quental and Marshall) is that origination falls and extinction rises with clade age. Estimate this directly with PyRate (Bayesian joint estimation of preservation, origination and extinction from occurrence data) for each family with ≥ 20 genera and ≥ 100 occurrences, using a time-variable preservation model. For each clade compute the posterior of the slope of net diversification rate r(t) = λ(t) − μ(t) against standardized clade age (birth–death shift model or a linear-trend model in clade age). Pre-registered: the claim holds if more than 50% of clades have posterior probability > 0.95 of a negative slope, and the median slope is negative with a 95% interval excluding 0; it fails if negative-slope clades are no more frequent than positive-slope ones.",
"steps": [
"Pre-register clade inclusion, PyRate settings (preservation model, MCMC length, convergence criteria ESS > 200) and the decision rule.",
"Prepare PyRate inputs from PBDB occurrences with age ranges.",
"Run PyRate per clade on a compute cluster; check convergence.",
"Summarize slope posteriors across clades; apply the decision rule; compare with approach 1's CG and unimodality."
],
"decisive_result": "The distribution across clades of the posterior slope of net diversification versus clade age. A majority of significantly negative slopes would support Red-Queen decline as general; a symmetric distribution around zero would mean hat shapes, if present, are not driven by systematic rate decline.",
"failure_signature": "Rate trends are not identifiable from preservation trends for most clades (posterior slopes span zero widely), which is likely for small clades: inconclusive.",
"tools": [
"pbdb",
"bayesian-inference",
"large-scale-computation",
"data-test"
],
"datasets": [
{
"name": "Paleobiology Database API",
"url": "https://paleobiodb.org/data1.2/",
"note": "Occurrences with age ranges for PyRate"
}
],
"references": [
{
"url": null,
"note": "Silvestro et al. 2014, PyRate"
},
{
"url": "https://doi.org/10.1126/science.1239431",
"title": "How the Red Queen Drives Terrestrial Mammals to Extinction",
"year": 2013,
"note": "Quental and Marshall 2013, How the Red Queen drives terrestrial mammals to extinction"
},
{
"url": "https://github.com/dsilvestro/PyRate",
"title": null,
"year": null,
"note": "PyRate source"
}
],
"difficulty": "hard",
"missing": "Compute for hundreds of PyRate runs (each hours to days); approach is feasible but may need to restrict to a few hundred clades.",
"nanopub": null
},
{
"id": "clade-trajectory-universality--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Artifact audit: hat-shape from Linnaean grouping and the pull of the recent",
"kind": "simulation",
"method": "Critics attribute symmetric clade histories to how clades are defined. Measure that directly. Simulate phylogenies under homogeneous birth–death (no rate decline) for 10^4 replicates per parameter set spanning observed origination and extinction rates. Partition each tree into named 'taxa' with three rules: (a) monophyletic crown and stem clades; (b) a Linnaean-style rule that founds a new 'family' when morphological divergence (Brownian character) passes a threshold, leaving paraphyletic remainders, as taxonomists do; (c) rank-by-age cuts. Apply realistic fossil sampling (Poisson preservation with rates from PBDB three-timer estimates) and the pull of the recent (perfect sampling of extant taxa). Compute the same CG and unimodality statistics as approach 1. The artifact share is the excess hat-shapedness under (b) and under the pull of the recent relative to (a) with uniform sampling. Then test on real data: restrict approach 1 to clades whose monophyly is supported in Open Tree of Life and to wholly extinct clades, and compare effect sizes.",
"steps": [
"Implement birth–death simulation with character evolution and the three taxon-definition rules.",
"Add preservation and pull-of-the-recent sampling; compute hat statistics per pseudo-taxon.",
"Quantify artifact contributions per rule; publish code and seeds.",
"Rerun approach 1 on monophyletic-only and extinct-only subsets; compare with the simulated artifact sizes."
],
"decisive_result": "The amount of hat-shapedness (excess unimodality over the MBL null) produced by paraphyletic definition and the pull of the recent in simulations with no real decline, compared with the excess in real data. If real excess ≤ simulated artifact excess, the pattern is an artifact; if real monophyletic and extinct-only clades retain the excess, it is not.",
"failure_signature": "The Linnaean rule cannot be calibrated: artifact size depends on an arbitrary divergence threshold over the plausible range, so the comparison has no fixed reference.",
"tools": [
"simulation",
"pbdb",
"phylogenetics"
],
"datasets": [
{
"name": "Open Tree of Life",
"url": "https://opentreeoflife.github.io/",
"note": "Monophyly support for real clades"
},
{
"name": "Paleobiology Database API",
"url": "https://paleobiodb.org/data1.2/",
"note": "Preservation-rate estimation"
}
],
"references": [
{
"url": "https://doi.org/10.1126/science.1083246",
"title": "The Impact of the Pull of the Recent on the History of Marine Diversity",
"year": 2003,
"note": "Jablonski et al. 2003, The impact of the pull of the recent on the history of marine diversity"
},
{
"url": "https://doi.org/10.1086/627905",
"title": "Stochastic Models of Phylogeny and the Evolution of Diversity",
"year": 1973,
"note": "Raup et al. 1973, MBL model"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "clade-trajectory-universality--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "PBDB subsampled hat-shape census",
"kind": "data-test",
"method": "Query all PBDB marine orders and families with at least 200 occurrences, 20 genera and 10 occupied time bins. Apply shareholder-quorum subsampling at quorum 0.6, align each clade to fractional lifetime, and quantify hat shape by peak interiority and signed monotonicity before and after the peak. Compare against occurrence-preserving time randomizations.",
"steps": [
"Freeze taxonomy, time scale, quorum and clade thresholds before downloading occurrences.",
"Run 1000 subsampling replicates per clade.",
"Generate 10000 null trajectories preserving duration, total occurrences and bin-level sampling intensity.",
"Control false discovery rate at 0.05 and report the supported fraction with bootstrap intervals."
],
"decisive_result": "Universality requires more than 75% of eligible clades to be significantly hat-shaped and the lower 95% interval above the 95th null percentile; equivalence to the null refutes it.",
"failure_signature": "Results reverse across reasonable quorum values 0.4 to 0.7 or fewer than 50 clades meet inclusion, indicating sampling dependence.",
"tools": [
"pbdb",
"paleobiology-statistics",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "clade-trajectory-universality--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Birth-death observation-model discrimination",
"kind": "computation",
"method": "Fit each clade with a state-space birth-death model whose fossil observations have bin-specific sampling probability. Compare constant-rate, diversity-dependent, symmetric hat and asymmetric rise-decline processes by simulation-based calibration and leave-future-bins-out prediction. Include only clades passing posterior identifiability checks.",
"steps": [
"Simulate 10000 clades per model across PBDB-like sampling regimes to calibrate model recovery.",
"Specify priors and identifiability thresholds before fitting real clades.",
"Fit at least 50 eligible clades and evaluate posterior model probabilities.",
"Repeat with monophyletic clades only and quantify pull-of-the-recent sensitivity."
],
"decisive_result": "A biological hat process is supported if calibrated posterior probability exceeds 0.9 in a majority of identifiable clades and outpredicts sampling-only models; failure to recover simulated hats first invalidates the test.",
"failure_signature": "Simulation-based calibration confuses hats with sampling heterogeneity in more than 20% of cases, so observed symmetry cannot identify the process.",
"tools": [
"pbdb",
"bayesian-computation",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Reliable monophyly annotations and computationally intensive state-space inference.",
"nanopub": null
},
{
"id": "clade-trajectory-universality--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "SQS hat-shape test of marine orders and families",
"kind": "data-test",
"method": "Pre-register on the Paleobiology Database occurrence API. Taxa: marine invertebrate orders and families with at least 50 occurrences and at least 8 occupied international stages, accepted names only. Richness in each stage is shareholder-quorum subsampling at quorum 0.6 with 1000 trials, using Alroy's sqs. A clade is hat-shaped if the stage of peak subsampled richness lies in the middle third of its occupied duration and richness at both endpoint stages is below half the peak. The null for each clade is a constant-rate birth-death trajectory conditioned on the same duration and the same number of occurrences, passed through the same quorum filter, 200 replicates. The clade p-value is the fraction of null trajectories with symmetry at least as strong. The summary is the fraction of clades that are hat-shaped. Universality, pre-registered, requires that fraction above 0.5 and a bootstrap interval for the excess over the null mean that excludes 0. Repeat with the two most recent Cenozoic stages removed. If that sensitivity flips the call, the result is a pull-of-the-recent artifact.",
"steps": [
"Freeze quorum, trials, occurrence floor, hat definition and the 0.5 fraction before download.",
"Download PBDB occurrences for accepted marine invertebrate orders and families.",
"Compute SQS richness curves and the hat indicator.",
"Simulate the conditioned birth-death null through the same filter.",
"Bootstrap the excess over the null and repeat after dropping the last two Cenozoic stages."
],
"decisive_result": "Fraction of qualifying clades hat-shaped, with a bootstrap interval for its excess over the null. Supports a general hat only if the fraction exceeds 0.5, the excess interval excludes 0, and the call survives dropping the last two Cenozoic stages.",
"failure_signature": "The excess interval contains 0, or the call flips when recent stages are dropped. The pattern is then indistinguishable from the null or is a sampling-edge artifact.",
"tools": [
"pbdb",
"paleobiology-statistics",
"subsampling",
"data-test"
],
"datasets": [
{
"name": "Paleobiology Database data service 1.2",
"url": "https://paleobiodb.org/data1.2/",
"note": "Occurrence lists for marine invertebrate orders and families."
},
{
"name": "Alroy shareholder quorum subsampling",
"url": "http://bio.mq.edu.au/~jalroy/SQS.html",
"note": "sqs implementation, quorum 0.6, 1000 trials."
}
],
"references": [
{
"url": "https://doi.org/10.1666/06084.1",
"title": "Symmetric waxing and waning of marine invertebrate genera",
"year": 2007,
"note": "Foote 2007, Symmetric waxing and waning of marine invertebrate genera"
},
{
"url": null,
"note": "Liow and Stenseth 2007, The rise and fall of species"
},
{
"url": "https://doi.org/10.1126/science.1239431",
"title": "How the Red Queen Drives Terrestrial Mammals to Extinction",
"year": 2013,
"note": "Quental and Marshall 2013, How the Red Queen drives terrestrial mammals to extinction"
}
],
"difficulty": "hard",
"missing": "PBDB taxonomic status is not a monophyly proof. The test excludes unaccepted names and still cannot guarantee monophyletic clades.",
"nanopub": null
},
{
"id": "clade-trajectory-universality--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "A sampling-intensity null that increases toward the present",
"kind": "computation",
"method": "The birth-death null above uses constant sampling. This null uses the empirical PBDB collection counts per international stage as a time-varying sampling probability, normalized once from the global marine-invertebrate collection curve and then frozen. Simulate origination and extinction at constant rates, sample occurrences with that curve, and apply the same SQS hat statistic. The empirical hat fraction is an artifact of sampling intensity if it lies inside the central 95 percent of this null. The collection curve is a property of the database, computed once, not fit to the clade shapes. This can exonerate a hat that the constant-sampling null flags, or the reverse, which is why it is a separate approach.",
"steps": [
"Compute the global marine collection count per stage and freeze it.",
"Simulate constant-rate origination and extinction sampled with that curve.",
"Pass simulations through SQS and the hat statistic.",
"Place the empirical hat fraction in the null predictive interval.",
"Do not refit the collection curve to clade shape."
],
"decisive_result": "The empirical hat fraction lies inside or outside the central 95 percent of the sampling-intensity null. Inside means the hat fraction is accounted for by the collection curve.",
"failure_signature": "The empirical fraction lies outside that interval. Sampling intensity of this form then does not explain the pattern, and the approach has failed as an artifact explanation.",
"tools": [
"pbdb",
"paleobiology-statistics"
],
"datasets": [
{
"name": "Paleobiology Database data service 1.2",
"url": "https://paleobiodb.org/data1.2/",
"note": "Global collection counts per stage, and the empirical hat fraction from the paired test."
}
],
"references": [
{
"url": "https://doi.org/10.1666/06084.1",
"title": "Symmetric waxing and waning of marine invertebrate genera",
"year": 2007,
"note": "Foote 2007, the symmetry claim the sampling null tries to reproduce"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "clade-trajectory-universality--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Diversity-dependent extinction versus constant rates",
"kind": "computation",
"method": "If hats survive the sampling nulls, ask whether diversity-dependent extinction fits the declines better than constant rates. On each SQS curve that passed the occurrence floor, fit two birth-death models by Poisson likelihood of the stage-to-stage richness changes: constant origination and extinction, and constant origination with extinction linear in standing SQS richness. Compare by AIC. Diversity dependence is supported only if it beats the constant model by more than 10 in AIC for a fraction of clades whose bootstrap interval lies above 0.5. This can fail while the hat-shape test succeeds: symmetric-looking curves need not have diversity-dependent rates. It is the Red Queen piece of the question, not a second hat statistic.",
"steps": [
"Freeze the linear extinction model and the AIC gap.",
"Fit both likelihoods on each SQS curve.",
"Record the fraction of clades with AIC gap above 10.",
"Bootstrap that fraction.",
"Require the interval to lie above 0.5."
],
"decisive_result": "Bootstrap interval, entirely above 0.5, for the fraction of clades in which diversity-dependent extinction beats constant rates by AIC of more than 10.",
"failure_signature": "The interval is not entirely above 0.5. Diversity-dependent extinction then does not explain the declines in this model class, whether or not the curves look hat-shaped.",
"tools": [
"pbdb",
"paleobiology-statistics",
"model-comparison"
],
"datasets": [
{
"name": "Paleobiology Database data service 1.2",
"url": "https://paleobiodb.org/data1.2/",
"note": "Stage-to-stage SQS richness used as the likelihood data."
}
],
"references": [
{
"url": "https://doi.org/10.1126/science.1239431",
"title": "How the Red Queen Drives Terrestrial Mammals to Extinction",
"year": 2013,
"note": "Quental and Marshall 2013, diversity-dependent decline is the hypothesis"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
}
]
},
{
"id": "empty-shell-morphospace",
"title": "Why is shell morphospace empty where it is?",
"field": "both",
"tier": "B",
"question": "In Raup's parameter space of coiled shells (whorl expansion W, distance from axis D, translation T), is the unoccupied region explained by developmental constraint, functional selection (strength, hydrodynamics, stability) or phylogenetic inertia, and can these be told apart quantitatively?",
"known_state": "Raup (1966) showed realized shells occupy a small, structured part of the theoretical space. Fifty years of theoretical morphology (reviewed by Gerber 2017) have proposed functional and constructional explanations, but no accepted test discriminates them, partly because digitized shell parameter data were sparse. Thompson's shell-form claims (ch11) assume the logarithmic-spiral generative model.",
"acceptance_criteria": [
"A pre-registered test on digitized museum shell datasets (parameters measured by the morpholib MorphoEval pipeline), fitting explicit occupancy models: a developmental-constraint model (parameter correlations imposed by the generating curve), a functional model (strength and stability thresholds) and a phylogenetic model, compared by predictive likelihood on held-out taxa.",
"An empty-region map with the winning model's predicted boundaries."
],
"falsifiers": [
"If no model beats a phylogenetic random walk on held-out taxa, the question stays open with present data."
],
"required_capabilities": [
"theoretical-morphology",
"morpholib",
"3d-shape-analysis",
"data-test"
],
"data_access": "Digitized shell collections (MorphoSource, iDigBio, open), Raup parameters re-measured; taxon resolution via test-data/organisms.",
"scales": [
"organism",
"evolutionary"
],
"frontier_problem": null,
"related_claims": [
{
"id": "thompson1917-ch11-09",
"label": "All differences of form between shells are differences of degree in a few growth ratios",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "thompson1917-ch11-11",
"label": "Separation or overlap of whorls reflects the growth-rate ratio of outer to inner border",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "thompson1917-ch11-05",
"label": "The shell grows in size without changing its shape",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": "refutes",
"theorem": false
},
{
"id": "thompson1917-ch11-15",
"label": "Dissimilar turbinate shells share similar spiral angles and differ mainly in apical angle",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": "supports",
"theorem": false
}
],
"references": [
{
"url": "https://www.jstor.org/stable/1301405",
"title": null,
"year": null,
"note": "Raup 1966, Geometric analysis of shell coiling: general problems"
},
{
"url": "https://doi.org/10.1111/brv.12276",
"title": "The geometry of morphospaces: lessons from the classic\n R\n aup shell coiling model",
"year": 2016,
"note": "Gerber 2017, The geometry of morphospaces: lessons from the classic Raup shell coiling model"
}
],
"nanopub": null,
"approaches": [
{
"id": "empty-shell-morphospace--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Measure Raup parameters from 3D scans and compare occupancy models on held-out taxa",
"kind": "data-test",
"method": "Build the dataset the field lacks. From MorphoSource CT and surface scans of gastropod, ammonoid (where 3D fossil scans exist) and bivalve shells, and iDigBio 2D images in standard apertural view for gastropods, extract the coiling axis and the generating-curve centroid trajectory, and fit Raup's W (whorl expansion per revolution), D (relative distance of generating curve from axis) and T (translation) by nonlinear least squares on the spiral. Fit generating-curve shape with elliptic Fourier coefficients. Use the morpholib MorphoEval pipeline for segmentation and quality scoring, and admit a specimen only if fit RMSE < 3% of shell height. Target: ≥ 500 species across ≥ 50 families, taxa resolved via the organisms layer and MolluscaBase. Occupancy models over (log W, D, T): (M_dev) a developmental-constraint density proportional to the volume of parameter space producing geometrically valid, non-self-intersecting shells, with correlations imposed by the generating curve; (M_fun) density decreasing with distance below functional thresholds, from approach 2; (M_phy) phylogenetic Brownian or OU model on the molluscan tree with no boundaries; and combinations. Score by predictive log-likelihood on held-out families (leave-one-family-out).",
"steps": [
"Pre-register specimen admission criteria, models and the held-out scoring rule.",
"Query MorphoSource and iDigBio for shell media; build the fetch list with licences and checksums.",
"Run segmentation and spiral fitting through morpholib; quality-filter.",
"Fit the occupancy models; compute held-out log-likelihood; produce the empty-region map under the winning model."
],
"decisive_result": "A ranked comparison of developmental, functional and phylogenetic occupancy models by held-out predictive log-likelihood (a difference > 10 units counts as decisive), plus a map of the empty region with the winning model's predicted boundary.",
"failure_signature": "Fewer than about 200 species pass quality filters, or no model beats the phylogenetic random walk on held-out families, which is the problem's stated falsifier.",
"tools": [
"morpholib",
"3d-shape-analysis",
"organisms-layer",
"data-test"
],
"datasets": [
{
"name": "MorphoSource",
"url": "https://www.morphosource.org/catalog/media",
"note": "3D CT and surface scans of shells"
},
{
"name": "iDigBio specimen records API",
"url": "https://search.idigbio.org/v2/search/records?rq=%7B%22family%22%3A%22conidae%22%7D&limit=1",
"note": "Specimen images and metadata for gastropods"
},
{
"name": "MolluscaBase",
"url": "https://www.molluscabase.org/",
"note": "Taxonomic resolution"
}
],
"references": [
{
"url": "https://www.jstor.org/stable/1301405",
"title": null,
"year": null,
"note": "Raup 1966, Geometric analysis of shell coiling: general problems"
},
{
"url": "https://doi.org/10.1111/brv.12276",
"title": "The geometry of morphospaces: lessons from the classic\n R\n aup shell coiling model",
"year": 2016,
"note": "Gerber 2017, The geometry of morphospaces: lessons from the classic Raup shell coiling model"
},
{
"url": "https://github.com/fractastical/morpholib",
"title": null,
"year": null,
"note": "morpholib MorphoEval pipeline"
}
],
"difficulty": "hard",
"missing": "Scan coverage of molluscs on MorphoSource is uneven; ammonoids are mostly 2D. Labelled training data for spiral-fitting quality control would need curation.",
"nanopub": null
},
{
"id": "empty-shell-morphospace--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Compute functional landscapes over the whole Raup space by generative modelling and FE",
"kind": "simulation",
"method": "Make the functional hypothesis predictive before looking at occupancy. Generate shells on a dense grid over (log W ∈ [0, 6], D ∈ [0, 0.9], T ∈ [0, 6]) with a circular or elliptical generating curve. For each, compute: (a) internal volume per unit shell material (constructional economy); (b) for planispiral forms, hydrostatic stability, the distance between centres of buoyancy and mass with a stated tissue density; (c) crushing resistance by linear-elastic finite-element analysis of a compression load across the aperture, normalized by material volume; (d) a self-overlap indicator (whorls that would intersect). Define the functional hypothesis as the Pareto front of (a)–(c). Its predicted occupied region is the set of points within δ of the front, with δ fixed from a pilot of 20 species chosen at random and excluded from approach 1's test. The output is a predicted occupancy surface supplied to approach 1 as M_fun.",
"steps": [
"Implement the Raup generator with mesh output; validate against published Raup figures.",
"Compute volume and material metrics analytically or numerically on the grid.",
"Run FE compression on a coarser grid (about 10^4 shells) and interpolate.",
"Compute the Pareto front and the predicted occupancy surface; freeze it before approach 1's scoring."
],
"decisive_result": "A pre-computed functional occupancy surface whose boundary is fixed before occupancy data are examined, evaluated in approach 1 against developmental and phylogenetic surfaces. If it predicts the empty regions (for example sparse occupancy at high D and low W), functional selection is supported quantitatively.",
"failure_signature": "The Pareto front is so broad that nearly the whole space is within δ, so it predicts no empty region, or results are dominated by the arbitrary choice of load case.",
"tools": [
"simulation",
"finite-element-analysis",
"theoretical-morphology"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1111/brv.12276",
"title": "The geometry of morphospaces: lessons from the classic\n R\n aup shell coiling model",
"year": 2016,
"note": "Gerber 2017, review of functional and constructional explanations"
},
{
"url": "https://doi.org/10.1016/j.jtbi.2012.07.009",
"title": "Mechanical growth and morphogenesis of seashells",
"year": 2012,
"note": "Moulton, Goriely and Chirat 2012, Mechanical growth and morphogenesis of seashells"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "empty-shell-morphospace--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Lean-proved constructional boundaries of Raup space plus a growth-rule reachability map",
"kind": "mixed",
"method": "Separate what geometry forbids outright from what development or selection disfavours. Formalize in Lean the logarithmic-spiral shell: the generating curve's centroid at angle θ is at radius r(θ) = r₀ W^{θ/2π}, with the curve's inner edge at distance proportional to D·r. Prove: (a) successive whorls just touch when D = 1/W, Raup's boundary between overlapping and separated coils, generalized to elliptical generating curves; (b) a whorl penetrates the previous one exactly when D < 1/W under stated conditions, with the conditions on T for helicospiral forms; (c) for T = 0, an umbilicus exists iff D > 0. These carve out the constructionally impossible region (self-intersection), which must not count as an 'unexplained empty region'. Second, derive developmental reachability from local aperture growth rules (Moulton–Goriely style, with growth rate varying around the aperture): compute which (W, D, T) combinations are produced by growth-rate profiles in a bounded family, giving the developmental-constraint surface M_dev for approach 1.",
"steps": [
"Define the spiral shell in Lean with ℝ parameters; prove the touching and penetration conditions with explicit hypotheses.",
"Report the theorems against Thompson ch11 claims (logarithmic spiral generative model).",
"Implement an aperture-growth model; sample growth-rate profiles; record resulting (W, D, T).",
"Produce the reachability density as M_dev for approach 1."
],
"decisive_result": "Lean theorems giving the exact self-intersection boundary of Raup space (D = 1/W and its elliptical and helicospiral generalizations), and a growth-rule reachability density. Together they separate 'impossible', 'developmentally unlikely' and 'possible but unoccupied' regions, the three-way distinction the problem needs.",
"failure_signature": "The growth-rule family is so flexible that every geometrically valid point is reachable, leaving M_dev uninformative; or the geometric theorems hold only under circular generating curves, too restrictive for real gastropods.",
"tools": [
"lean4",
"mathlib",
"simulation",
"theoretical-morphology"
],
"datasets": [],
"references": [
{
"url": "https://www.jstor.org/stable/1301405",
"title": null,
"year": null,
"note": "Raup 1966, the W = 1/D boundary"
},
{
"url": "https://doi.org/10.1016/j.jtbi.2012.07.009",
"title": "Mechanical growth and morphogenesis of seashells",
"year": 2012,
"note": "Moulton, Goriely and Chirat 2012, growth-based shell morphogenesis"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "empty-shell-morphospace--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Held-out shell occupancy mechanism tournament",
"kind": "data-test",
"method": "Measure Raup W, D and T from at least 1000 openly digitized adult shells spanning 100 genera with MorphoEval. Fit three preregistered occupancy models: developmental covariance from mantle growth, finite-element failure and stability thresholds, and a phylogenetic diffusion model. Hold out entire families to distinguish interpolation from mechanism.",
"steps": [
"Blindly validate automated parameters against 100 manual measurements with ICC above 0.9.",
"Specify feasible regions for each mechanistic model before inspecting occupancy.",
"Fit models to 80% of families and predict voxel occupancy in held-out families.",
"Compare proper scoring rules and map false-positive empty regions."
],
"decisive_result": "A mechanism wins if its held-out log score exceeds phylogenetic diffusion by at least 20 and predicts at least 80% of occupied voxels while falsely allowing under 20% of robustly empty voxels.",
"failure_signature": "Parameter ICC below 0.9 or no model improves family-held-out prediction over phylogenetic diffusion.",
"tools": [
"morpholib",
"morphometrics",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "A sufficiently large consistently oriented set of open 3D shell meshes.",
"nanopub": null
},
{
"id": "empty-shell-morphospace--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Mechanics-derived forbidden-region simulation",
"kind": "simulation",
"method": "Generate shells across a dense W-D-T lattice and simulate growth, self-intersection, center-of-mass stability and pressure loading. Use interval geometry for self-intersection and validated finite-element convergence for failure. Compare the predicted infeasible boundary to observed morphospace without fitting boundary parameters to occupancy.",
"steps": [
"Evaluate at least one million W-D-T points with adaptive refinement near constraints.",
"Certify self-intersection and aperture occlusion geometrically.",
"Compute dimensionless stress and stability under preregistered material ranges.",
"Test whether observed taxa lie inside the feasible set and empty voxels outside it."
],
"decisive_result": "Functional constraint explains emptiness if at least 90% of observed shells are feasible and at least 70% of well-sampled empty voxels are infeasible across the full material range.",
"failure_signature": "Large connected empty regions remain mechanically feasible, or more than 10% of observed shells are classified impossible.",
"tools": [
"mechanobiology-simulation",
"large-scale-computation",
"morphometrics"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Species-specific shell material properties and loading regimes.",
"nanopub": null
},
{
"id": "empty-shell-morphospace--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Prove the planispiral non-intersection boundary",
"kind": "formalization",
"method": "Raup's parameters are whorl expansion W, relative distance D of the generating curve from the coiling axis, and translation T. For the planispiral case T = 0, define the shell as the Thompson logarithmic spiral of expansion W with a circular generating curve of Raup's D. Prove the necessary and sufficient algebraic inequality on W and D such that the generating circle at angle θ does not meet the circle at angle θ minus 2π. The theorem is that inequality, derived inside Lean from the definitions, not copied from a secondary source. State the open solid-of-revolution case T ≠ 0 as out of scope. This is the developmental-constraint boundary the occupancy test will use.",
"steps": [
"Define W, D and the planispiral logarithmic spiral.",
"Write the two successive generating circles.",
"Prove they are disjoint if and only if the derived inequality holds.",
"Leave T ≠ 0 unformalized and say so.",
"Audit axioms."
],
"decisive_result": "A no-sorry equivalence between non-intersection of successive planispiral whorls and an explicit inequality in W and D.",
"failure_signature": "A numerically non-intersecting (W, D) pair that violates the inequality, or the converse. The statement is then false.",
"tools": [
"lean4",
"theoretical-morphology"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1111/brv.12276",
"title": "The geometry of morphospaces: lessons from the classic\n R\n aup shell coiling model",
"year": 2016,
"note": "Gerber 2017, The geometry of morphospaces, used for the parameter definitions and the occupancy question"
},
{
"url": null,
"note": "Raup 1966, Geometric analysis of shell coiling: general problems. https://www.jstor.org/stable/1301405"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "empty-shell-morphospace--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Held-out occupancy models on museum meshes",
"kind": "data-test",
"method": "Measure W, D and T with the morpholib MorphoEval pipeline on open gastropod and cephalopod meshes from MorphoSource, supplemented by iDigBio specimen records for provenance. A taxon enters only if it resolves to a GBIF key or an Open Tree taxon. Pre-register a minimum of 30 species; below that the verdict is underpowered. Three predictive models, scored by log predictive density on families held out one at a time. Developmental constraint: the planispiral inequality from the Lean theorem, with a Gaussian likelihood inside the allowed region and a fixed penalty outside. Functional: from each mesh, the center of mass must project inside the aperture-down contact footprint, and a thickness proxy from the mesh, if the mesh has a wall, supplies a strength score; meshes without a wall use only the stability score. Phylogenetic: a phylogenetic Gaussian process on the Open Tree subtree. The developmental or functional model wins only if its expected log predictive density beats the phylogenetic model by an interval that excludes 0. If neither does, the question stays open, which is the stated falsifier.",
"steps": [
"Freeze the 30-species floor and the scoring rule before measuring meshes.",
"Measure W, D, T on MorphoSource shells with MorphoEval and keep provenance from iDigBio.",
"Fit the geometric, functional and phylogenetic models.",
"Score them by family-held-out log predictive density.",
"Declare a winner only if it beats the phylogenetic model with an interval excluding 0."
],
"decisive_result": "A held-out log-predictive comparison. A non-phylogenetic model wins only if its advantage over the phylogenetic Gaussian process has an interval excluding 0, on at least 30 species.",
"failure_signature": "Neither geometric nor functional model beats the phylogenetic model, or fewer than 30 species have measurable meshes. The question then stays open on present public scans.",
"tools": [
"morpholib",
"3d-shape-analysis",
"data-test"
],
"datasets": [
{
"name": "MorphoSource",
"url": "https://www.morphosource.org/",
"note": "Open 3D meshes of gastropod and cephalopod shells."
},
{
"name": "iDigBio",
"url": "https://www.idigbio.org/",
"note": "Specimen provenance for the meshes."
},
{
"name": "Open Tree of Life",
"url": null,
"note": "Induced subtree for the phylogenetic Gaussian process."
}
],
"references": [
{
"url": "https://doi.org/10.1111/brv.12276",
"title": "The geometry of morphospaces: lessons from the classic\n R\n aup shell coiling model",
"year": 2016,
"note": "Gerber 2017, the occupancy question and Raup-space geometry"
},
{
"url": null,
"note": "Raup 1966, Geometric analysis of shell coiling: general problems, parameter definitions. https://www.jstor.org/stable/1301405"
}
],
"difficulty": "hard",
"missing": "Most shells are not scanned. The test is built to return underpowered if the open meshes do not reach 30 species.",
"nanopub": null
},
{
"id": "empty-shell-morphospace--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Grid the generating curve and compare empty cells",
"kind": "simulation",
"method": "Fill a rectangular grid in (W, D, T) at the resolution of Raup's published figures, generate the logarithmic-spiral solid, and mark cells whose successive whorls intersect. Overlay the empirical (W, D, T) points from the mesh measurements. Developmental constraint explains an empty region only where the empty cells are exactly the intersecting cells, up to a pre-registered boundary strip of one grid cell. Empirical points inside the intersecting region falsify the generating model. Empty cells that do not intersect remain unexplained by this geometric filter and are passed to the functional model rather than called a success. This approach never sees phylogeny, so it cannot be rescued by a phylogenetic random walk.",
"steps": [
"Choose the grid from Raup's figure limits and freeze it.",
"Mark intersecting cells by the same circle test used in the Lean theorem, extended numerically to T ≠ 0.",
"Overlay measured shells.",
"Count empirical points in intersecting cells and empty non-intersecting cells.",
"Accept developmental constraint only for regions where those two sets match within one grid cell."
],
"decisive_result": "A cell-wise map. Developmental constraint accounts for a named empty region only if that region matches the intersecting cells within a one-cell strip and no measured shell falls in an intersecting cell.",
"failure_signature": "A measured shell lies in an intersecting cell, or a large empty region is non-intersecting. Geometry then does not explain that emptiness.",
"tools": [
"theoretical-morphology",
"simulation"
],
"datasets": [
{
"name": "MorphoSource",
"url": "https://www.morphosource.org/",
"note": "Empirical shells overlaid on the grid."
}
],
"references": [
{
"url": null,
"note": "Raup 1966, Geometric analysis of shell coiling: general problems, the parameter box being re-gridded. https://www.jstor.org/stable/1301405"
},
{
"url": "https://doi.org/10.1111/brv.12276",
"title": "The geometry of morphospaces: lessons from the classic\n R\n aup shell coiling model",
"year": 2016,
"note": "Gerber 2017, review of proposed explanations for the empty region"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
}
]
},
{
"id": "somite-number-control",
"title": "What sets vertebrate somite number?",
"field": "biology",
"tier": "B",
"question": "Across vertebrates, is total somite (and hence vertebral) number set by the ratio of the duration of axis elongation to the segmentation-clock period, as clock-and-wavefront predicts, and if so what evolutionary changes in clock period and elongation explain the range from about 30 to over 300?",
"known_state": "Cooke and Zeeman (1976) proposed clock-and-wavefront; Gomez et al. (2008) showed in zebrafish, chick, mouse and corn snake that snakes make many more somites by slowing axis elongation relative to the clock; Oates, Morelli and Soroldoni (2012) review the clock. Clock period and elongation rate are measured in only a handful of species, so the comparative prediction has not been tested broadly. Already a Frontier Problem.",
"acceptance_criteria": [
"A pre-registered comparative test: for every species with a measured clock period and elongation duration, compare predicted and observed vertebral counts (from museum and literature counts via the organisms layer); state in advance the tolerance for 'supports'.",
"A ranked list of species whose clock measurement would most reduce uncertainty about the mechanism."
],
"falsifiers": [
"Measured clock period and elongation duration whose ratio mispredicts somite number by more than the stated tolerance in several species."
],
"required_capabilities": [
"developmental-biology",
"comparative-data",
"data-test"
],
"data_access": "Vertebral counts in literature and museum databases (open); clock periods from published time-lapse studies.",
"scales": [
"developmental",
"evolutionary"
],
"frontier_problem": "Somite number control",
"related_claims": [
{
"id": "thompson1917-ch17-04",
"label": "Related organisms differ by simple, symmetrical Cartesian transformations",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1016/S0022-5193(76)80131-2",
"title": "A clock and wavefront model for control of the number of repeated structures during animal morphogenesis",
"year": 1976,
"note": "Cooke and Zeeman 1976, A clock and wavefront model for control of the number of repeated structures during animal morphogenesis"
},
{
"url": "https://doi.org/10.1038/nature07020",
"title": "Control of segment number in vertebrate embryos",
"year": 2008,
"note": "Gomez et al. 2008, Control of segment number in vertebrate embryos"
},
{
"url": "https://doi.org/10.1242/dev.063735",
"title": "Patterning embryos with oscillations: structure, function and dynamics of the vertebrate segmentation clock",
"year": 2012,
"note": "Oates, Morelli and Soroldoni 2012, Patterning embryos with oscillations"
}
],
"nanopub": null,
"approaches": [
{
"id": "somite-number-control--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Pre-registered test: clock period and PSM dynamics versus vertebral counts",
"kind": "data-test",
"method": "Avoid the tautology. 'One somite per clock cycle' makes N = (duration of somitogenesis)/(period) trivially true, so the test must predict N from quantities measured independently of the count: the clock period τ, the rate at which the presomitic mesoderm (PSM) is replenished by the tailbud, and the PSM length at onset, which together determine when the PSM is exhausted (Gomez et al. 2008). Compile from time-lapse literature every species with τ and PSM dynamics measured: zebrafish, medaka, chick, mouse, corn snake, and human and other mammals from in vitro PSM systems (Matsuda et al. 2020), at stated temperatures. Predict total somite number N̂ from the Gomez-type balance model with parameters measured, not fitted. Compare with observed somite or vertebral counts from literature and museum data via the organisms layer (FishBase for fish vertebrae; Head and Polly 2015 for squamates). Pre-registered tolerance: supports if |N̂ − N|/N ≤ 15% in every species with complete data; refutes if > 30% in two or more.",
"steps": [
"Write result.json with the prediction formula, tolerance and admissible measurement types.",
"Compile τ, PSM length and elongation rates per species with temperature; record sources.",
"Compute N̂ and compare with observed counts.",
"Compute, for unmeasured species, the expected reduction in posterior uncertainty about the mechanism from measuring τ (value-of-information ranking), using the variance of predicted N across plausible τ."
],
"decisive_result": "Per-species predicted versus observed somite numbers within or outside the 15% tolerance for all species with independent measurements (n ≈ 5–8), plus a ranked list of species whose clock measurement would most reduce uncertainty: long-bodied fishes, caecilians, and snakes other than corn snake.",
"failure_signature": "Too few species have PSM dynamics measured independently of somite counts (n < 4), so the test is underpowered; or published measurements were taken at temperatures incompatible with the counts.",
"tools": [
"comparative-data",
"organisms-layer",
"data-test"
],
"datasets": [
{
"name": "FishBase",
"url": "https://www.fishbase.se/",
"note": "Vertebral counts for fishes"
},
{
"name": "Head and Polly 2015 vertebral data (supplementary to the Nature article)",
"url": "https://doi.org/10.1038/nature14042",
"note": "Squamate and amniote vertebral counts"
}
],
"references": [
{
"url": "https://doi.org/10.1016/S0022-5193(76)80131-2",
"title": "A clock and wavefront model for control of the number of repeated structures during animal morphogenesis",
"year": 1976,
"note": "Cooke and Zeeman 1976, clock and wavefront"
},
{
"url": "https://doi.org/10.1038/nature07020",
"title": "Control of segment number in vertebrate embryos",
"year": 2008,
"note": "Gomez et al. 2008, Control of segment number in vertebrate embryos"
},
{
"url": "https://doi.org/10.1126/science.aba7668",
"title": "Species-specific segmentation clock periods are due to differential biochemical reaction speeds",
"year": 2020,
"note": "Matsuda et al. 2020, Species-specific segmentation clock periods"
},
{
"url": "https://doi.org/10.1242/dev.063735",
"title": "Patterning embryos with oscillations: structure, function and dynamics of the vertebrate segmentation clock",
"year": 2012,
"note": "Oates, Morelli and Soroldoni 2012, review of the segmentation clock"
},
{
"url": "https://doi.org/10.1038/nature14042",
"title": "Evolution of the snake body form reveals homoplasy in amniote Hox gene function",
"year": 2015,
"note": "Head and Polly 2015, snake body form and vertebral counts"
}
],
"difficulty": "hard",
"missing": "Clock and PSM measurements exist for only a handful of species; the decisive species need new time-lapse experiments the swarm cannot perform.",
"nanopub": null
},
{
"id": "somite-number-control--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Fit a PSM balance model across species and predict held-out species",
"kind": "mixed",
"method": "Formalize the clock-and-wavefront count as a dynamical model and test it out of sample. Model: PSM length L(t) obeys dL/dt = v_g(t) − v_s(t), where v_g is tailbud addition and v_s = S(t)/τ(t) is segmentation (segment length S over period τ), with S set by the wavefront speed, S = v_w τ. Somitogenesis ends at t* when L(t*) = L_min. N = ∫₀^{t*} dt/τ(t). Prove in Lean the qualitative theorems: (a) with constant τ and v_g decaying exponentially at rate k, N is an explicit closed-form function of τ k and L₀/S₀; (b) N increases when τ decreases at fixed elongation and v_w (the snake mechanism), and the elasticity of N to τ is exactly −1 only when elongation is independent of the clock. Then fit the free parameters (k and L_min) on four well-measured species and predict N for a held-out fifth (leave-one-out), comparing elasticity predictions with snake–lizard contrasts.",
"steps": [
"Prove the closed form and elasticity theorems in Lean (ODE solution by explicit integration, monotonicity).",
"Extract L(t), S(t) and τ(t) from published time courses (zebrafish, chick, mouse, corn snake).",
"Fit k and L_min; predict the held-out species; report leave-one-out errors.",
"State which comparative contrast (for example snake versus lizard) would most sharply test the elasticity prediction."
],
"decisive_result": "Leave-one-out prediction errors for N across the measured species (decisive if all within 15%), and a Lean-proved elasticity statement making precise how much of the snake's high count is due to τ versus elongation.",
"failure_signature": "Leave-one-out errors exceed 30% for any species, meaning species differ in mechanisms the model lacks (for example termination by Hox or Gdf11 timing rather than PSM depletion).",
"tools": [
"lean4",
"ode-simulation",
"comparative-data"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nature07020",
"title": "Control of segment number in vertebrate embryos",
"year": 2008,
"note": "Gomez et al. 2008, PSM dynamics in four species"
},
{
"url": "https://doi.org/10.1038/nature11804",
"title": "Scaling of embryonic patterning based on phase-gradient encoding",
"year": 2012,
"note": "Lauschke et al. 2013 (online 2012), Scaling of embryonic patterning based on phase-gradient encoding"
}
],
"difficulty": "hard",
"missing": "Digitized PSM time courses exist only as published figures; extraction is laborious and introduces measurement error.",
"nanopub": null
},
{
"id": "somite-number-control--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Temperature test: do clock and elongation Q10s predict meristic plasticity?",
"kind": "data-test",
"method": "Use temperature as a natural perturbation the swarm can exploit from existing data. In ectotherms, temperature changes both clock period and elongation rate. Clock-and-wavefront predicts that total somite number varies with rearing temperature in proportion to the difference of their thermal sensitivities: d ln N / dT ≈ (E_τ − E_elong)/(kT²) in Arrhenius form. If both scale identically, N is temperature-invariant, as reported for zebrafish somitogenesis by Schröter et al. 2008. Compile (a) clock-period and elongation-rate temperature series for zebrafish and any other species with them, and (b) laboratory rearing-temperature experiments reporting vertebral counts in fishes (meristic plasticity literature), plus (c) the latitudinal vertebral-count gradient (Jordan's rule) from FishBase across species and populations. Pre-registered predictions: the measured E difference predicts the sign and magnitude of rearing-temperature effects within a factor of 2, and species whose measured E difference is near zero show no latitudinal gradient.",
"steps": [
"Pre-register the Arrhenius prediction and tolerance.",
"Extract temperature series for τ and elongation from published studies; estimate E values with CIs.",
"Compile rearing-temperature vertebral-count experiments; compute d ln N/dT per study.",
"Compare predicted and observed slopes; run a phylogenetically controlled test of Jordan's rule against predicted sensitivity where data allow."
],
"decisive_result": "Agreement or disagreement, within a factor of 2, between d ln N/dT predicted from measured clock and elongation thermal sensitivities and the observed rearing-temperature effect on vertebral counts. A sign mismatch in two or more species refutes the clock-and-wavefront account of meristic plasticity.",
"failure_signature": "No species has both a clock-temperature series and a rearing-temperature vertebral-count experiment, so predictions and observations come from different species and the test is indirect.",
"tools": [
"comparative-data",
"data-test",
"organisms-layer"
],
"datasets": [
{
"name": "FishBase",
"url": "https://www.fishbase.se/",
"note": "Vertebral counts and distribution latitudes"
}
],
"references": [
{
"url": "https://doi.org/10.1002/dvdy.21458",
"title": "Dynamics of zebrafish somitogenesis",
"year": 2008,
"note": "Schröter et al. 2008, Dynamics of zebrafish somitogenesis (temperature dependence)"
},
{
"url": "https://doi.org/10.1038/nature07020",
"title": "Control of segment number in vertebrate embryos",
"year": 2008,
"note": "Gomez et al. 2008"
}
],
"difficulty": "hard",
"missing": "Matched clock and meristic data for the same species are rare; new temperature-series time-lapse experiments would make the test decisive.",
"nanopub": null
},
{
"id": "somite-number-control--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Cross-species clock-ratio prediction test",
"kind": "data-test",
"method": "Pre-register a comparative dataset of every vertebrate species with independently measured segmentation-clock period, somitogenesis duration and final somite count. Predict count as duration divided by period with no fitted slope, propagate measurement error, and use phylogenetic residual analysis. Require at least 15 species spanning fish, amphibians, reptiles, birds and mammals.",
"steps": [
"Define admissible live-imaging measurements and exclude inferred clock periods.",
"Blindly extract clock, duration and count in duplicate.",
"Compute posterior predictive intervals from measurement uncertainties only.",
"Test absolute percentage error and clade-structured bias."
],
"decisive_result": "Clock-and-wavefront count control is supported if at least 80% of species fall within 15% of the parameter-free prediction and phylogenetic residual SD is below 0.1 log units.",
"failure_signature": "Three or more independently measured species miss by over 30% in the same direction, or fewer than 10 species have all three direct measurements.",
"tools": [
"data-test",
"phylogenetic-comparative-methods",
"organisms-layer"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Direct elongation-duration and clock-period measurements exist for too few species.",
"nanopub": null
},
{
"id": "somite-number-control--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Clock-wavefront perturbation identifiability map",
"kind": "simulation",
"method": "Build a stochastic delayed-oscillator plus moving-wavefront model and perform global sensitivity analysis over clock period, axis elongation, coupling delay and arrest-front speed. Reproduce published perturbation time courses, then identify perturbations whose somite-count outcomes discriminate period control from elongation control.",
"steps": [
"Implement a spatial oscillator chain with explicit observation model.",
"Calibrate separately to each species using only time-course phase and axis length.",
"Run Sobol sensitivity and profile-likelihood identifiability analyses.",
"Rank feasible temperature, FGF and Notch perturbations by expected information gain."
],
"decisive_result": "A perturbation is decisive if competing mechanisms predict nonoverlapping 95% intervals separated by at least three somites; publish the top five preregisterable experiments.",
"failure_signature": "All feasible perturbations have overlapping predictions because period and elongation parameters remain structurally nonidentifiable.",
"tools": [
"developmental-simulation",
"large-scale-computation",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": "Executing the discriminating perturbations requires laboratory access.",
"nanopub": null
},
{
"id": "somite-number-control--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Clock period over elongation time versus vertebral counts",
"kind": "data-test",
"method": "Pre-register the clock-and-wavefront prediction N = T_elongation / T_clock. A species enters only when a published time-lapse study reports both the segmentation-clock period and the duration of axis elongation, and an independent count of vertebrae or somites exists. Gomez et al. 2008 is the seed list, not a cap: later papers enter under the same measurement rule. The tolerance is |log(N_predicted / N_observed)| < log(1.25), that is, within 25 percent. The claim is supported only if every qualifying species meets the tolerance. If fewer than 6 species qualify, the verdict is inconclusive, underpowered, and is not converted into support by relaxing the tolerance. Vertebral counts and clock measurements must come from different publications so the count is not the number the clock paper already compared.",
"steps": [
"Freeze the 25 percent tolerance and the 6-species floor.",
"Admit a species only with both clock measurements and an independent vertebral count.",
"Compute N_predicted = T_elongation / T_clock.",
"Apply the log-ratio tolerance to each species.",
"Support the claim only if every qualifying species passes and at least 6 qualify."
],
"decisive_result": "Every qualifying species within 25 percent, with at least 6 species. Otherwise the comparative prediction is not supported at the pre-registered bar.",
"failure_signature": "Two or more qualifying species miss the 25 percent tolerance. A single miss with at least 6 species is a mixed result, not support. Fewer than 6 species is underpowered.",
"tools": [
"developmental-biology",
"comparative-data",
"data-test"
],
"datasets": [
{
"name": "FishBase",
"url": "https://www.fishbase.se/",
"note": "Vertebral counts for fishes, used only when a clock measurement also exists and the count's source is not the clock paper."
}
],
"references": [
{
"url": "https://doi.org/10.1016/S0022-5193(76)80131-2",
"title": "A clock and wavefront model for control of the number of repeated structures during animal morphogenesis",
"year": 1976,
"note": "Cooke and Zeeman 1976, the clock-and-wavefront model"
},
{
"url": "https://doi.org/10.1038/nature07020",
"title": "Control of segment number in vertebrate embryos",
"year": 2008,
"note": "Gomez et al. 2008, Control of segment number in vertebrate embryos"
},
{
"url": "https://doi.org/10.1242/dev.063735",
"title": "Patterning embryos with oscillations: structure, function and dynamics of the vertebrate segmentation clock",
"year": 2012,
"note": "Oates, Morelli and Soroldoni 2012, Patterning embryos with oscillations"
}
],
"difficulty": "tractable",
"missing": "Clock period and elongation duration are published for only a handful of species. The test is expected to hit the underpowered floor, and that is an acceptable outcome.",
"nanopub": null
},
{
"id": "somite-number-control--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Rank unmeasured species by expected variance reduction",
"kind": "computation",
"method": "The second acceptance criterion is a ranked list of species whose clock measurement would most reduce uncertainty. Use FishBase vertebral counts as the response and the Open Tree vertebrate subtree as the covariance. The model is N = α T / τ with log α a phylogenetic Gaussian process. Species that already have a clock measurement anchor the process. For every FishBase species without a clock measurement, compute the expected reduction in posterior variance of α from observing its clock period, using the current phylogenetic covariance. Rank by that reduction. Extremes that are phylogenetically redundant with zebrafish, chick, mouse or snake should rank low. The list is the result; it does not test the mechanism.",
"steps": [
"Assemble FishBase vertebral counts and the Open Tree distances.",
"Anchor the phylogenetic regression at species with published clock data.",
"Compute expected variance reduction for each unmeasured species.",
"Publish the ranked list with the variance reductions.",
"Note species that cannot be ranked because they are absent from the tree."
],
"decisive_result": "A ranked list of unmeasured species with the expected reduction in variance of the clock-and-wavefront coefficient.",
"failure_signature": "The ranking is unchanged if phylogenetic covariance is replaced by a star phylogeny, in which case the list is only a ranking by vertebral-count extremity and does not use the tree. Report that degeneration rather than hiding it.",
"tools": [
"comparative-data"
],
"datasets": [
{
"name": "FishBase",
"url": "https://www.fishbase.se/",
"note": "Vertebral counts used as the response for the value-of-information ranking."
},
{
"name": "Open Tree of Life",
"url": null,
"note": "Vertebrate subtree supplying phylogenetic covariance."
}
],
"references": [
{
"url": "https://doi.org/10.1038/nature07020",
"title": "Control of segment number in vertebrate embryos",
"year": 2008,
"note": "Gomez et al. 2008, the measured species that anchor the ranking"
}
],
"difficulty": "tractable",
"missing": "FishBase does not cover tetrapods. Tetrapod counts enter only from literature sources named in the result, and species without a count are omitted.",
"nanopub": null
},
{
"id": "somite-number-control--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Hold out mouse, chick and snake from a zebrafish-fit clock",
"kind": "simulation",
"method": "Fit a delayed-coupling phase-oscillator model of the segmentation clock to zebrafish only: coupling delay, intrinsic period, and elongation speed set so the simulated somite count matches the zebrafish count within the same 25 percent tolerance. Freeze those parameters. Predict chick, mouse and corn snake by changing only the two quantities Gomez et al. identify as the evolutionary variables, clock period and elongation speed, to the values published for those species. The model fails if any of those three predictions misses the 25 percent tolerance. No parameter may be touched after the zebrafish fit. This fails differently from the comparative ratio test: the ratio test can pass while this oscillator, with its frozen coupling, misses the other species.",
"steps": [
"Specify the delayed-coupling oscillator and fit it to zebrafish only.",
"Freeze coupling and intrinsic parameters.",
"Insert published clock period and elongation speed for chick, mouse and corn snake.",
"Simulate somite number.",
"Apply the 25 percent tolerance to each held-out species."
],
"decisive_result": "Chick, mouse and corn snake somite counts each within 25 percent of the zebrafish-calibrated oscillator, with no refitting.",
"failure_signature": "Any one of the three held-out species misses the tolerance. The oscillator then does not carry the comparative prediction, even if the bare ratio N = T/τ does.",
"tools": [
"simulation",
"developmental-biology"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nature07020",
"title": "Control of segment number in vertebrate embryos",
"year": 2008,
"note": "Gomez et al. 2008, the four species and the two evolutionary variables"
},
{
"url": "https://doi.org/10.1242/dev.063735",
"title": "Patterning embryos with oscillations: structure, function and dynamics of the vertebrate segmentation clock",
"year": 2012,
"note": "Oates, Morelli and Soroldoni 2012, the oscillator phenomenology being implemented"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
}
]
},
{
"id": "murrays-law-vascular-optimality",
"title": "Does real vasculature obey Murray's law?",
"field": "both",
"tier": "B",
"question": "At vascular branch points, does the cube of the parent radius equal the sum of the cubes of the daughter radii (Murray's minimum-work law), does the exponent shift toward 2 in high-shear or turbulent beds, and does plant xylem obey the same law?",
"known_state": "Murray (1926) derived the cubic law from minimizing pumping work plus blood-volume maintenance; Sherman (1981) found good agreement in arterial trees; McCulloh, Sperry and Adler (2003) found plant xylem conduits obey Murray's law where conduits are not load-bearing. Modern retinal and whole-organ imaging datasets allow exponent estimation at thousands of branch points, but no standardized cross-tissue, cross-kingdom test exists. Thompson's claim (ch15-13) is the minimum-effort statement.",
"acceptance_criteria": [
"A pre-registered test estimating the branching exponent across at least three open vascular datasets (retinal fundus segmentations, micro-CT of organ vasculature, xylem anatomy), with the hypotheses 'exponent 3', 'exponent 7/3 (turbulent)' and 'no single exponent' stated in advance."
],
"falsifiers": [
"Exponent intervals excluding 3 across tissues would refute the general law; a shear-dependent shift would support the amended form."
],
"required_capabilities": [
"image-segmentation",
"vascular-morphometry",
"data-test"
],
"data_access": "Open retinal vessel datasets (DRIVE, HRF), open micro-CT vascular casts, xylem data from McCulloh et al. 2003.",
"scales": [
"tissue",
"organ"
],
"frontier_problem": null,
"related_claims": [
{
"id": "thompson1917-ch15-13",
"label": "Blood-vessels are arranged for minimum effort and minimum wall-surface",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": null,
"theorem": true
},
{
"id": "thompson1917-ch16-03",
"label": "Structures that must resist bending are built as hollow tubes",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": null,
"theorem": true
}
],
"references": [
{
"url": "https://doi.org/10.1073/pnas.12.3.207",
"title": "The Physiological Principle of Minimum Work",
"year": 1926,
"note": "Murray 1926, The physiological principle of minimum work I"
},
{
"url": "https://doi.org/10.1085/jgp.78.4.431",
"title": "On connecting large vessels to small. The meaning of Murray's law.",
"year": 1981,
"note": "Sherman 1981, On connecting large vessels to small: the meaning of Murray's law"
},
{
"url": "https://doi.org/10.1038/nature01444",
"title": "Water transport in plants obeys Murray's law",
"year": 2003,
"note": "McCulloh, Sperry and Adler 2003, Water transport in plants obeys Murray's law"
}
],
"nanopub": null,
"approaches": [
{
"id": "murrays-law-vascular-optimality--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Pre-registered multi-tissue branching-exponent estimation with errors-in-variables",
"kind": "data-test",
"method": "Datasets: retinal fundus segmentations (DRIVE, HRF, manual vessel annotations), whole mouse brain vasculature (VesSAP), pig coronary morphometry (Kassab et al. 1993 tables) and xylem conduit data (McCulloh et al. 2003). At each bifurcation estimate radii from the segmentation distance transform averaged over a segment length of 5–10 radii away from the junction, and solve r₀^x = r₁^x + r₂^x for x by root-finding. Estimate per-tissue exponent distributions with a hierarchical errors-in-variables model: observed radii are true radii plus heteroscedastic noise with variance calibrated from approach 3. Hypotheses, stated in advance: H3, x = 3 (Murray); H7/3, x = 7/3 (turbulent, large arteries); H_none, no single x across tissues (between-tissue SD of x > 0.3). Decision: a tissue supports H3 if the 95% interval of its mean x contains 3 and excludes 7/3; it supports H7/3 conversely; H_none holds if tissue intervals are disjoint. Also test shear dependence: x versus vessel diameter class within retina and brain.",
"steps": [
"Write result.json with datasets, estimator, noise calibration source and decision rules.",
"Download DRIVE and HRF; obtain VesSAP graphs; transcribe Kassab and McCulloh tables; record checksums.",
"Extract bifurcations and radii; fit the hierarchical model with noise priors from approach 3.",
"Report per-tissue x intervals, the diameter-class trend and the verdict."
],
"decisive_result": "Per-tissue posterior intervals for the branching exponent across at least four tissue types, and a verdict among x = 3, x = 7/3, or no single exponent. Thousands of bifurcations per dataset give intervals of about ±0.1 once measurement error is modelled.",
"failure_signature": "Calibrated measurement noise in image-derived radii (2D retinal images especially) inflates x intervals beyond ±0.5, so 3 and 7/3 cannot be distinguished: inconclusive, measurement-limited.",
"tools": [
"image-segmentation",
"vascular-morphometry",
"bayesian-hierarchical-models",
"data-test"
],
"datasets": [
{
"name": "DRIVE retinal vessel segmentation dataset",
"url": "https://drive.grand-challenge.org/",
"note": "40 fundus images with manual vessel segmentations"
},
{
"name": "HRF high-resolution fundus database",
"url": "https://www5.cs.fau.de/research/data/fundus-images/",
"note": "High-resolution fundus images with vessel annotations"
},
{
"name": "VesSAP whole mouse brain vasculature",
"url": "https://discotechnologies.org/VesSAP/",
"note": "Vessel graphs with radii from cleared mouse brains"
}
],
"references": [
{
"url": "https://doi.org/10.1073/pnas.12.3.207",
"title": "The Physiological Principle of Minimum Work",
"year": 1926,
"note": "Murray 1926, The physiological principle of minimum work"
},
{
"url": "https://doi.org/10.1085/jgp.78.4.431",
"title": "On connecting large vessels to small. The meaning of Murray's law.",
"year": 1981,
"note": "Sherman 1981, On connecting large vessels to small"
},
{
"url": "https://doi.org/10.1038/nature01444",
"title": "Water transport in plants obeys Murray's law",
"year": 2003,
"note": "McCulloh, Sperry and Adler 2003, Water transport in plants obeys Murray's law"
},
{
"url": "https://doi.org/10.1038/s41592-020-0792-1",
"title": "Machine learning analysis of whole mouse brain vasculature",
"year": 2020,
"note": "Todorov et al. 2020, Machine learning analysis of whole mouse brain vasculature (VesSAP)"
},
{
"url": "https://doi.org/10.1152/ajpheart.1993.265.1.h350",
"title": "Morphometry of pig coronary arterial trees",
"year": 1993,
"note": "Kassab, Rider, Tang and Fung 1993, Morphometry of pig coronary arterial trees"
},
{
"url": "https://doi.org/10.1098/rsif.2011.0270",
"title": "Intraspecific scaling laws of vascular trees",
"year": 2011,
"note": "Huo and Kassab 2012 (online 2011), Intraspecific scaling laws of vascular trees"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "murrays-law-vascular-optimality--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Lean formalization of Murray's cubic law and its turbulent and load-bearing variants",
"kind": "formalization",
"method": "Formalize the optimization so that each empirical exponent has a precise theoretical antecedent. Cost per unit length of a vessel with flow Q and radius r: C(r) = 8μQ²/(πr⁴) + b π r², Poiseuille dissipation plus metabolic maintenance proportional to volume. Theorem 1: C has a unique minimizer on r > 0, with r³ = (16μ/(bπ²))^{1/2} · Q, so Q ∝ r³. Theorem 2: with flow conservation Q₀ = Q₁ + Q₂ at a bifurcation and each segment at its optimum, r₀³ = r₁³ + r₂³. Theorem 3 (turbulent): with dissipation ∝ Q^{7/4}r^{−19/4} (Blasius friction), optimal radius r ∝ Q^{3/7}, giving exponent 7/3 at bifurcations. Theorem 4 (xylem): if conduits also bear mechanical load with cost ∝ r² independent of flow, the optimum shifts away from the cubic law (explicit exponent), matching McCulloh et al.'s observation that load-bearing conduits deviate. Proofs use derivative sign analysis (Mathlib's deriv and monotonicity lemmas) and rpow algebra.",
"steps": [
"Define the cost functions with ℝ and positivity hypotheses.",
"Prove existence, uniqueness and the closed form of the minimizer (first-derivative test, strict convexity).",
"Derive bifurcation exponents for laminar and turbulent cases with flow conservation.",
"Report each theorem linked to thompson1917-ch15-13 with scope notes on what data must decide."
],
"decisive_result": "A Lean module proving the cubic law, the 7/3 turbulent law and the load-bearing deviation from explicit cost functions, compiled with only standard axioms. This makes approach 1's hypotheses theorem-backed.",
"failure_signature": "None mathematically. The risk is rpow manipulations in Mathlib taking longer than expected; low risk.",
"tools": [
"lean4",
"mathlib"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1073/pnas.12.3.207",
"title": "The Physiological Principle of Minimum Work",
"year": 1926,
"note": "Murray 1926"
},
{
"url": "https://doi.org/10.1038/nature01444",
"title": "Water transport in plants obeys Murray's law",
"year": 2003,
"note": "McCulloh, Sperry and Adler 2003"
},
{
"url": "https://doi.org/10.1073/pnas.1012194108",
"title": "Hydraulic trade-offs and space filling enable better predictions of vascular structure and function in plants",
"year": 2010,
"note": "Savage et al. 2010, Hydraulic trade-offs and space filling in plant vascular networks"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "murrays-law-vascular-optimality--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Synthetic-tree calibration: can image-derived radii distinguish exponent 3 from 7/3?",
"kind": "simulation",
"method": "Exponent estimates from images are biased: estimating x from r₀^x = r₁^x + r₂^x is nonlinear, and radius noise near junctions inflates or deflates x. Before trusting approach 1, generate synthetic vascular trees with known exponent x_true ∈ {2, 7/3, 2.7, 3}, using constrained constructive optimization or a simple binary tree with stated asymmetry and length ratios. Render them through realistic image formation for each modality: a fundus model (2D projection, Gaussian blur, vessel central reflex, noise at matched SNR); light-sheet volumes for VesSAP (anisotropic PSF, voxel size); micro-CT. Run the same segmentation and radius pipeline as approach 1. Output: the bias and variance of estimated x as a function of x_true, vessel size in pixels and modality. These become the noise priors in approach 1's model, and determine the minimum vessel diameter (in pixels) at which 3 and 7/3 are distinguishable.",
"steps": [
"Implement the synthetic tree generator and modality-specific forward image models.",
"Run approach 1's segmentation pipeline on 10^3 synthetic trees per setting.",
"Fit bias and variance curves for x as functions of x_true and diameter in pixels.",
"Publish calibration tables; set approach 1's minimum-diameter inclusion threshold from them."
],
"decisive_result": "Calibration curves giving, for each modality, the smallest vessel diameter at which estimated x separates 3 from 7/3 with 95% power. If no real dataset reaches that resolution, approach 1 is underpowered by design and the question needs better imaging.",
"failure_signature": "Forward models fail validation: synthetic images differ visibly from real ones in vessel-contrast statistics, so the calibration does not transfer.",
"tools": [
"simulation",
"image-segmentation",
"vascular-morphometry"
],
"datasets": [
{
"name": "DRIVE",
"url": "https://drive.grand-challenge.org/",
"note": "Real images to validate the fundus forward model"
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.cell.2020.01.028",
"title": "Mapping the Fine-Scale Organization and Plasticity of the Brain Vasculature",
"year": 2020,
"note": "Kirst et al. 2020, Mapping the fine-scale organization and plasticity of the brain vasculature (imaging resolution for the forward model)"
},
{
"url": "https://doi.org/10.1085/jgp.78.4.431",
"title": "On connecting large vessels to small. The meaning of Murray's law.",
"year": 1981,
"note": "Sherman 1981"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "murrays-law-vascular-optimality--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Errors-in-variables branching exponent test",
"kind": "data-test",
"method": "Analyze at least three independent vascular modalities: retinal images, segmented 3D organ vessels and plant xylem. At each bifurcation fit r_parent^p = r_1^p + r_2^p with hierarchical errors-in-variables likelihood, accounting for pixel resolution, taper and branch-order dependence. Hold out specimens, not branch points.",
"steps": [
"Pre-register segmentation quality and minimum 500 bifurcations per modality.",
"Estimate radius uncertainty from repeated segmentation and exclude unresolved vessels.",
"Fit shared p = 3, p = 7/3, p = 2 and varying-p models.",
"Compare specimen-held-out predictive density and residual shear dependence."
],
"decisive_result": "General Murray scaling requires p's 95% interval within 0.2 of 3 in every modality and no varying-p model ELPD gain above 10; exclusion of 3 in two modalities refutes generality.",
"failure_signature": "Radius uncertainty exceeds 20% at most branches or exponent estimates shift by more than 0.3 under preregistered taper corrections.",
"tools": [
"morpholib",
"image-segmentation",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Open, quality-controlled 3D vascular trees with calibrated voxel dimensions.",
"nanopub": null
},
{
"id": "murrays-law-vascular-optimality--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Formal derivation and robust optimality envelope",
"kind": "formalization",
"method": "Formalize Poiseuille pumping cost plus volume-maintenance cost for a bifurcating vessel in Lean. Prove strict convexity, derive the optimal radius-flow relation, and derive the cubic junction law under flow conservation. Then vary the drag exponent and maintenance cost to obtain a theorem mapping physical assumptions to branching exponent.",
"steps": [
"Define positive radii, flows and cost with explicit viscosity and length hypotheses.",
"Prove existence and uniqueness of the radius minimizer.",
"Derive the generalized exponent for resistance proportional to r^-k.",
"Construct counterexamples with load-bearing or nonlaminar cost terms."
],
"decisive_result": "A no-sorry Lean theorem proving p = 3 exactly under named assumptions and a generalized theorem quantifying how p changes when each assumption changes.",
"failure_signature": "The optimization requires an unacknowledged flow-radius premise equivalent to the cubic law, making the standard derivation circular.",
"tools": [
"lean4",
"calculus",
"fluid-mechanics"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "murrays-law-vascular-optimality--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Derive the junction exponent from a stated cost",
"kind": "formalization",
"method": "Minimize, for one vessel, a Poiseuille pumping cost proportional to Q squared over r to the fourth plus a blood-volume cost proportional to r squared. Prove the optimum is r cubed proportional to Q. At a junction with Q_parent = Q_1 + Q_2 and the same constant of proportionality, prove r_parent cubed equals r_1 cubed plus r_2 cubed. Then replace the resistance exponent by a parameter b and the flow exponent by a parameter a, and prove the junction exponent is (b + 2) / a for a volume cost proportional to r squared. State which (a, b) would yield 7/3, and do not assert that turbulent flow has those exponents unless the algebra produces them from a stated friction law. The laminar cubic law and the general exponent are the theorems.",
"steps": [
"Define Poiseuille power and volume cost.",
"Prove the cubic optimum and the junction law.",
"Generalize to resistance exponent b and flow exponent a.",
"Solve for the (a, b) pair that would give junction exponent 7/3.",
"Audit axioms."
],
"decisive_result": "A no-sorry cubic junction law under Poiseuille-plus-volume cost, and a general exponent (b + 2) / a with the unique (a, b) pair that equals 7/3.",
"failure_signature": "The cubic law does not follow from those costs, or more than one (a, b) pair on the stated family yields 7/3. The derivation would then not pin down the turbulent alternative.",
"tools": [
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1073/pnas.12.3.207",
"title": "The Physiological Principle of Minimum Work",
"year": 1926,
"note": "Murray 1926, The physiological principle of minimum work I"
},
{
"url": "https://doi.org/10.1085/jgp.78.4.431",
"title": "On connecting large vessels to small. The meaning of Murray's law.",
"year": 1981,
"note": "Sherman 1981, On connecting large vessels to small"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "murrays-law-vascular-optimality--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Estimate the branching exponent on retina and large vessels",
"kind": "data-test",
"method": "Pre-register three sources: the DRIVE gold-standard vessel segmentations (40 images), the HRF gold-standard segmentations (45 images: 15 healthy, 15 glaucoma, 15 diabetic retinopathy), and the cardiovascular models in the Vascular Model Repository. Extract branch points from the existing segmentations and from the repository centerlines. At each bifurcation estimate α by minimizing the squared difference between log(r_parent to the α) and log(r_1 to the α plus r_2 to the α). Pool α in a Gaussian model with a random intercept for image or model. Hypotheses, decided by AIC and by the 95 percent interval: a single α whose interval contains 3, a single α whose interval contains 7/3, or no single exponent. No single exponent wins if a model with α depending on source (retina versus large vessel) beats the single-α model by AIC of more than 10, or if the single-α interval has width greater than 1. A source with fewer than 100 branch points is dropped. The test proceeds only if at least two sources remain.",
"steps": [
"Freeze the estimator, the AIC gap and the 100-branch minimum.",
"Extract bifurcations from DRIVE, HRF and the Vascular Model Repository.",
"Estimate per-junction α and pool with a random intercept.",
"Compare α = 3, α = 7/3 and a source-dependent α by AIC and intervals.",
"Drop any source under 100 branch points and stop if fewer than two sources remain."
],
"decisive_result": "A 95 percent interval for α and an AIC comparison. Murray's cubic law is supported only if the interval contains 3 and excludes 7/3, and the source-dependent model does not win.",
"failure_signature": "The interval excludes 3 in both retina and large vessels, which refutes a general cubic law, or the interval contains both 3 and 7/3 and the source model does not win, which leaves the exponent unresolved.",
"tools": [
"image-segmentation",
"vascular-morphometry",
"data-test"
],
"datasets": [
{
"name": "DRIVE retinal vessel segmentation",
"url": "https://drive.grand-challenge.org/",
"note": "Forty fundus images with manual vessel masks; branch points taken from the masks."
},
{
"name": "High-Resolution Fundus image database",
"url": "https://www5.cs.fau.de/research/data/fundus-images/index.html",
"note": "Forty-five images with binary vessel gold standards."
},
{
"name": "Vascular Model Repository",
"url": "https://www.vascularmodel.com/",
"note": "Cardiovascular models whose centerline radii supply large-vessel bifurcations."
}
],
"references": [
{
"url": "https://doi.org/10.1073/pnas.12.3.207",
"title": "The Physiological Principle of Minimum Work",
"year": 1926,
"note": "Murray 1926, the cubic prediction"
},
{
"url": "https://doi.org/10.1085/jgp.78.4.431",
"title": "On connecting large vessels to small. The meaning of Murray's law.",
"year": 1981,
"note": "Sherman 1981, arterial agreement with the cubic law"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "murrays-law-vascular-optimality--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Xylem conduits and a radius-dependent exponent",
"kind": "data-test",
"method": "Two pre-registered extensions that can support an amended law even if the pooled animal exponent excludes 3. First, transcribe conduit radii at branch points from the tables of McCulloh, Sperry and Adler 2003, with a checksum of the transcription. Do not digitize figures. If the tables are not numerically usable, drop the xylem arm and say so. Estimate α by the same one-dimensional fit. Plant xylem agrees with Murray only if its interval contains 3. Second, on the retinal junctions only, regress per-junction α on log parent radius, with a random intercept for image. A shear-dependent amendment is supported if the slope's 95 percent interval excludes 0. These two tests are not pooled with the animal single-exponent test.",
"steps": [
"Transcribe McCulloh tabled radii or drop xylem if the tables are not numeric.",
"Estimate xylem α and its interval.",
"Regress retinal per-junction α on log parent radius.",
"Support a radius-dependent exponent only if the slope interval excludes 0.",
"Report xylem and the slope as separate calls."
],
"decisive_result": "A xylem interval that contains or excludes 3, and a retinal slope of α on log parent radius whose interval contains or excludes 0.",
"failure_signature": "Xylem excludes 3, which refutes a cross-kingdom cubic law, or the retinal slope interval contains 0, which fails the shear-dependent amendment. Unusable tables make xylem inconclusive rather than a refutation.",
"tools": [
"vascular-morphometry",
"data-test"
],
"datasets": [
{
"name": "DRIVE retinal vessel segmentation",
"url": "https://drive.grand-challenge.org/",
"note": "Retinal junctions for the radius regression."
},
{
"name": "High-Resolution Fundus image database",
"url": "https://www5.cs.fau.de/research/data/fundus-images/index.html",
"note": "Additional retinal junctions for the same regression."
}
],
"references": [
{
"url": "https://doi.org/10.1038/nature01444",
"title": "Water transport in plants obeys Murray's law",
"year": 2003,
"note": "McCulloh, Sperry and Adler 2003, Water transport in plants obeys Murray's law; tabled radii only"
}
],
"difficulty": "hard",
"missing": "A public xylem file of branch-point radii. If the 2003 tables cannot be transcribed without guessing, the xylem arm stops.",
"nanopub": null
}
]
},
{
"id": "morphogenesis-cognition-gene-overlap",
"title": "Do morphogenesis and learning share a gene set beyond pleiotropy?",
"field": "biology",
"tier": "C",
"question": "Do genes annotated to morphogenesis and genes annotated to learning and memory overlap more than expected from gene-set size, pleiotropy and shared housekeeping components, as Levin's TAME framework predicts?",
"known_state": "Levin (2022) predicts wide overlap because, on his account, morphogenesis uses the same problem-solving machinery as cognition. The repo's one test (levin-2022-tame-19) returned refutes at confidence 0.7 using GO term overlap, but did not control for pleiotropy, annotation bias or gene-set size. INDRA CoGEx allows a mechanistically grounded, permutation-controlled test.",
"acceptance_criteria": [
"A pre-registered test using INDRA CoGEx gene sets for morphogenesis and learning/memory (GO and HPO), a permutation null matched on gene-set size, pleiotropy degree and annotation count, and a stated effect-size threshold for 'wide overlap'.",
"The same test on ion-channel and gap-junction subsets, which Levin names specifically."
],
"falsifiers": [
"Overlap within the matched null refutes the prediction; overlap well above it (especially in the bioelectric subset) supports it and would revise the current verdict."
],
"required_capabilities": [
"indra-cogex",
"gene-set-statistics",
"data-test"
],
"data_access": "INDRA CoGEx MCP (public), Gene Ontology, HPO.",
"scales": [
"molecular",
"cellular",
"developmental",
"neural"
],
"frontier_problem": null,
"related_claims": [
{
"id": "levin-2022-tame-19",
"label": "Genes for morphogenesis and for learning and memory should overlap widely",
"thinker": "Michael Levin",
"status": "untested",
"evidence": "refutes",
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.3389/fnsys.2022.768201",
"title": "Technological Approach to Mind Everywhere: An Experimentally-Grounded Framework for Understanding Diverse Bodies and Minds",
"year": 2022,
"note": "Levin 2022, Technological approach to mind everywhere (TAME)"
}
],
"nanopub": null,
"approaches": [
{
"id": "morphogenesis-cognition-gene-overlap--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Matched permutation test of GO/HPO morphogenesis–learning overlap via INDRA CoGEx",
"kind": "data-test",
"method": "Gene sets: M = human genes annotated (experimental evidence codes only, to limit annotation-cascade bias) to GO 'anatomical structure morphogenesis' (GO:0009653) or descendants, or to HPO 'abnormality of body morphology'-type branches; L = genes annotated to GO 'learning or memory' (GO:0007611) or descendants, or to HPO memory and learning phenotypes. Retrieve via INDRA CoGEx (GO and HPO edges are in the graph). Statistic: observed |M ∩ L| over the expected value under a matched null. Null: 10^5 permutations drawing gene sets of the same sizes in which each gene is replaced by a random gene from the same stratum of (number of distinct GO biological-process annotations: pleiotropy, in deciles; publication count from CoGEx literature edges: annotation bias, in deciles; GTEx expression breadth τ: housekeeping, in tertiles). Pre-registered 'wide overlap': observed/expected ≥ 1.5 with permutation p < 0.001. Refuted: observed/expected ≤ 1.1 or p > 0.05. Repeat on the bioelectric subset (ion channels and gap junctions: HGNC groups for voltage-gated channels, K2P, connexins, innexins, pannexins).",
"steps": [
"Write result.json with GO and HPO identifiers, evidence-code filter, strata, threshold and the bioelectric subset definition.",
"Query INDRA CoGEx for annotations and publication counts; fetch GTEx breadth; record query hashes.",
"Run the stratified permutation test for all genes and for the bioelectric subset.",
"Compare with the repo's existing levin-2022-tame-19 verdict and record a revision if the verdict changes."
],
"decisive_result": "Observed/expected overlap with permutation p under a null matched for pleiotropy, annotation depth and expression breadth, for all genes and for ion-channel and gap-junction genes, with pre-registered thresholds for 'wide overlap'.",
"failure_signature": "Strata are so collinear (highly studied genes are also highly pleiotropic and broadly expressed) that the matched null has too few genes per stratum to permute, making p-values unstable.",
"tools": [
"indra-cogex",
"gene-set-statistics",
"data-test"
],
"datasets": [
{
"name": "INDRA CoGEx",
"url": "https://discovery.indra.bio/",
"note": "GO and HPO annotations and literature evidence counts"
},
{
"name": "Gene Ontology",
"url": "https://geneontology.org/",
"note": "Term definitions and evidence codes"
},
{
"name": "Human Phenotype Ontology",
"url": "https://hpo.jax.org/",
"note": "Phenotype annotations"
},
{
"name": "GTEx portal",
"url": "https://gtexportal.org/",
"note": "Tissue expression for breadth (tau) stratification"
}
],
"references": [
{
"url": "https://doi.org/10.3389/fnsys.2022.768201",
"title": "Technological Approach to Mind Everywhere: An Experimentally-Grounded Framework for Understanding Diverse Bodies and Minds",
"year": 2022,
"note": "Levin 2022, TAME"
},
{
"url": "https://doi.org/10.1371/journal.pone.0017258",
"title": "The Impact of Multifunctional Genes on \"Guilt by Association\" Analysis",
"year": 2011,
"note": "Gillis and Pavlidis 2011, The impact of multifunctional genes on guilt-by-association analysis"
},
{
"url": "https://doi.org/10.1038/s41598-018-19333-x",
"title": "Gene annotation bias impedes biomedical research",
"year": 2018,
"note": "Haynes, Tomczak and Khatri 2018, Gene annotation bias impedes biomedical research"
},
{
"url": "https://github.com/gyorilab/indra_cogex",
"title": null,
"year": null,
"note": "INDRA CoGEx source"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "morphogenesis-cognition-gene-overlap--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Annotation-independent test using mouse knockout phenotyping (IMPC/MGI)",
"kind": "data-test",
"method": "GO and HPO overlap inherits curation bias. The International Mouse Phenotyping Consortium phenotypes knockouts systematically through a fixed pipeline, so which genes are tested for which phenotypes does not depend on prior interest. For all IMPC knockout lines that completed both morphological (skeletal, craniofacial, organ morphology) and behavioural or neurological procedures, build a 2 × 2 table: abnormal morphology yes/no versus abnormal learning-related or behavioural phenotype yes/no. Fit logistic regression of the behavioural call on the morphology call with covariates: number of other phenotype systems hit (pleiotropy), viability class and sex. Supplement with MGI curated knockouts restricted to studies reporting both kinds. Pre-registered: an adjusted odds ratio ≥ 2 (95% CI excluding 1.3) supports a shared gene set beyond pleiotropy; OR CI including 1 refutes it. Run the ion-channel and gap-junction subset separately.",
"steps": [
"Pre-register phenotype term sets (MP ontology IDs), covariates and thresholds.",
"Download IMPC statistical results; filter to lines with both procedure classes; record version and checksums.",
"Fit adjusted logistic models for all genes and the bioelectric subset.",
"Compare with approach 1 to see whether annotation-based and screen-based answers agree."
],
"decisive_result": "An adjusted odds ratio, with CI, for co-occurrence of morphological and behavioural or learning phenotypes in systematically phenotyped knockouts. This is the strongest available test free of curation bias.",
"failure_signature": "IMPC behavioural procedures measure activity and anxiety rather than learning per se, so the 'learning' category cannot be operationalized; the test then reverts to a broader behavioural proxy, which must be stated as a reinterpretation.",
"tools": [
"gene-set-statistics",
"data-test",
"indra-cogex"
],
"datasets": [
{
"name": "International Mouse Phenotyping Consortium",
"url": "https://www.mousephenotype.org/",
"note": "Systematic knockout phenotype calls"
},
{
"name": "Mouse Genome Informatics",
"url": "https://www.informatics.jax.org/",
"note": "Curated knockout phenotypes (MP ontology)"
}
],
"references": [
{
"url": "https://doi.org/10.3389/fnsys.2022.768201",
"title": "Technological Approach to Mind Everywhere: An Experimentally-Grounded Framework for Understanding Diverse Bodies and Minds",
"year": 2022,
"note": "Levin 2022, TAME"
},
{
"url": "https://doi.org/10.1371/journal.pbio.2006643",
"title": "Large-scale investigation of the reasons why potentially important genes are ignored",
"year": 2018,
"note": "Stoeger, Gerlach, Morimoto and Nunes Amaral 2018, Large-scale investigation of the reasons why potentially important genes are ignored (study-bias context)"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "morphogenesis-cognition-gene-overlap--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Mechanistic proximity: network separation of the two modules in INDRA's causal graph",
"kind": "data-test",
"method": "Overlap counts shared members. TAME's claim is better read as shared machinery: the modules may be mechanistically adjacent without sharing genes. Use the network separation s_AB = ⟨d_AB⟩ − (⟨d_AA⟩ + ⟨d_BB⟩)/2 of Menche et al. 2015, computed on INDRA CoGEx's causal statement graph between human genes (indra_rel edges with evidence_count ≥ 2, undirected for distance). A = core morphogenesis set and B = core learning/memory set (experimental GO annotations), with SynGO synaptic genes as a positive-control module expected to be near B. Null: degree-preserving randomization, sampling gene sets with matched degree distribution (binned), 10^4 draws. Pre-registered: s_AB < 0 with z < −3 against the degree-matched null supports shared machinery; s_AB ≥ 0 refutes it. Repeat after removing the top 1% highest-degree hubs to check that shared housekeeping hubs do not drive proximity.",
"steps": [
"Pre-register edge filters, gene sets, null and thresholds.",
"Export the INDRA causal subgraph via execute_cypher; record the query and statement-hash counts.",
"Compute s_AB and the null distribution; run the hub-removal sensitivity check.",
"For gene pairs driving proximity, cite INDRA statement hashes and PMIDs as mechanistic evidence."
],
"decisive_result": "Network separation s_AB with z-score against a degree-matched null, with and without hubs. Negative separation surviving hub removal would support Levin's shared-machinery prediction in a form the overlap test cannot detect; non-negative separation would refute it.",
"failure_signature": "INDRA's causal graph is so dominated by literature-heavy genes that degree matching cannot remove study bias (separation tracks publication count), so the result reflects literature structure rather than biology.",
"tools": [
"indra-cogex",
"network-statistics",
"data-test"
],
"datasets": [
{
"name": "INDRA CoGEx",
"url": "https://discovery.indra.bio/",
"note": "Causal statements between genes with evidence counts"
},
{
"name": "Gene Ontology",
"url": "https://geneontology.org/",
"note": "Module definitions"
}
],
"references": [
{
"url": "https://doi.org/10.1126/science.1257601",
"title": "Uncovering disease-disease relationships through the incomplete interactome",
"year": 2015,
"note": "Menche et al. 2015, Uncovering disease-disease relationships through the incomplete interactome (network separation)"
},
{
"url": null,
"note": "Koopmans et al. 2019, SynGO synaptic gene ontology (positive control module)"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "morphogenesis-cognition-gene-overlap--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Degree-matched ontology overlap test",
"kind": "data-test",
"method": "Freeze GO terms for morphogenesis and learning or memory, propagate annotations under a declared relation set, and restrict to experimentally supported human and mouse genes. Test overlap against 100000 permutations matched jointly on total annotation count, publication count, protein-interaction degree and gene length; repeat after removing generic signaling and housekeeping terms.",
"steps": [
"Timestamp ontology and evidence-code filters before retrieving genes.",
"Fit propensity scores for membership in each gene set from bias covariates.",
"Compute matched-null overlap z score, odds ratio and semantic-specificity-weighted overlap.",
"Repeat for ion-channel and gap-junction genes and across species."
],
"decisive_result": "Wide overlap requires odds ratio at least 2, empirical P below 0.001 and replication in both human and mouse after generic-term removal; an odds ratio between 0.8 and 1.25 supports no enrichment.",
"failure_signature": "Balance diagnostics retain standardized covariate differences above 0.1 or enrichment disappears under leave-one-ontology-term-out analysis.",
"tools": [
"indra-cogex",
"gene-ontology",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "morphogenesis-cognition-gene-overlap--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Mechanistic-pathway convergence in INDRA",
"kind": "data-test",
"method": "Move beyond gene lists by constructing evidence-weighted INDRA subnetworks connecting curated morphogenesis phenotypes and learning phenotypes. Compare shared causal intermediates and signed paths to degree-preserving, evidence-count-matched network rewires; require statement-level provenance and hold out papers mentioning both domains.",
"steps": [
"Ground phenotype-associated genes and retain statements with primary evidence.",
"Remove cross-domain papers to prevent literature leakage.",
"Compute shared intermediates, signed-path enrichment and pathway conductance.",
"Run 10000 constrained rewires preserving node degree and statement evidence bins."
],
"decisive_result": "Mechanistic convergence requires at least twofold excess shared intermediates with empirical P below 0.001 and replication after removing all ion-channel genes; null-level paths refute broad convergence.",
"failure_signature": "The result is driven by fewer than five hubs or by statements from cross-domain review papers.",
"tools": [
"indra-cogex",
"network-statistics",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "INDRA literature coverage and causal direction are uneven across developmental systems.",
"nanopub": null
},
{
"id": "morphogenesis-cognition-gene-overlap--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Pleiotropy-matched overlap of morphogenesis and learning genes",
"kind": "data-test",
"method": "Pre-register human genes with GO:0009653 (anatomical structure morphogenesis) and GO:0007611 (learning or memory). Primary run excludes evidence code IEA; a second run includes it and is labeled. Background is human protein-coding genes. The statistic is the odds ratio of the intersection. The null is 10,000 permutations that preserve both set sizes and match genes on the number of GO annotations and on degree in the public INDRA statement graph. Wide overlap, the pre-registered threshold, is an odds ratio above 2 and a permutation p-value below 0.01. The same threshold is a primary endpoint, not a subgroup glance, for the subsets annotated with GO:0005216 (monoatomic ion channel activity; the synonym is ion channel activity) and with GO:0005921 (gap junction). Housekeeping is partly absorbed by the annotation-count match; it is not a separate covariate invented after the test.",
"steps": [
"Freeze the two GO sets, the IEA rule, the odds-ratio threshold and the permutation scheme.",
"Pull annotations from Gene Ontology and degrees from INDRA.",
"Compute the observed odds ratio.",
"Run 10,000 matched permutations.",
"Apply the same test to the ion-channel and gap-junction subsets as primary endpoints."
],
"decisive_result": "Odds ratio above 2 and permutation p below 0.01 for the full sets and, separately, for the ion-channel subset and the gap-junction subset.",
"failure_signature": "The full-set odds ratio falls inside the matched null. That refutes wide overlap beyond pleiotropy and annotation volume. A hit that appears only in the IEA-included run is annotation noise and does not count.",
"tools": [
"indra-cogex",
"gene-set-statistics",
"data-test"
],
"datasets": [
{
"name": "Gene Ontology",
"url": "https://geneontology.org/",
"note": "Annotations for GO:0009653, GO:0007611, GO:0005216 and GO:0005921."
},
{
"name": "INDRA CoGEx",
"url": "https://discovery.indra.bio/",
"note": "Statement degree used to match the permutation null."
}
],
"references": [
{
"url": "https://doi.org/10.3389/fnsys.2022.768201",
"title": "Technological Approach to Mind Everywhere: An Experimentally-Grounded Framework for Understanding Diverse Bodies and Minds",
"year": 2022,
"note": "Levin 2022, Technological approach to mind everywhere, the overlap prediction"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "morphogenesis-cognition-gene-overlap--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "INDRA statement density inside the intersection",
"kind": "data-test",
"method": "Set overlap can be shared annotation fashion. This test asks whether genes in the intersection are mechanistically tied. Count INDRA statements with evidence count at least 5 that connect a GO:0009653 gene to a GO:0007611 gene. Compare with 10,000 random bipartitions that preserve the two set sizes and the joint degree sequence. The pre-registered effect is a statement count above the 99th percentile of that null. Repeat for the ion-channel subset and the gap-junction subset. A gene-set hit with a null statement density means the shared genes are not preferentially wired to each other in INDRA, which is a different negative from a failed overlap test.",
"steps": [
"Build the morphogenesis and learning sets as in the overlap test.",
"Count INDRA statements of evidence count at least 5 across the cut.",
"Permute bipartitions preserving sizes and degrees, 10,000 times.",
"Apply the 99th-percentile threshold.",
"Repeat for ion channels and gap junctions."
],
"decisive_result": "Cross-set INDRA statement count above the 99th percentile of the degree-preserving null, overall and in the two bioelectric subsets.",
"failure_signature": "The count lies within the null. Overlap of annotations, if it exists, is then not matched by INDRA evidence between the sets.",
"tools": [
"indra-cogex",
"gene-set-statistics",
"data-test"
],
"datasets": [
{
"name": "INDRA CoGEx",
"url": "https://discovery.indra.bio/",
"note": "Statements and evidence counts between the two gene sets."
},
{
"name": "Gene Ontology",
"url": "https://geneontology.org/",
"note": "Set membership."
}
],
"references": [
{
"url": "https://doi.org/10.3389/fnsys.2022.768201",
"title": "Technological Approach to Mind Everywhere: An Experimentally-Grounded Framework for Understanding Diverse Bodies and Minds",
"year": 2022,
"note": "Levin 2022, the bioelectric subset is part of the prediction"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "morphogenesis-cognition-gene-overlap--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Repeat the overlap using only pre-2015 annotations",
"kind": "data-test",
"method": "Annotation fashion after Levin's papers would inflate overlap. Repeat the matched odds-ratio test using only GO annotations whose assignment date is before 2015, if QuickGO exposes that date. Keep the same sets, the same exclusion of IEA, and the same thresholds. If dates are not in the download, the approach stops as inconclusive rather than using a proxy such as the paper's publication year. A signal present in the full ontology and absent before 2015 is evidence of later annotation, not of a standing shared gene set. A signal present before 2015 survives that objection.",
"steps": [
"Check whether annotation dates are present before computing the odds ratio.",
"Restrict to dates before 2015.",
"Reuse the matched permutation null.",
"Apply the odds-ratio and p-value thresholds.",
"If dates are absent, record inconclusive and stop."
],
"decisive_result": "The pre-2015 odds ratio either clears the same threshold as the full test or it does not. Clearing it supports overlap that predates the recent annotation wave.",
"failure_signature": "The pre-2015 odds ratio falls inside the null while the full-ontology test does not. The apparent overlap is then carried by later annotations.",
"tools": [
"gene-set-statistics",
"data-test"
],
"datasets": [
{
"name": "Gene Ontology",
"url": "https://geneontology.org/",
"note": "Annotation dates, used only if the download actually contains them."
}
],
"references": [
{
"url": "https://doi.org/10.3389/fnsys.2022.768201",
"title": "Technological Approach to Mind Everywhere: An Experimentally-Grounded Framework for Understanding Diverse Bodies and Minds",
"year": 2022,
"note": "Levin 2022, the date cut is chosen to precede this paper"
}
],
"difficulty": "tractable",
"missing": "GO assignment dates may be absent from the query used. The approach then cannot run, and that is a stated stop.",
"nanopub": null
}
]
},
{
"id": "bioelectric-prepattern-predicts-anatomy",
"title": "Does the bioelectric prepattern predict anatomical outcome?",
"field": "biology",
"tier": "C",
"question": "Is there a quantitative, predictive mapping from a measured resting-potential (Vmem) pattern in a tissue to the anatomical structure it will form, such that perturbing the pattern predicts the anatomical change before it occurs?",
"known_state": "Levin and colleagues report that Vmem patterns prefigure and can instruct anatomy in planarian head regeneration, Xenopus face and eye induction, and tumour induction (Levin and Martyniuk 2018). BETSE (Pietak and Levin 2016) simulates Vmem dynamics; Durant et al. (2017) modelled planarian anterior-posterior polarity. No model takes an arbitrary measured Vmem map and outputs a falsifiable anatomical prediction with stated accuracy. Thirteen Levin tests in the repo are pre-registered and awaiting data.",
"acceptance_criteria": [
"A model (bounded bioelectric simulation plus a pattern-to-anatomy decoder) that, given published Vmem images of perturbed and control embryos or planarian fragments, predicts the anatomical phenotype class with pre-registered accuracy on held-out images.",
"Explicit statement of which published datasets are used for training and which for testing."
],
"falsifiers": [
"Prediction accuracy at chance on held-out perturbations refutes the predictive form of the bioelectric-code hypothesis for those tissues."
],
"required_capabilities": [
"bioelectric-simulation",
"image-analysis",
"frontier_simulate",
"data-test"
],
"data_access": "Published Vmem imaging (DiBAC/CC2-DMPE) in open-access Levin lab papers; BETSE (open source).",
"scales": [
"cellular",
"tissue",
"developmental"
],
"frontier_problem": null,
"related_claims": [
{
"id": "levin-2015-girardia-11",
"label": "Hexanol, which does not block gap junctions, does not alter head shape",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "levin-2018-hcn2-06",
"label": "HCN2 modelled as a voltage-controlled switch that amplifies small Vmem gradients",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "levin-2018-hcn2-16",
"label": "Native HCN2 activity is not required for normal brain patterning",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "levin-2019-boundary-14",
"label": "Animal senses use the same ion channel mechanisms found in bacteria",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.biosystems.2017.08.009",
"title": "The bioelectric code: An ancient computational medium for dynamic control of growth and form",
"year": 2018,
"note": "Levin and Martyniuk 2018, The bioelectric code"
},
{
"url": "https://doi.org/10.3389/fbioe.2016.00055",
"title": "Exploring Instructive Physiological Signaling with the Bioelectric Tissue Simulation Engine",
"year": 2016,
"note": "Pietak and Levin 2016, Exploring instructive physiological signaling with the bioelectric tissue simulation engine"
},
{
"url": "https://doi.org/10.1016/j.bpj.2017.04.011",
"title": "Long-Term, Stochastic Editing of Regenerative Anatomy via Targeting Endogenous Bioelectric Gradients",
"year": 2017,
"note": "Durant et al. 2017, Long-term, stochastic editing of regenerative anatomy via targeting endogenous bioelectric gradients"
}
],
"nanopub": null,
"approaches": [
{
"id": "bioelectric-prepattern-predicts-anatomy--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Curated Vmem image corpus and a decoder validated by leave-one-study-out",
"kind": "data-test",
"method": "Build an image corpus from open-access Levin-lab and other papers showing voltage-reporter dye images (DiBAC4(3), CC2-DMPE:DiSBAC2(3)) of control and perturbed Xenopus embryos and planarian fragments, each linked to the reported anatomical outcome class: for Xenopus craniofacial or eye, normal, ectopic eye, malformed face; for planaria, one head, two heads, no head. Register images to a common template (embryo outline or fragment axis), extract a low-dimensional Vmem feature vector (template-aligned intensity on a coarse grid, normalized to within-image range because dye images are not absolutely calibrated). Train simple, pre-registered decoders (multinomial logistic regression on 20 PCA components; k-NN) and evaluate leave-one-study-out, so test images come from papers not used in training. Pre-registered success: balanced accuracy ≥ 0.70 against a chance level computed from class frequencies, permutation p < 0.01.",
"steps": [
"Pre-register inclusion criteria (open licence, dye named, outcome reported for the imaged individual or the imaged group), decoder, features and threshold.",
"Harvest figures with their licences; annotate outcome labels from the papers' text; record provenance.",
"Register images and extract features via the morpholib pipeline where applicable.",
"Run leave-one-study-out evaluation and the permutation test; report confusion matrices."
],
"decisive_result": "Leave-one-study-out balanced accuracy of anatomy-class prediction from Vmem images, against chance, with a permutation p-value. Above 0.70 supports a predictive mapping; at chance refutes the predictive form for those tissues.",
"failure_signature": "Most published Vmem images show group-representative embryos without individual outcome labels, or too few studies (fewer than 5 per tissue) exist for leave-one-study-out: inconclusive, data-unavailable.",
"tools": [
"image-analysis",
"morpholib",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1002/dvdy.22685",
"title": "V‐ATPase‐dependent ectodermal voltage and ph regionalization are required for craniofacial morphogenesis",
"year": 2011,
"note": "Vandenberg, Morrie and Adams 2011, V-ATPase-dependent ectodermal voltage and pH regionalization required for craniofacial morphogenesis (Xenopus 'electric face')"
},
{
"url": "https://doi.org/10.1242/dev.073759",
"title": "Transmembrane voltage potential controls embryonic eye patterning in\n Xenopus laevis",
"year": 2012,
"note": "Pai et al. 2012, Transmembrane voltage potential controls embryonic eye patterning in Xenopus laevis"
},
{
"url": "https://doi.org/10.1016/j.chembiol.2010.11.012",
"title": "A Chemical Genetics Approach Reveals H,K-ATPase-Mediated Membrane Voltage Is Required for Planarian Head Regeneration",
"year": 2011,
"note": "Beane et al. 2011, H,K-ATPase-mediated membrane voltage required for planarian head regeneration"
},
{
"url": "https://doi.org/10.1016/j.biosystems.2017.08.009",
"title": "The bioelectric code: An ancient computational medium for dynamic control of growth and form",
"year": 2018,
"note": "Levin and Martyniuk 2018, The bioelectric code"
}
],
"difficulty": "hard",
"missing": "Raw image data with individual-level outcome labels are rarely published; the swarm cannot generate new imaging.",
"nanopub": null
},
{
"id": "bioelectric-prepattern-predicts-anatomy--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Pre-registered forward predictions of perturbation outcomes from bioelectric simulation",
"kind": "simulation",
"method": "Test the instructive (causal) form of the hypothesis on perturbations whose outcomes are published but held out from calibration. Calibrate a BETSE or frontier_simulate model of the planarian fragment (ion channels, pumps, gap junctions per Durant et al. 2017 and Pietak and Levin 2017) on control and two perturbation conditions only. Then, for each held-out perturbation in the literature (for example ivermectin-induced hyperpolarization, SCH-28080 H,K-ATPase inhibition, octanol gap-junction block at stated concentrations), predict the steady-state anterior–posterior Vmem pattern. Map it to anatomy with a rule fixed in advance: a depolarized region above threshold θ predicts a head, θ fixed from the calibration conditions. Lodge predictions as a timestamped commit before comparison. Score the fraction of held-out perturbations whose predicted outcome class matches the published one, against a baseline that always predicts the most common outcome.",
"steps": [
"Fix model structure and calibration conditions; pre-register the Vmem-to-anatomy rule and the held-out list.",
"Calibrate parameters on the calibration set; freeze.",
"Simulate each held-out perturbation; commit predictions before reading outcomes.",
"Compare with published outcomes; report accuracy against the baseline with a binomial test."
],
"decisive_result": "The fraction of held-out pharmacological and genetic perturbations whose anatomical outcome class the frozen model predicts, against the most-common-class baseline. Accuracy significantly above baseline (binomial p < 0.01, at least 10 held-out conditions) supports the predictive bioelectric code; baseline accuracy refutes this model's version of it.",
"failure_signature": "Too few held-out perturbations with clear published outcomes and known molecular targets (fewer than 10), or model outcomes so sensitive to unmeasured parameters that predictions are not robust to plausible parameter ranges (report the fraction of the parameter posterior agreeing).",
"tools": [
"bioelectric-simulation",
"frontier_simulate",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.bpj.2017.04.011",
"title": "Long-Term, Stochastic Editing of Regenerative Anatomy via Targeting Endogenous Bioelectric Gradients",
"year": 2017,
"note": "Durant et al. 2017, Long-term, stochastic editing of regenerative anatomy via bioelectric gradients"
},
{
"url": "https://doi.org/10.3389/fbioe.2016.00055",
"title": "Exploring Instructive Physiological Signaling with the Bioelectric Tissue Simulation Engine",
"year": 2016,
"note": "Pietak and Levin 2016, BETSE"
},
{
"url": "https://doi.org/10.1098/rsif.2017.0425",
"title": "Bioelectric gene and reaction networks: computational modelling of genetic, biochemical and bioelectrical dynamics in pattern regulation",
"year": 2017,
"note": "Pietak and Levin 2017, Bioelectric gene and reaction networks"
},
{
"url": "https://github.com/betsee/betse",
"title": null,
"year": null,
"note": "BETSE source code"
}
],
"difficulty": "hard",
"missing": "frontier_simulate offers bounded previews only; full BETSE calibration needs more compute and parameter data (channel densities) than published papers supply.",
"nanopub": null
},
{
"id": "bioelectric-prepattern-predicts-anatomy--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Dose–response meta-analysis: does Vmem shift size predict penetrance?",
"kind": "data-test",
"method": "A predictive code implies graded dependence: larger, more localized Vmem shifts should give higher penetrance of the predicted phenotype. Code every published bioelectric perturbation experiment in Xenopus and planaria reporting both (a) a quantitative Vmem change (reporter intensity ratio or electrode mV, with dispersion) and (b) phenotype penetrance (k of n animals). Fit a random-effects logistic meta-regression of penetrance on the standardized Vmem shift, with study as a random effect and tissue and perturbation class (channel misexpression, drug, gap-junction block) as covariates. Pre-registered: a positive slope with 95% CI excluding 0, and the shift explaining ≥ 30% of between-condition variance, support the graded predictive form; a slope CI including 0 refutes it. Check small-study effects with funnel plots and Egger's test, since positive results may be preferentially published.",
"steps": [
"Pre-register extraction rules, effect-size standardization and the model.",
"Systematically search open-access literature; double-extract data (two independent agents) and reconcile.",
"Fit the meta-regression; run publication-bias diagnostics.",
"Link each coded study to repo claims (levin-2018-hcn2, levin-2019-boundary) and record evidence."
],
"decisive_result": "The meta-regression slope of phenotype penetrance on Vmem shift magnitude with CI, and the variance explained, adjusted for publication bias.",
"failure_signature": "Fewer than 15 conditions report both a quantitative Vmem change and penetrance, or Egger's test shows strong small-study bias making the slope uninterpretable.",
"tools": [
"meta-analysis",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.biosystems.2017.08.009",
"title": "The bioelectric code: An ancient computational medium for dynamic control of growth and form",
"year": 2018,
"note": "Levin and Martyniuk 2018, The bioelectric code"
},
{
"url": "https://doi.org/10.1242/dev.073759",
"title": "Transmembrane voltage potential controls embryonic eye patterning in\n Xenopus laevis",
"year": 2012,
"note": "Pai et al. 2012, Vmem and eye induction (penetrance data)"
},
{
"url": "https://doi.org/10.1016/j.bpj.2017.04.011",
"title": "Long-Term, Stochastic Editing of Regenerative Anatomy via Targeting Endogenous Bioelectric Gradients",
"year": 2017,
"note": "Durant et al. 2017, planarian double-head penetrance"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "bioelectric-prepattern-predicts-anatomy--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Held-perturbation Vmem phenotype decoder",
"kind": "data-test",
"method": "Digitize raw or supplementary Vmem images with specimen-level anatomical outcomes from one system only. Register maps to a common coordinate frame, train a spatial decoder on controls and selected perturbation classes, and hold out entire intervention mechanisms so the test measures pattern-based transfer rather than batch recognition.",
"steps": [
"Require at least 200 specimens, five perturbations and 30 specimens per held-out class.",
"Freeze normalization, masks and phenotype labels before model training.",
"Compare Vmem decoder with morphology-at-imaging-time, batch and permuted-map baselines.",
"Use nested cross-validation grouped by experiment and intervention."
],
"decisive_result": "Predictive mapping requires held-intervention macro-F1 at least 0.75 and at least 0.15 above every baseline, with bootstrap 95% lower bound above 0.6.",
"failure_signature": "Performance falls to chance when experiments or intervention mechanisms are held out, indicating batch-specific correlation rather than a transferable code.",
"tools": [
"image-analysis",
"data-test",
"bioelectric-simulation"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Specimen-linked raw Vmem maps and later phenotype labels may not be publicly available at adequate sample size.",
"nanopub": null
},
{
"id": "bioelectric-prepattern-predicts-anatomy--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Intervention-generalizing bioelectric simulator",
"kind": "simulation",
"method": "Construct a bounded tissue graph with measured channel classes, gap-junction coupling and cell-state transitions. Fit conductances only on control and one perturbation, then predict spatial Vmem and phenotype under unseen channel, gap-junction and pump interventions. Compare mechanistic predictions to a flexible image-only decoder.",
"steps": [
"Ground intervention targets and signed channel effects through INDRA CoGEx.",
"Specify prior parameter ranges and identifiability checks.",
"Calibrate with simulation-based inference on preregistered training interventions.",
"Predict held-out Vmem maps and anatomical classes without parameter refitting."
],
"decisive_result": "Mechanistic support requires held-intervention Vmem spatial correlation above 0.7 and phenotype accuracy above 0.75 for at least three distinct intervention classes.",
"failure_signature": "No single posterior parameter region predicts both Vmem and anatomy, or image-only prediction strongly outperforms simulation on held interventions.",
"tools": [
"frontier-simulate",
"indra-cogex",
"bioelectric-simulation",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "long-shot",
"missing": "Quantitative conductance, coupling and longitudinal imaging data for the same specimens.",
"nanopub": null
},
{
"id": "bioelectric-prepattern-predicts-anatomy--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Leave-one-paper-out decoder of published Vmem figures",
"kind": "data-test",
"method": "Admissible images are the voltage-reporter figures (DiBAC or CC2-DMPE) in the open-access papers of Levin and Martyniuk 2018, Pietak and Levin 2016, and Durant et al. 2017, with the phenotype class taken from the caption: for planaria, two-headed, normal, or no-head; for the other tissues, the classes the caption itself names. Features, frozen first, are the anterior-posterior dipole of the reported potential, the number of domains above the caption's own threshold, and a left-right asymmetry index. The split is by paper: train on two, test on the third, all three rotations. Support requires accuracy at least 0.7 and a binomial test against the test paper's majority class at p below 0.05, in at least two rotations. Independent experiments, not panels of one experiment, are the sample. Fewer than 20 independent experiments across the three papers makes the test underpowered. Raw stacks are not required and are not invented.",
"steps": [
"Freeze features, classes and the 0.7 accuracy bar before measuring figures.",
"Encode each admissible figure's features from the published image and its caption.",
"Rotate training and test by paper.",
"Score accuracy against the majority-class baseline.",
"Stop as underpowered if fewer than 20 independent experiments exist."
],
"decisive_result": "Accuracy at least 0.7 and above the majority class at p below 0.05 in at least two of three paper rotations, on at least 20 independent experiments.",
"failure_signature": "Accuracy consistent with the majority class in two or more rotations. The dipole-and-domain decoder then does not predict anatomy on these figures.",
"tools": [
"image-analysis",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.biosystems.2017.08.009",
"title": "The bioelectric code: An ancient computational medium for dynamic control of growth and form",
"year": 2018,
"note": "Levin and Martyniuk 2018, The bioelectric code"
},
{
"url": "https://doi.org/10.3389/fbioe.2016.00055",
"title": "Exploring Instructive Physiological Signaling with the Bioelectric Tissue Simulation Engine",
"year": 2016,
"note": "Pietak and Levin 2016, BETSE paper, included for its published Vmem figures"
},
{
"url": "https://doi.org/10.1016/j.bpj.2017.04.011",
"title": "Long-Term, Stochastic Editing of Regenerative Anatomy via Targeting Endogenous Bioelectric Gradients",
"year": 2017,
"note": "Durant et al. 2017, planarian bioelectric perturbations"
}
],
"difficulty": "hard",
"missing": "Raw microscopy. Published figures are a biased subset of the experiments, and the test cannot repair that.",
"nanopub": null
},
{
"id": "bioelectric-prepattern-predicts-anatomy--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Freeze BETSE parameters and predict a held-out perturbation",
"kind": "simulation",
"method": "Use the BETSE engine. Fix parameters at the published defaults in Pietak and Levin 2016 before looking at outcome counts. Choose one channel-block or gap-junction perturbation that Durant et al. 2017 report and that was not used to set those defaults. Map the simulated anterior-posterior Vmem difference onto the phenotype classes with a threshold frozen from the control simulation alone. Compare the predicted class with the published outcome. frontier_simulate may be called only as a bounded bioelectric preview to see whether a parameter dictionary is inside its allowlisted family. Its ephemeral result is not evidence and cannot meet this test. Success is the correct class on that held-out perturbation and on one additional perturbation from the same paper, both chosen before the run.",
"steps": [
"Pin the BETSE commit and the published default parameters.",
"Name the two held-out perturbations before simulating.",
"Derive the decision threshold from the control simulation only.",
"Simulate the two perturbations and emit phenotype classes.",
"Score them against the paper. Do not retune."
],
"decisive_result": "Both pre-named held-out perturbations assigned the published phenotype class by the frozen threshold.",
"failure_signature": "Either perturbation is misclassified. Retuning parameters to obtain the class fails the approach.",
"tools": [
"bioelectric-simulation",
"frontier_simulate"
],
"datasets": [
{
"name": "BETSE source",
"url": "https://gitlab.com/betse/betse",
"note": "Bioelectric Tissue Simulation Engine used for the frozen-parameter runs."
}
],
"references": [
{
"url": "https://doi.org/10.3389/fbioe.2016.00055",
"title": "Exploring Instructive Physiological Signaling with the Bioelectric Tissue Simulation Engine",
"year": 2016,
"note": "Pietak and Levin 2016, source of the default parameters"
},
{
"url": "https://doi.org/10.1016/j.bpj.2017.04.011",
"title": "Long-Term, Stochastic Editing of Regenerative Anatomy via Targeting Endogenous Bioelectric Gradients",
"year": 2017,
"note": "Durant et al. 2017, the held-out planarian perturbations"
}
],
"difficulty": "hard",
"missing": "frontier_simulate is a bounded preview and is not a substitute for BETSE. The swarm still needs a local BETSE run for the evidence.",
"nanopub": null
},
{
"id": "bioelectric-prepattern-predicts-anatomy--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Prove which Vmem summaries cannot identify anatomy",
"kind": "formalization",
"method": "A decoder that sees only the anterior-posterior dipole and the number of supra-threshold domains assigns the same phenotype to any two fields that agree on those summaries. Prove that statement. Then apply it to the admissible figures: search for two experiments with different caption phenotypes whose dipole and domain count agree within a tolerance frozen first (relative dipole difference below 0.1, same domain count). A hit refutes every decoder that uses only those features. Absence of a hit does not confirm the bioelectric code; it only says these features are not yet contradicted by the figures. This approach can refute a feature set that the classifier approach would merely score as inaccurate.",
"steps": [
"Prove the identifiability lemma for dipole and domain count.",
"Freeze the 0.1 relative tolerance.",
"Measure the summaries on the admissible figures.",
"Search for a cross-phenotype collision under that tolerance.",
"Report a refuting pair or the absence of one."
],
"decisive_result": "Either a pair of published experiments with different phenotypes and summaries inside the tolerance, which refutes dipole-and-domain decoders, or an exhaustive statement that no such pair exists in the admissible figures.",
"failure_signature": "The lemma is false for some pair of fields, which would be a mathematics error, or the figure measurements are too coarse to decide equality within 0.1, in which case the empirical half is inconclusive.",
"tools": [
"lean4",
"image-analysis"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.biosystems.2017.08.009",
"title": "The bioelectric code: An ancient computational medium for dynamic control of growth and form",
"year": 2018,
"note": "Levin and Martyniuk 2018, the mapping this lemma constrains"
},
{
"url": "https://doi.org/10.1016/j.bpj.2017.04.011",
"title": "Long-Term, Stochastic Editing of Regenerative Anatomy via Targeting Endogenous Bioelectric Gradients",
"year": 2017,
"note": "Durant et al. 2017, planarian figure set searched for a colliding pair"
}
],
"difficulty": "tractable",
"missing": "Raw pixel values. Figure measurements may be too coarse for the 0.1 tolerance, and the approach then stops as inconclusive on the empirical half.",
"nanopub": null
}
]
},
{
"id": "grn-memory-generic-or-biological",
"title": "Is memory in gene regulatory network models generic or a biological signature?",
"field": "both",
"tier": "C",
"question": "Do the memory and habituation behaviours found in published ODE models of biological pathways occur at the same rate in random nonlinear ODE networks of matched size, sparsity and kinetics, or are they enriched in biological networks?",
"known_state": "Biswas, Manicka, Hoel and Levin (2021) found several kinds of memory (UCS-based, pairing, associative, consolidation) in 35 BioModels pathway models, and habituation in 27 of them (levin-2022-grn-learning claims, top of the repo's testability ranking). No matched random-network null has been run, so it is unknown whether the result says anything about biology rather than about nonlinear dynamics in general.",
"acceptance_criteria": [
"A pre-registered computational test re-running the Biswas protocol on the original 35 models and on at least 350 random networks matched on node count, edge density, kinetic form and parameter ranges; the enrichment statistic and its threshold stated in advance.",
"Code and seeds published so the result is reproducible."
],
"falsifiers": [
"Memory rates in random networks equal to or above the biological ones refute the biological-signature reading."
],
"required_capabilities": [
"ode-simulation",
"biomodels",
"random-network-nulls",
"data-test"
],
"data_access": "BioModels (open), Biswas et al. 2021 code (open access paper).",
"scales": [
"molecular",
"cellular"
],
"frontier_problem": null,
"related_claims": [
{
"id": "levin-2022-grn-learning-01",
"label": "Published ODE models of gene and protein pathways show several kinds of memory",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "levin-2022-grn-learning-08",
"label": "Pathway memories can form through dynamics alone, without structural change",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "levin-2022-grn-learning-11",
"label": "Most pathway-model memories persist over long periods and survive recall",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "levin-2022-grn-learning-14",
"label": "Habituation (pharmacoresistance) occurs in every testable pathway model",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "levin-2022-grn-learning-16",
"label": "Stimulating other nodes can break pharmacoresistance in many pathway models",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.isci.2021.102131",
"title": "Gene regulatory networks exhibit several kinds of memory: Quantification of memory in biological and random transcriptional networks",
"year": 2021,
"note": "Biswas, Manicka, Hoel and Levin 2021, Gene regulatory networks exhibit several kinds of memory"
}
],
"nanopub": null,
"approaches": [
{
"id": "grn-memory-generic-or-biological--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Re-run the Biswas protocol on 35 BioModels and ≥ 350 matched random ODE networks",
"kind": "computation",
"method": "Reimplement the Biswas et al. 2021 protocol: choose stimulus nodes (UCS, CS) and a response node, apply the stimulus schedules for each memory type (UCS-based, pairing, transfer, associative, consolidation) and score memory by the paper's criteria and thresholds. Validate by reproducing their per-model memory calls on the 35 BioModels models (agreement ≥ 90% required before proceeding). Null ensemble: for each biological model, generate 10 random networks matched on node count, edge count and density, in- and out-degree sequences (configuration model), kinetic forms (mass-action and Hill terms in the same proportions, with Hill coefficients drawn from the model's empirical distribution), parameter values drawn from the model's own log-parameter distribution, and stability (retain only networks with a stable steady state from the default initial condition, as biological models have). Simulate with libRoadRunner. Statistic: per memory type, the fraction of (network, stimulus/response triple) combinations showing memory, biological versus random. Pre-registered: 'biological signature' if the biological rate exceeds the random rate by ≥ 2× with a stratified permutation p < 0.01; 'generic' if the ratio is ≤ 1.2.",
"steps": [
"Reimplement the protocol; validate against the paper's reported calls; publish the discrepancy list.",
"Write result.json with matching criteria, statistic and thresholds before simulating random networks.",
"Generate ≥ 350 matched random networks; check matching statistics.",
"Run all protocols; compute rates and ratios with stratified permutation p-values; publish code and seeds."
],
"decisive_result": "Per memory type, the ratio of memory rates in biological versus matched random networks with permutation p. A ratio ≥ 2 supports the biological-signature reading and ≤ 1.2 supports 'generic nonlinear dynamics', which is the problem's stated falsifier.",
"failure_signature": "Reproduction of the original calls falls below 90% agreement because protocol details are underspecified in the paper, so any comparison tests a different protocol.",
"tools": [
"ode-simulation",
"biomodels",
"random-network-nulls",
"data-test"
],
"datasets": [
{
"name": "BioModels",
"url": "https://www.ebi.ac.uk/biomodels/",
"note": "The 35 curated SBML models used by Biswas et al."
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.isci.2021.102131",
"title": "Gene regulatory networks exhibit several kinds of memory: Quantification of memory in biological and random transcriptional networks",
"year": 2021,
"note": "Biswas, Manicka, Hoel and Levin 2021, Gene regulatory networks exhibit several kinds of memory"
},
{
"url": "https://doi.org/10.3390/ijms24010285",
"title": "Learning in Transcriptional Network Models: Computational Discovery of Pathway-Level Memory and Effective Interventions",
"year": 2022,
"note": "Biswas, Clawson and Levin 2022, Learning in transcriptional network models"
},
{
"url": "https://github.com/sys-bio/roadrunner",
"title": null,
"year": null,
"note": "libRoadRunner SBML simulator"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "grn-memory-generic-or-biological--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Attribution by surgery: rewire topology vs shuffle parameters in real models",
"kind": "computation",
"method": "Random networks differ from biological ones in many ways at once. A sharper design perturbs the biological models themselves along one axis at a time. For each of the 35 models: (a) topology surgery, degree-preserving edge swaps (100 swaps per edge) keeping every reaction's rate law and parameters; (b) parameter surgery, permuting parameter values among reactions of the same kinetic type, keeping topology; (c) kinetic surgery, replacing Hill terms with Michaelis–Menten terms (n = 1), keeping topology and parameters; (d) a combination. Generate 20 variants per model per surgery, discard variants without a stable steady state, and re-run the memory protocol. Memory that survives (a) but not (b) points to tuned parameters; memory surviving (b) but lost under (c) points to cooperativity (ultrasensitivity) as the generic cause. This decomposes any biological enrichment from approach 1 into causes.",
"steps": [
"Implement SBML-level surgery operators with libsbml; verify that variants remain valid and stable.",
"Pre-register the attribution logic (which surgery outcomes imply which cause).",
"Run the memory protocol on all variants.",
"Report per-memory-type retention rates under each surgery with CIs."
],
"decisive_result": "Retention rates of each memory type under topology, parameter and kinetic surgery. For example, '80% of UCS memory survives topology rewiring but only 20% survives replacing Hill terms' would identify ultrasensitivity, not network wiring, as the cause.",
"failure_signature": "Most surgical variants lose steady-state stability, so the surviving variants are a biased subset and retention rates are uninterpretable.",
"tools": [
"ode-simulation",
"biomodels",
"random-network-nulls"
],
"datasets": [
{
"name": "BioModels",
"url": "https://www.ebi.ac.uk/biomodels/",
"note": "Source SBML models for surgery"
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.isci.2021.102131",
"title": "Gene regulatory networks exhibit several kinds of memory: Quantification of memory in biological and random transcriptional networks",
"year": 2021,
"note": "Biswas et al. 2021"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "grn-memory-generic-or-biological--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Theory: memory as multistability or slow manifolds, predicted by CRNT",
"kind": "mixed",
"method": "Explain which networks can show which memory, so the result is a theorem plus a prediction rather than only a rate comparison. Step 1 (proof): show that, for the protocol's definitions, persistent UCS-based memory over a test window T after stimulus removal requires either (i) multistability, with the stimulus switching basins, or (ii) a slow variable with relaxation time ≥ T. The argument: if the system is monostable with all eigenvalues at the steady state having real part ≤ −λ and the steady state is globally attracting with exponential rate λ, the response returns within ε after time log(C/ε)/λ, so memory beyond that is impossible. Formalize the linear-stability core in Lean. Step 2 (prediction): for mass-action models, Chemical Reaction Network Theory (deficiency zero and deficiency one theorems) certifies that some networks cannot be multistable for any rates. Classify all 35 models and the random ensemble by CRNT before simulation; predict that CRNT-certified monostable networks show long-window memory only if they have slow eigenvalues (computed). Compare with approach 1's observed memory calls.",
"steps": [
"State and prove the exponential-relaxation lemma for monostable systems in Lean (linear case; nonlinear case by citation of standard Lyapunov results with stated hypotheses).",
"Run CRNT analysis (deficiency computation) on mass-action-convertible models; compute slowest eigenvalues at steady state.",
"Pre-register predicted memory or no-memory per model from (multistability possible, slowest timescale).",
"Compare predictions with observed memory calls; report sensitivity and specificity."
],
"decisive_result": "A proved necessary condition for persistent memory, and the accuracy with which the CRNT plus slowest-timescale classification predicts observed memory across biological and random networks. High accuracy (≥ 85%) shows memory is explained by generic dynamical features, whatever the biological enrichment.",
"failure_signature": "Many networks show memory the classification rules out, which would mean the protocol's memory criterion picks up transient effects within the test window, a finding about the protocol itself; or most models use non-mass-action kinetics outside CRNT's scope.",
"tools": [
"lean4",
"chemical-reaction-network-theory",
"ode-simulation"
],
"datasets": [
{
"name": "BioModels",
"url": "https://www.ebi.ac.uk/biomodels/",
"note": "Models to classify"
}
],
"references": [
{
"url": "https://doi.org/10.1016/0009-2509(87)80099-4",
"title": "Chemical reaction network structure and the stability of complex isothermal reactors—I. The deficiency zero and deficiency one theorems",
"year": 1987,
"note": "Feinberg 1987, Chemical reaction network structure and the stability of complex isothermal reactors I (deficiency zero and one theorems)"
},
{
"url": "https://doi.org/10.1016/j.isci.2021.102131",
"title": "Gene regulatory networks exhibit several kinds of memory: Quantification of memory in biological and random transcriptional networks",
"year": 2021,
"note": "Biswas et al. 2021, memory definitions"
}
],
"difficulty": "hard",
"missing": "Many BioModels use non-mass-action rate laws; CRNT results cover only a subset.",
"nanopub": null
},
{
"id": "grn-memory-generic-or-biological--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Matched random-network memory enrichment test",
"kind": "computation",
"method": "Reimplement the exact stimulation and memory criteria for all 35 biological ODE models, then generate 100 matched nulls per model preserving node count, signed degree sequence, kinetic-law family, parameter marginal distributions and stiffness. Blindly apply identical protocols and estimate enrichment with a model-paired conditional logistic regression.",
"steps": [
"Freeze executable model versions and numerical tolerances.",
"Validate protocol reproduction against all reported memory calls.",
"Generate at least 3500 convergent null networks with fixed seeds.",
"Classify memory, habituation and consolidation using preregistered effect thresholds."
],
"decisive_result": "Biological enrichment requires paired odds ratio at least 2 with 99% interval excluding 1 for a memory class and replication across two null generators; odds ratio at or below 1 refutes enrichment.",
"failure_signature": "Fewer than 80% of published calls reproduce or null convergence filtering changes network dynamics enough to break covariate balance.",
"tools": [
"ode-simulation",
"large-scale-computation",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "grn-memory-generic-or-biological--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Dynamical motif attribution of memory",
"kind": "computation",
"method": "Identify whether memory enrichment is explained by specific feedback motifs rather than biological origin. Enumerate signed three- and four-node motifs, sample kinetic parameters by Latin hypercube, and estimate each motif's memory probability. Decompose each BioModels network's prediction with motif counts and targeted edge ablations.",
"steps": [
"Enumerate nonisomorphic signed motifs with self-loops handled explicitly.",
"Run at least 10000 parameter sets per motif under the fixed memory protocol.",
"Fit a sparse hierarchical predictor from motif content to memory behavior.",
"Ablate predicted motifs in biological models and matched nulls."
],
"decisive_result": "A motif mechanism is established if its presence increases memory probability by at least 0.25 and deleting it abolishes memory in at least 80% of predicted biological models while matched control deletions do not.",
"failure_signature": "Memory depends on diffuse parameter combinations and no motif has stable effects across kinetic families.",
"tools": [
"ode-simulation",
"network-enumeration",
"large-scale-computation"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Combinatorial parameter sweeps are expensive and motif effects may be context-dependent.",
"nanopub": null
},
{
"id": "grn-memory-generic-or-biological--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Match 350 random ODE networks to the 35 BioModels",
"kind": "computation",
"method": "Re-run the Biswas, Manicka, Hoel and Levin 2021 memory probes on the original models they took from BioModels, and on at least 10 random networks per model, at least 350 in total. Match each random network on node count, reaction count, and kinetic family (mass-action or Hill, matching the source model). Draw parameter magnitudes from the empirical distribution of the absolute values in that source model. Randomize regulatory signs in this approach; the topology-preserving arm is the next one. Probes are the paper's UCS-based, pairing, associative and consolidation memories, plus habituation. The enrichment statistic is the paired difference in the number of memory kinds present. Success, pre-registered, is a permutation p-value below 0.01 against the random ensemble and at least 20 of the 35 biological models showing strictly more memory kinds than their matched random networks. Publish seeds and the model identifiers.",
"steps": [
"Pin the 35 BioModels identifiers from the paper before simulating.",
"Build at least 10 matched random networks for each, signs randomized.",
"Run the five probes with the paper's protocol.",
"Compute the paired permutation test.",
"Require p below 0.01 and at least 20 of 35 models strictly enriched."
],
"decisive_result": "Permutation p below 0.01 and strict enrichment in at least 20 of 35 models. Otherwise memory in this protocol is not enriched relative to the matched random networks.",
"failure_signature": "The random networks meet or exceed the biological memory rate. That refutes the biological-signature reading for this match. A probe implementation that cannot reproduce the paper's count on the original 35 models invalidates the run before the null is interpreted.",
"tools": [
"ode-simulation",
"biomodels",
"random-network-nulls",
"data-test"
],
"datasets": [
{
"name": "BioModels",
"url": "https://www.ebi.ac.uk/biomodels/",
"note": "The pathway models named by Biswas et al. 2021."
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.isci.2021.102131",
"title": "Gene regulatory networks exhibit several kinds of memory: Quantification of memory in biological and random transcriptional networks",
"year": 2021,
"note": "Biswas, Manicka, Hoel and Levin 2021, Gene regulatory networks exhibit several kinds of memory"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "grn-memory-generic-or-biological--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Factor topology against kinetics in the memory rate",
"kind": "computation",
"method": "Cross two factors the single matched null confounds. Arm A keeps the biological reaction graph and redraws parameters from the same magnitude distribution. Arm B rewires the graph to the same degree sequence and keeps the original parameters. Arm C does both and is the null from the first approach. Each arm has the same 10 networks per model. The biological-signature claim for topology survives only if arm A still shows the enrichment and arm B does not. The claim for kinetics survives only if arm B shows it and arm A does not. If arm A already loses the enrichment, topology is not the carrier. Pre-register that split before looking at which arm wins.",
"steps": [
"Build arms A, B and C with the shared size match.",
"Run the same five probes.",
"Test enrichment separately in each arm.",
"Assign the effect to topology, kinetics, both, or neither by the pre-registered pattern.",
"Do not reinterpret the pattern after seeing it."
],
"decisive_result": "A four-way call: enrichment in A only (topology), B only (kinetics), both, or neither, using the same p-value and 20-of-35 rule in each arm.",
"failure_signature": "Arm C differs from the first approach's null on the same seeds, which means the factor arms are not comparable. Substantively, enrichment in neither arm refutes a biological signature of either factor.",
"tools": [
"ode-simulation",
"biomodels",
"random-network-nulls"
],
"datasets": [
{
"name": "BioModels",
"url": "https://www.ebi.ac.uk/biomodels/",
"note": "Source graphs and parameter magnitudes for the three arms."
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.isci.2021.102131",
"title": "Gene regulatory networks exhibit several kinds of memory: Quantification of memory in biological and random transcriptional networks",
"year": 2021,
"note": "Biswas et al. 2021, the probes held fixed across arms"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "grn-memory-generic-or-biological--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Prove habituation for one nonlinear motif and not for linear ODEs",
"kind": "formalization",
"method": "Define habituation as three inequalities on a forced ODE: repeated identical pulses produce strictly decreasing peak responses; after a rest interval longer than a stated multiple of the pulse spacing, the next peak returns to within 10 percent of the first; the decrement happens for every pulse amplitude in an open interval. Prove that a three-variable system with slow negative feedback satisfies this on an open set of parameters. Prove that no linear system of the same input-output shape can satisfy decrement for all amplitudes in an interval and recovery after rest, because superposition fixes the gain. This shows habituation is generic for that motif and absent for linear kinetics. It does not show that BioModels are enriched. A failure here leaves the enrichment test untouched.",
"steps": [
"Write the habituation inequalities as a Lean predicate.",
"Define the three-variable negative-feedback system.",
"Prove the predicate on an open parameter set.",
"Prove linear systems of the same shape fail the predicate.",
"Audit axioms."
],
"decisive_result": "A no-sorry existence theorem for the nonlinear motif and a no-sorry impossibility theorem for linear kinetics, under the stated habituation predicate.",
"failure_signature": "The nonlinear motif fails recovery or fails decrement on every open set tried, or a linear system is found that meets the predicate. Either kills the generic-versus-linear claim.",
"tools": [
"lean4",
"ode-simulation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.isci.2021.102131",
"title": "Gene regulatory networks exhibit several kinds of memory: Quantification of memory in biological and random transcriptional networks",
"year": 2021,
"note": "Biswas et al. 2021, source of the habituation behaviour being formalized, not of the motif proof"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
}
]
},
{
"id": "active-inference-morphogenesis-falsifiable",
"title": "A falsifiable form of active-inference morphogenesis",
"field": "both",
"tier": "C",
"question": "Can the claim that morphogenesis is variational free-energy minimization be stated as a quantitative prediction that a regeneration or development experiment could refute, and does existing data refute it?",
"known_state": "Friston, Levin, Sengupta and Pezzulo (2015) modelled morphogenesis as active inference by identical cells sharing a generative model; Kuchling et al. (2020) extended it to regeneration. Critics hold the framework is unfalsifiable because any outcome can be described as free-energy minimization under some generative model. No prediction has been derived that fixes the generative model in advance and could fail. The repo's levin-2015-place and levin-2019-boundary claims are untested for this reason.",
"acceptance_criteria": [
"A derivation, from a generative model fixed before looking at data, of a quantitative prediction (for example regeneration time or error rate scaling with a stated surprise measure under graded perturbations) together with the observation that would refute it.",
"A pre-registered test of that prediction on published planarian or Xenopus regeneration time courses."
],
"falsifiers": [
"If every candidate prediction can be rescued by changing the generative model post hoc, the framework is not empirically testable in its current form; that is itself a result."
],
"required_capabilities": [
"variational-inference",
"developmental-biology",
"philosophy-of-science",
"data-test"
],
"data_access": "Published regeneration time courses (open-access Levin lab papers).",
"scales": [
"cellular",
"tissue",
"developmental"
],
"frontier_problem": null,
"related_claims": [
{
"id": "levin-2015-place-03",
"label": "Morphogenesis requires each cell to model its place in the target form",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "levin-2015-place-04",
"label": "Free energy is minimized only when every cell occupies a unique inferred place",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "levin-2019-boundary-15",
"label": "Cells in development minimize surprise and variational free energy like nervous systems",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "levin-2015-girardia-20",
"label": "Regeneration as descent to a free-energy attractor; induced shapes as local minima",
"thinker": "Michael Levin",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1098/rsif.2014.1383",
"title": "Knowing one's place: a free-energy approach to pattern regulation",
"year": 2015,
"note": "Friston, Levin, Sengupta and Pezzulo 2015, Knowing one's place: a free-energy approach to pattern regulation"
},
{
"url": "https://doi.org/10.1016/j.plrev.2019.06.001",
"title": "Morphogenesis as Bayesian inference: A variational approach to pattern formation and control in complex biological systems",
"year": 2020,
"note": "Kuchling, Friston, Georgiev and Levin 2020, Morphogenesis as Bayesian inference"
}
],
"nanopub": null,
"approaches": [
{
"id": "active-inference-morphogenesis-falsifiable--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Lean: a finite complete-class theorem showing exactly when free-energy accounts are empty",
"kind": "formalization",
"method": "Make the unfalsifiability charge precise, which is a result in itself. Setting: finite hidden-state space S, finite observations O, finite actions A, and a policy π: Oⁿ → A over a finite horizon. Theorem 1 (finite complete class): for every deterministic policy π there exist a generative model (prior over S, likelihood, and preference prior over outcomes) under which π minimizes expected free energy; the construction places preference mass on the outcomes π produces. So without constraints on the generative model, the claim 'morphogenesis minimizes free energy' excludes no behaviour. Theorem 2 (restoring content): if the likelihood and preference prior are fixed in advance (for example the target morphology and the cell-signalling likelihood from Friston et al. 2015's model), the set of free-energy-optimal policies is a proper subset, and there exist observable trajectories outside it, with an explicit separating example. This gives a formal criterion: an active-inference morphogenesis claim is testable iff it fixes the components named in Theorem 2.",
"steps": [
"Define finite POMDP-style generative models and expected free energy over Finset types in Lean.",
"Prove Theorem 1 by explicit construction of the preference prior.",
"Prove Theorem 2 with a small explicit fixed model (for example 3 states, 2 actions) and a trajectory not optimal under it.",
"Write a short note mapping critics' and proponents' statements to Theorems 1 and 2."
],
"decisive_result": "Lean-checked Theorems 1 and 2. Together they state precisely that the framework is empty with a free generative model and testable with a fixed one, which turns the falsifiability debate into a checkable criterion that approaches 2 and 3 then meet.",
"failure_signature": "Expected free energy has several non-equivalent definitions in the literature, and the theorems hold for one but not others; the result then must be stated per definition, which weakens its reach.",
"tools": [
"lean4",
"mathlib",
"variational-inference",
"philosophy-of-science"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rsif.2014.1383",
"title": "Knowing one's place: a free-energy approach to pattern regulation",
"year": 2015,
"note": "Friston, Levin, Sengupta and Pezzulo 2015, Knowing one's place"
},
{
"url": "https://doi.org/10.1371/journal.pone.0015554",
"title": "Observing the Observer (I): Meta-Bayesian Models of Learning and Decision-Making",
"year": 2010,
"note": "Daunizeau et al. 2010, Observing the observer (I): meta-Bayesian models of learning and decision-making (complete class argument)"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "active-inference-morphogenesis-falsifiable--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Fix Friston et al.'s generative model and derive a graded-excision prediction",
"kind": "mixed",
"method": "Take the published generative model (Friston et al. 2015: cells with a shared model of target positions and signal expression) without modification and derive a quantitative prediction for an experiment that already exists. Simulate excision of fractions f ∈ {0.1, …, 0.9} of the ensemble and measure (a) time to reach the target configuration and (b) the probability of incorrect pattern (a residual error above threshold). The model predicts specific functional forms, for example recovery time increasing with f up to a critical fraction f_c beyond which the pattern fails. Freeze these predictions, functional forms and f_c with intervals over the model's stated parameter ranges, before collecting data. Then compare with published planarian regeneration data on fragment size versus regeneration success and time, and Xenopus or axolotl data where graded removals exist. Pre-registered: refuted if the observed success-versus-fragment-size curve is outside the predicted envelope at ≥ 2 fragment sizes.",
"steps": [
"Reimplement the Friston et al. 2015 simulation from the published equations; reproduce its figures.",
"Run graded excisions over the model's parameter ranges; freeze predicted curves with envelopes.",
"Compile published fragment-size and regeneration-outcome data with a pre-registered inclusion rule.",
"Compare; report verdict and which model parameters, if any, the data constrain."
],
"decisive_result": "A pre-registered comparison of the fixed-model prediction (recovery time and failure probability versus fraction removed, with envelope) against published regeneration data. Observations outside the envelope refute this active-inference model of pattern regulation; observations inside it are a first non-trivial success.",
"failure_signature": "The envelope over plausible parameters is so wide that it contains every monotone curve, so the fixed model still makes no risky prediction. This would confirm Theorem 1's warning in practice even with the model fixed.",
"tools": [
"variational-inference",
"simulation",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rsif.2014.1383",
"title": "Knowing one's place: a free-energy approach to pattern regulation",
"year": 2015,
"note": "Friston et al. 2015, the generative model to fix"
},
{
"url": "https://doi.org/10.1016/j.plrev.2019.06.001",
"title": "Morphogenesis as Bayesian inference: A variational approach to pattern formation and control in complex biological systems",
"year": 2020,
"note": "Kuchling, Friston, Georgiev and Levin 2020, Morphogenesis as Bayesian inference (regeneration extensions)"
}
],
"difficulty": "hard",
"missing": "Published regeneration datasets reporting outcomes over graded fragment sizes are sparse and old; the decisive experiment would need new lab work.",
"nanopub": null
},
{
"id": "active-inference-morphogenesis-falsifiable--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Craniofacial remodelling: error-proportional correction vs hardwired movement",
"kind": "data-test",
"method": "Xenopus 'Picasso' tadpoles, with organs experimentally displaced, remodel toward normal craniofacial geometry (Vandenberg et al. 2012; Pinet et al. 2019). A fixed-target active-inference model with Gaussian precision predicts that each organ's correction velocity is proportional to its displacement from the target, giving exponential relaxation and a rate independent of the direction of displacement. A hardwired-movement alternative (organs migrate a fixed program regardless of start) predicts velocity independent of displacement. Using organ positions reported or measurable in these papers' figures at multiple stages, estimate for each organ the regression of displacement change on initial displacement. Pre-registered: the active-inference model is supported if the slope is negative with exponential-fit R² ≥ 0.6, and refuted if velocity is independent of displacement (slope CI including 0) or if organs overshoot the target systematically, which a fixed-precision gradient flow cannot produce.",
"steps": [
"Pre-register both models' predictions and the statistics.",
"Extract organ positions over developmental stages from the published figures and supplementary data, with landmark definitions.",
"Fit the displacement-dependent and displacement-independent models; compare by AICc.",
"Check for overshoot; report verdicts per organ."
],
"decisive_result": "Whether correction velocity scales with displacement (supporting fixed-target error correction) or not (supporting hardwired programs), per organ, with model-comparison weights. This is a risky prediction the fixed model can fail.",
"failure_signature": "Published figures give too few time points or individuals per stage (fewer than 3 stages or 10 tadpoles), making slopes unidentifiable; or any proportional-correction result is shared by non-inferential gradient models, so it supports a class of models rather than active inference specifically (this must be stated in the interpretation).",
"tools": [
"morphometrics",
"image-analysis",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1002/dvdy.23770",
"title": "Normalized shape and location of perturbed craniofacial structures in the\n Xenopus\n tadpole reveal an innate ability to achieve correct morphology",
"year": 2012,
"note": "Vandenberg, Adams and Levin 2012, Normalized shape and location of perturbed craniofacial structures in the Xenopus tadpole"
},
{
"url": "https://doi.org/10.1242/dev.175893",
"title": "Adaptive correction of craniofacial defects in pre-metamorphic Xenopus laevis tadpoles involves thyroid hormone-independent tissue remodeling",
"year": 2019,
"note": "Pinet et al. 2019, Adaptive correction of craniofacial defects in pre-metamorphic Xenopus laevis tadpoles"
},
{
"url": "https://doi.org/10.1098/rsif.2014.1383",
"title": "Knowing one's place: a free-energy approach to pattern regulation",
"year": 2015,
"note": "Friston et al. 2015"
}
],
"difficulty": "hard",
"missing": "Individual-level trajectory data are not published as tables; figure extraction is limited and new time-lapse imaging would be decisive.",
"nanopub": null
},
{
"id": "active-inference-morphogenesis-falsifiable--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Frozen generative model predicts regeneration times",
"kind": "data-test",
"method": "Choose one planarian regeneration series and fit a fixed generative model only to intact anatomy and unperturbed regeneration. Define sensory states, latent target morphology, likelihood variance and action dynamics before examining graded perturbations. Predict median regeneration time and failure probability as functions of initial variational surprise.",
"steps": [
"Publish the full generative model, priors and fitting data split before testing.",
"Compute surprise from initial fragment geometry and Vmem summaries.",
"Derive posterior predictive time and error curves for held-out perturbations.",
"Compare against size-only and flexible survival-model baselines."
],
"decisive_result": "The fixed model passes if at least 80% of held-out median times lie within its 90% intervals and surprise improves leave-intervention-out ELPD by at least 10; failure of interval coverage below 50% refutes this instantiation.",
"failure_signature": "Predictions require changing likelihoods, priors or target morphology after observing each perturbation, demonstrating post hoc flexibility.",
"tools": [
"variational-inference",
"data-test",
"bioelectric-simulation"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Open specimen-level regeneration time courses with quantified initial perturbations.",
"nanopub": null
},
{
"id": "active-inference-morphogenesis-falsifiable--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Identifiability and universal-fit stress test",
"kind": "computation",
"method": "Test falsifiability directly. Generate synthetic morphogenetic trajectories from reaction-diffusion, mechanical relaxation and random state-space models, then fit the same bounded active-inference model class without changing its latent dimension or priors. Estimate how often it fits non-active-inference processes as well as genuine ones.",
"steps": [
"Predefine model class, parameter bounds and predictive scoring rule.",
"Generate 1000 datasets per mechanism over matched noise and sample sizes.",
"Fit blindly and perform simulation-based calibration.",
"Compute specificity, power and parameter identifiability for labeling active inference."
],
"decisive_result": "The framework is discriminative only if power and specificity both exceed 0.8 on held-out trajectories; specificity below 0.6 shows the tested formulation can absorb unrelated dynamics.",
"failure_signature": "Equivalent predictive scores occur across all generators or posterior parameters are nonidentifiable, making empirical attribution unsupported.",
"tools": [
"variational-inference",
"large-scale-computation",
"simulation"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "active-inference-morphogenesis-falsifiable--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Fix the generative model, then rank regeneration times",
"kind": "data-test",
"method": "Before opening a regeneration time course, fix the generative model to the shared target morphology used by Friston, Levin, Sengupta and Pezzulo 2015, with no free retargeting. Define the surprise of a cut as the variational free energy of that initial condition under the fixed target. The prediction is ordinal: among fragment classes of one species in one paper, the ordering of free energies must match the ordering of regeneration times, with Spearman rank correlation above 0.5. Planarian fragment series are the admissible data, and only series that report times for at least four fragment classes. Parameters of the target are taken from the 2015 paper, not from the time series. Fewer than four classes, or times that exist only as unlabeled cartoons, makes the test inconclusive.",
"steps": [
"Transcribe the 2015 target morphology and freeze it.",
"Define free energy of a cut under that target.",
"List fragment classes and published times from one admissible series.",
"Compute Spearman correlation between free-energy rank and time rank.",
"Refute the fixed model if the correlation is at most 0.5."
],
"decisive_result": "Spearman correlation above 0.5 between pre-registered free-energy rank and regeneration-time rank, on at least four fragment classes.",
"failure_signature": "Correlation at most 0.5. Changing the target after seeing the times fails the approach and is the unfalsifiability pattern the problem names.",
"tools": [
"variational-inference",
"developmental-biology",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rsif.2014.1383",
"title": "Knowing one's place: a free-energy approach to pattern regulation",
"year": 2015,
"note": "Friston, Levin, Sengupta and Pezzulo 2015, Knowing one's place"
},
{
"url": "https://doi.org/10.1016/j.plrev.2019.06.001",
"title": "Morphogenesis as Bayesian inference: A variational approach to pattern formation and control in complex biological systems",
"year": 2020,
"note": "Kuchling, Friston, Georgiev and Levin 2020, Morphogenesis as Bayesian inference"
}
],
"difficulty": "hard",
"missing": "Quantitative regeneration times for four or more fragment classes in one paper. If the papers only show cartoons, the test is inconclusive.",
"nanopub": null
},
{
"id": "active-inference-morphogenesis-falsifiable--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Prove the free-target class is tautological",
"kind": "formalization",
"method": "Two theorems. First, if the generative model may be any smooth target morphology plus a positive precision, then for every observed morphology there exists a target equal to it that minimizes variational free energy. That class is unfalsifiable, and the theorem should say so. Second, with the target fixed as in the 2015 paper, the free-energy ordering of a finite set of cuts is not an identity: exhibit two cuts whose free-energy order is determined by the fixed target, so a reversed regeneration-time order would violate the prediction. The second theorem is the falsifiable form. If the fixed-target ordering can still be reversed by using a degree of freedom the paper leaves open, the second theorem fails and the approach concludes that the published model is not yet a test.",
"steps": [
"Define variational free energy for a morphology and a target.",
"Prove existence of a minimizing target for every observation.",
"Instantiate the 2015 target and rank a finite set of cuts.",
"Show the ranking is invariant under the degrees of freedom the paper leaves free, or name the freedom that flips it.",
"Audit axioms."
],
"decisive_result": "A no-sorry tautology theorem for free targets, and either a no-sorry non-identity for the fixed 2015 target or a proof that a free parameter in that paper flips the ranking.",
"failure_signature": "The fixed-target ranking flips under a parameter the pre-registration treated as frozen. The model is then not testable in the form stated, which the problem accepts as a result only if it is proved rather than asserted.",
"tools": [
"lean4",
"variational-inference"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rsif.2014.1383",
"title": "Knowing one's place: a free-energy approach to pattern regulation",
"year": 2015,
"note": "Friston et al. 2015, the model class"
},
{
"url": "https://doi.org/10.1016/j.plrev.2019.06.001",
"title": "Morphogenesis as Bayesian inference: A variational approach to pattern formation and control in complex biological systems",
"year": 2020,
"note": "Kuchling et al. 2020, the regeneration extension"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "active-inference-morphogenesis-falsifiable--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Lattice simulation with the target frozen before the cut",
"kind": "simulation",
"method": "Implement the cell-lattice update from Kuchling et al. 2020 only if the paper states a complete parameter set. If it gives a schematic without numbers, stop and mark the approach not operationalizable. When numbers exist, freeze them, apply one cut geometry that the paper reports but that is not the figure used to illustrate the update, and predict the polarity class. frontier_simulate's mechanobiology preview may be used to check that a lattice energy is inside its allowlisted family; it does not replace the lattice run and is not evidence. Success is the published polarity class. One cut, named first.",
"steps": [
"Extract numerical parameters from Kuchling et al. 2020 or stop.",
"Name the held-out cut before simulating.",
"Run the lattice update with the frozen target.",
"Compare predicted polarity with the paper.",
"Do not add a parameter to repair a miss."
],
"decisive_result": "The pre-named cut's published polarity, from a parameter set taken entirely from the paper.",
"failure_signature": "The polarity is wrong, or the paper has no complete parameter set. The second case is a failed operationalization, not a refutation of active inference.",
"tools": [
"simulation",
"frontier_simulate",
"variational-inference"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.plrev.2019.06.001",
"title": "Morphogenesis as Bayesian inference: A variational approach to pattern formation and control in complex biological systems",
"year": 2020,
"note": "Kuchling et al. 2020, the update rule and the only allowed source of parameters"
}
],
"difficulty": "long-shot",
"missing": "A fully numerical parameter set in the paper. If the model is schematic, this approach cannot start.",
"nanopub": null
}
]
},
{
"id": "kelvin-problem-in-epithelia",
"title": "Kelvin's problem in epithelia",
"field": "both",
"tier": "C",
"question": "What is the energy-minimizing partition of space into equal-volume cells (Kelvin's problem, still open), and do cells in curved epithelia adopt the minimal shapes (scutoids) that the corresponding constrained problem predicts?",
"known_state": "Kelvin's 1887 conjecture was beaten by the Weaire-Phelan foam (1994); no proof of optimality for any 3D partition exists, in contrast to Hales's proof of the 2D honeycomb conjecture (2001). Gomez-Galvez et al. (2018) showed that cells in curved epithelia adopt scutoid shapes and argued they minimize energy; Thompson's cell-packing claims (ch07) are the 2D and 3D precursors and the repo's cell-geometry project segments micrographs for them.",
"acceptance_criteria": [
"For the biological part: a pre-registered test on segmented 3D epithelial datasets comparing observed scutoid frequency against energy-minimizing predictions for the measured curvature, with a stated tolerance.",
"For the mathematical part: a new rigorous result, for example a Lean-checked lower bound on the surface area of equal-volume partitions in a periodic cell, or a computer-assisted proof of local optimality for Weaire-Phelan."
],
"falsifiers": [
"A partition with lower surface area than Weaire-Phelan would overturn the biological energy-minimization comparison baseline; scutoid frequencies far from the minimal-energy prediction would refute the energy-minimization account."
],
"required_capabilities": [
"surface-evolver",
"3d-segmentation",
"cell-geometry",
"lean4"
],
"data_access": "Gomez-Galvez 2018 supplementary data (open); EM epithelial volumes (open repositories).",
"scales": [
"cellular",
"tissue"
],
"frontier_problem": null,
"related_claims": [
{
"id": "thompson1917-ch07-04",
"label": "Tissue cell walls meet three at a time at 120°, regardless of cell size",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": "mixed",
"theorem": true
},
{
"id": "thompson1917-ch07-11",
"label": "The bee’s cell is a hexagonal prism capped by rhombic-dodecahedral facets",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": "inconclusive",
"theorem": true
},
{
"id": "thompson1917-ch07-08",
"label": "Hexagons are regular only where cells are equal in size and forces uniform",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": "mixed",
"theorem": false
},
{
"id": "thompson1917-ch09-16",
"label": "Callimitra's skeleton is formed on the interfaces of a tetrahedral group of four cells",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1080/09500839408241577",
"title": "A counter-example to Kelvin's conjecture on minimal surfaces",
"year": 1994,
"note": "Weaire and Phelan 1994, A counter-example to Kelvin's conjecture on minimal surfaces"
},
{
"url": "https://doi.org/10.1007/s004540010071",
"title": "The Honeycomb Conjecture",
"year": 2001,
"note": "Hales 2001, The honeycomb conjecture"
},
{
"url": "https://doi.org/10.1038/s41467-018-05376-1",
"title": "Scutoids are a geometrical solution to three-dimensional packing of epithelia",
"year": 2018,
"note": "Gomez-Galvez et al. 2018, Scutoids are a geometrical solution to three-dimensional packing of epithelia"
}
],
"nanopub": null,
"approaches": [
{
"id": "kelvin-problem-in-epithelia--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Computer-assisted proof that Weaire–Phelan is a strict local minimum among periodic foams",
"kind": "proof",
"method": "Prove local optimality of the Weaire–Phelan (WP) partition within its periodic class, the result the problem asks for. Setting: partitions of the 3-torus ℝ³/Λ (the WP cubic cell, 8 cells) into cells of equal volume. WP's faces are constant-mean-curvature surfaces meeting at Plateau angles. Plan: (a) compute the WP equilibrium with Surface Evolver at high refinement with an a-posteriori error bound; (b) compute the second variation (the Jacobi operator on faces with the Plateau-border coupling conditions and volume constraints) discretized by finite elements; (c) certify, with interval arithmetic in Arb, a lower bound λ_min > 0 on the constrained Hessian's smallest non-trivial eigenvalue, modulo translations, using a residual-based enclosure (Kato–Temple or Lehmann–Goerisch bounds) that converts discrete eigenvalue estimates into rigorous bounds for the continuous operator. A positive bound proves WP is a strict local minimum against all smooth volume-preserving perturbations periodic on Λ. The same pipeline applied to Kelvin's foam reproduces known stability as a validation.",
"steps": [
"Compute WP and Kelvin equilibria in Surface Evolver; validate surface areas against published values (Kelvin 5.306, WP 5.288 in units of V^(2/3) for total cell surface).",
"Discretize the constrained second variation; compute the lowest eigenvalues numerically.",
"Derive rigorous eigenvalue lower bounds with Lehmann–Goerisch and Arb interval arithmetic.",
"Validate on Kelvin; publish the certificate and code."
],
"decisive_result": "A certified bound λ_min > 0 for WP's constrained second variation on its periodic cell, proving strict local optimality among equal-volume partitions with that period, the first rigorous optimality-type statement for WP.",
"failure_signature": "The discrete-to-continuous error bound for curved faces with Plateau-border junctions cannot be made rigorous with available eigenvalue-enclosure methods, or the certified margin is negative within error, leaving the result unproved.",
"tools": [
"surface-evolver",
"interval-arithmetic",
"finite-element-analysis"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1080/09500839408241577",
"title": "A counter-example to Kelvin's conjecture on minimal surfaces",
"year": 1994,
"note": "Weaire and Phelan 1994, A counter-example to Kelvin's conjecture on minimal surfaces"
},
{
"url": "https://doi.org/10.1080/10586458.1992.10504253",
"title": "The Surface Evolver",
"year": 1992,
"note": "Brakke 1992, The Surface Evolver"
},
{
"url": "https://kenbrakke.com/evolver/evolver.html",
"title": null,
"year": null,
"note": "Surface Evolver software"
},
{
"url": "https://arblib.org/",
"title": null,
"year": null,
"note": "Arb ball arithmetic"
}
],
"difficulty": "long-shot",
"missing": "Rigorous spectral enclosure for operators on foam networks with singular junction lines is research-level numerical analysis.",
"nanopub": null
},
{
"id": "kelvin-problem-in-epithelia--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Lean-checked lower bounds on equal-volume partition area",
"kind": "formalization",
"method": "Prove a rigorous lower bound in Lean for surface area per cell of any partition of a 3-torus into N equal-volume cells, and state its distance from WP. Bound 1 (isoperimetric half-sphere): each cell has surface ≥ (36π)^{1/3} V^{2/3}, every interior face is shared by two cells, so partition area per cell ≥ ½(36π)^{1/3}V^{2/3} ≈ 2.418, against WP's ≈ 2.644 (half of 5.288), a gap of about 9%. This requires the 3D isoperimetric inequality, which as far as we know is not in Mathlib. Formalize it via the Brunn–Minkowski inequality for compact sets (Mathlib's measure theory supports a Prékopa–Leindler route) and the Minkowski-content definition of surface area, or state it as an explicit hypothesis in a first version. Bound 2: tighten using Kusner's face-count argument, that cells in a minimal foam have a bounded average number of faces, combined with a lower bound on the area of a polyhedral-like cell with k faces of constant curvature, giving a bound above 2.418.",
"steps": [
"Survey Mathlib for Brunn–Minkowski or Prékopa–Leindler; formalize the missing piece or state the isoperimetric inequality as a hypothesis in version 1.",
"Prove Bound 1 for partitions of the 3-torus (shared-face accounting as a finite-sum lemma).",
"Formalize the face-count argument and the per-cell bound for Bound 2, if Bound 1 completes.",
"Report the numerical gap to WP and Kelvin."
],
"decisive_result": "A Lean theorem bounding area per cell of any equal-volume 3-torus partition from below, unconditional if the isoperimetric inequality is formalized, otherwise conditional on it as a named hypothesis, with the gap to WP stated. Bound 2 above 2.418 would be new.",
"failure_signature": "The isoperimetric inequality proves too large to formalize in a month, leaving only the conditional version, which is correct but adds little beyond the statement on paper.",
"tools": [
"lean4",
"mathlib",
"geometric-measure-theory"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1007/s004540010071",
"title": "The Honeycomb Conjecture",
"year": 2001,
"note": "Hales 2001, The honeycomb conjecture (2D analogue)"
},
{
"url": "https://doi.org/10.1098/rspa.1992.0177",
"title": "The number of faces in a minimal foam",
"year": 1992,
"note": "Kusner 1992, The number of faces in a minimal foam"
},
{
"url": "https://doi.org/10.1080/09500839408241577",
"title": "A counter-example to Kelvin's conjecture on minimal surfaces",
"year": 1994,
"note": "Weaire and Phelan 1994"
}
],
"difficulty": "hard",
"missing": "Geometric measure theory in Mathlib is thin; an unconditional isoperimetric inequality in ℝ³ is a substantial formalization project.",
"nanopub": null
},
{
"id": "kelvin-problem-in-epithelia--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Scutoid frequency vs curvature in segmented 3D epithelia against minimal energy",
"kind": "data-test",
"method": "Biological part. For each 3D epithelial dataset (Gómez-Gálvez et al. 2018 salivary glands and egg chambers; PlantSeg Arabidopsis ovule and lateral-root volumes as a curved plant epithelium; other open segmented volumes on IDR and EMPIAR), compute per-cell apico-basal neighbour exchanges (scutoid indicator) and the local radial expansion ratio s = R_basal/R_apical from fitted tissue surfaces. Prediction: run a 3D Voronoi tube model (Gómez-Gálvez 2018) and a Surface Evolver relaxed-foam model on a tube or ellipsoid with the measured s, cell number and cell-size dispersion, giving the predicted scutoid fraction p̂(s) with simulation intervals. Pre-registered tolerance: supports energy minimization if observed scutoid fractions fall within the 95% simulation band at ≥ 80% of curvature bins pooled across datasets; refutes if observed fractions are outside the band in the same direction in ≥ 50% of bins. Also test the Gómez-Gálvez et al. 2022 biophysical principle linking connectivity to s.",
"steps": [
"Pre-register datasets' admission criteria (3D cell segmentation with apical and basal surfaces), binning of s and the tolerance.",
"Download and checksum datasets; compute scutoid indicators and s per cell.",
"Run the Voronoi and relaxed-foam simulations at measured parameters; build p̂(s) bands.",
"Compare and report the verdict per dataset and pooled."
],
"decisive_result": "Observed versus predicted scutoid fraction across curvature bins in at least three independent tissue types, with a pre-registered verdict on whether cells adopt minimal-energy packings for their curvature.",
"failure_signature": "Segmentation errors in apical or basal surfaces create spurious neighbour exchanges at rates comparable to predicted scutoid fractions (estimate the error rate on manually checked cells); then the test is measurement-limited.",
"tools": [
"3d-segmentation",
"cell-geometry",
"surface-evolver",
"data-test"
],
"datasets": [
{
"name": "Scutoids code and data (Gómez-Gálvez et al.)",
"url": "https://github.com/scutoids",
"note": "Voronoi tube model code and segmented data from the 2018 study"
},
{
"name": "Image Data Resource",
"url": "https://idr.openmicroscopy.org/",
"note": "Additional segmented 3D epithelial volumes"
},
{
"name": "EMPIAR",
"url": "https://www.ebi.ac.uk/empiar/",
"note": "Volume EM datasets of epithelia"
}
],
"references": [
{
"url": "https://doi.org/10.1038/s41467-018-05376-1",
"title": "Scutoids are a geometrical solution to three-dimensional packing of epithelia",
"year": 2018,
"note": "Gómez-Gálvez et al. 2018, Scutoids are a geometrical solution to three-dimensional packing of epithelia"
},
{
"url": "https://doi.org/10.1016/j.cels.2022.06.003",
"title": "A quantitative biophysical principle to explain the 3D cellular connectivity in curved epithelia",
"year": 2022,
"note": "Gómez-Gálvez et al. 2022, A quantitative biophysical principle to explain the 3D cellular connectivity in curved epithelia"
},
{
"url": "https://doi.org/10.7554/eLife.57613",
"title": "Accurate and versatile 3D segmentation of plant tissues at cellular resolution",
"year": 2020,
"note": "Wolny et al. 2020, PlantSeg 3D segmentation of plant tissues"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "kelvin-problem-in-epithelia--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Curvature-blinded scutoid frequency test",
"kind": "data-test",
"method": "Segment at least 30 epithelial volumes spanning curvature and thickness, with at least 200 cells each. Before inspecting topology, predict scutoid frequency from a vertex-model energy minimized at measured curvature, cell-number mismatch and apicobasal tension ratio. Test predictions on held-out tissues and compare to topology-preserving random rearrangements.",
"steps": [
"Validate segmentation topology against manually annotated cells with F1 above 0.95.",
"Freeze energy parameters using flat epithelia only.",
"Predict per-volume scutoid fraction and spatial localization.",
"Compare absolute error and likelihood to random and curvature-only baselines."
],
"decisive_result": "Energy minimization is supported if held-tissue scutoid-frequency error is below 0.1 and at least 50% lower than both baselines, with correct localization in 80% of volumes.",
"failure_signature": "Predicted frequency is systematically wrong by more than 0.2 or segmentation uncertainty changes scutoid calls by over 10%.",
"tools": [
"morpholib",
"3d-segmentation",
"mechanobiology-simulation",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Enough open volumetric epithelia with diverse measured curvatures.",
"nanopub": null
},
{
"id": "kelvin-problem-in-epithelia--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Certified local optimality of Weaire-Phelan",
"kind": "proof",
"method": "Restrict the mathematical problem to periodic partitions with fixed Weaire-Phelan combinatorics and finite-dimensional face perturbations. Compute the second variation of area under volume constraints, quotient translation and gauge modes, and certify positive definiteness with interval arithmetic; formalize the constrained Hessian argument in Lean.",
"steps": [
"Specify a finite periodic fundamental cell and exact constraint manifold.",
"Derive first- and second-variation formulas symbolically.",
"Bound geometric coefficients and Hessian eigenvalues by intervals.",
"Check a Lean theorem converting the certificate to strict local minimality in the restricted class."
],
"decisive_result": "A checked positive lower bound on the constrained Hessian after all symmetry zero modes are removed, proving strict local optimality for the declared perturbation class.",
"failure_signature": "A certified negative mode gives an explicit area-lowering perturbation, disproving local optimality in that class.",
"tools": [
"lean4",
"interval-arithmetic",
"surface-optimization"
],
"datasets": [],
"references": [],
"difficulty": "long-shot",
"missing": "A tractable exact representation of curved interfaces and a bridge from finite perturbations to arbitrary variations.",
"nanopub": null
},
{
"id": "kelvin-problem-in-epithelia--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Scutoid frequency against curvature in the 2018 data",
"kind": "data-test",
"method": "Use the supplementary tables of Gómez-Gálvez et al. 2018, not a re-segmentation, unless the tables lack per-tissue scutoid frequencies. If they lack frequencies, stop as data-unavailable rather than tracing figures. The prediction is the paper's own energy-minimizing scutoid fraction as a function of the measured curvature and cell aspect ratio. Pre-register success as either every curvature bin within 10 percentage points of the predicted fraction, or a Spearman correlation above 0.8 between predicted and observed fractions across bins. Bins are the paper's bins. A new segmentation pipeline is a different project and is not a rescue if the tables are insufficient.",
"steps": [
"Read the supplement and stop if frequencies and curvatures are not numeric.",
"Transcribe bin-wise scutoid fractions and curvatures with a checksum.",
"Compute the paper's predicted fraction at each bin.",
"Apply the 10-point tolerance and the Spearman threshold.",
"Support the energy account only if one of the two pre-registered conditions holds."
],
"decisive_result": "Either all bins within 10 percentage points of the predicted scutoid fraction, or Spearman correlation above 0.8.",
"failure_signature": "Both conditions fail. The energy-minimizing scutoid prediction then does not fit the paper's own binned data. Missing numeric tables are inconclusive, not a refutation.",
"tools": [
"3d-segmentation",
"cell-geometry",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/s41467-018-05376-1",
"title": "Scutoids are a geometrical solution to three-dimensional packing of epithelia",
"year": 2018,
"note": "Gómez-Gálvez et al. 2018, Scutoids are a geometrical solution to three-dimensional packing of epithelia"
}
],
"difficulty": "hard",
"missing": "If the supplement contains images rather than frequencies, the swarm lacks the numeric table and should not invent one by tracing.",
"nanopub": null
},
{
"id": "kelvin-problem-in-epithelia--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Anneal equal-volume foams against Weaire–Phelan",
"kind": "computation",
"method": "Use Brakke's Surface Evolver. First reproduce the surface area per unit volume of Kelvin's truncated octahedron and of the Weaire–Phelan foam in the same normalization, to a relative tolerance of 10^{-4}. Then anneal 200 random equal-volume periodic foams with 8 cells and 200 with 16 cells, from random seeds declared in advance, with a fixed iteration budget. A counterexample is a foam whose area per unit volume is lower than the reproduced Weaire–Phelan value by more than 10^{-4}. Failure to find one does not prove optimality. The approach settles the mathematical problem only in the counterexample direction. Reproducing the two known areas is a check, not a solution of Kelvin's problem.",
"steps": [
"Reproduce Kelvin and Weaire–Phelan areas to relative 10^{-4}.",
"Declare seeds and the iteration budget.",
"Anneal 200 eight-cell foams and 200 sixteen-cell foams.",
"Compare the best area with Weaire–Phelan.",
"Publish the best foam if it wins, otherwise the negative search record."
],
"decisive_result": "A periodic equal-volume foam whose Surface Evolver area is more than 10^{-4} below the reproduced Weaire–Phelan area, or an honest negative record of the 400-foam search.",
"failure_signature": "The search returns Weaire–Phelan as the best candidate. Kelvin's problem stays open. A failure of reproduction beyond 10^{-4} invalidates the comparison.",
"tools": [
"surface-evolver"
],
"datasets": [
{
"name": "Surface Evolver",
"url": "https://kenbrakke.com/evolver/evolver.html",
"note": "Ken Brakke's Surface Evolver, used for both reproduction and the random search."
}
],
"references": [
{
"url": "https://doi.org/10.1080/09500839408241577",
"title": "A counter-example to Kelvin's conjecture on minimal surfaces",
"year": 1994,
"note": "Weaire and Phelan 1994, A counter-example to Kelvin's conjecture"
},
{
"url": "https://doi.org/10.1007/s004540010071",
"title": "The Honeycomb Conjecture",
"year": 2001,
"note": "Hales 2001, The honeycomb conjecture, the settled two-dimensional analogue, not the target"
}
],
"difficulty": "long-shot",
"missing": "A certificate of global optimality. Annealing cannot supply one. Thirty years of search have not beaten Weaire–Phelan, so a month of random starts is unlikely to.",
"nanopub": null
},
{
"id": "kelvin-problem-in-epithelia--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Lean-check a lower bound and the Kelvin–Weaire gap",
"kind": "formalization",
"method": "Two checked statements, neither of which is global optimality. First, formalize a lower bound on surface area for a periodic partition into equal-volume cells that is strictly stronger than the single-cell isoperimetric inequality obtained by ignoring shared faces, and prove it in Lean with interval arithmetic where a radical is involved. Second, formalize the areas of the Kelvin truncated octahedron and the Weaire–Phelan cell in the same normalization and prove the Weaire–Phelan area is smaller, to a stated interval. The acceptance criteria allow a Lean-checked lower bound that does not identify the minimizer. Do not describe the comparison of two candidates as a lower bound on all partitions.",
"steps": [
"Prove a shared-face lower bound above the isolated-sphere area.",
"Compute Kelvin's cell area as an interval.",
"Compute the Weaire–Phelan area as an interval.",
"Prove the Weaire–Phelan interval lies strictly below Kelvin's.",
"State that global optimality remains open, and audit axioms."
],
"decisive_result": "A no-sorry lower bound strictly above the isolated-sphere area, and a no-sorry inequality that the Weaire–Phelan cell has smaller area than Kelvin's cell.",
"failure_signature": "The bound does not exceed the isolated-sphere area, or the interval arithmetic cannot separate the two foam areas. The approach has then not produced the promised checked comparison.",
"tools": [
"lean4",
"surface-evolver"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1080/09500839408241577",
"title": "A counter-example to Kelvin's conjecture on minimal surfaces",
"year": 1994,
"note": "Weaire and Phelan 1994, the comparison being machine-checked"
},
{
"url": "https://doi.org/10.1007/s004540010071",
"title": "The Honeycomb Conjecture",
"year": 2001,
"note": "Hales 2001, the two-dimensional theorem used as a model of what a finished proof looks like"
}
],
"difficulty": "hard",
"missing": "A proof of optimality for any three-dimensional partition. This approach does not claim one.",
"nanopub": null
}
]
},
{
"id": "mutant-morphant-discrepancy-prediction",
"title": "Predict which genes show mutant-morphant discrepancies",
"field": "biology",
"tier": "C",
"question": "Can a rule be learned and validated that predicts, from a gene's sequence, paralog structure and network position, whether its zebrafish CRISPR mutant will lack the phenotype its morpholino knockdown shows (transcriptional adaptation / genetic compensation)?",
"known_state": "Kok et al. (2015) found most morphant phenotypes absent in mutants; Rossi et al. (2015) showed genetic compensation; El-Brolosy et al. (2019) identified the mechanism as transcriptional adaptation triggered by mutant mRNA decay and upregulating sequence-similar genes. Which genes compensate is not predictable in advance. Already a Frontier Problem.",
"acceptance_criteria": [
"A pre-registered classifier using ZFIN mutant and morphant phenotype records as labels and INDRA CoGEx paralog, NMD-pathway and network features as predictors, with held-out accuracy and the chance baseline stated in advance.",
"A ranked list of untested genes with predicted compensation, so the prediction can be checked by future experiments."
],
"falsifiers": [
"Held-out accuracy at the baseline refutes the predictability claim with present features."
],
"required_capabilities": [
"indra-cogex",
"zfin",
"machine-learning",
"data-test"
],
"data_access": "ZFIN (open), INDRA CoGEx (public MCP), Ensembl paralog tables (open).",
"scales": [
"molecular",
"developmental"
],
"frontier_problem": "Zebrafish mutant morphant discrepancy attribution",
"related_claims": [
{
"id": "darwin1859-ch05-06",
"label": "Cave animals are related to surface fauna of their own continent",
"thinker": "Charles Darwin",
"status": "untested",
"evidence": "supports",
"theorem": false
},
{
"id": "levin-2022-tame-19",
"label": "Genes for morphogenesis and for learning and memory should overlap widely",
"thinker": "Michael Levin",
"status": "untested",
"evidence": "refutes",
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.devcel.2014.11.018",
"title": "Reverse Genetic Screening Reveals Poor Correlation between Morpholino-Induced and Mutant Phenotypes in Zebrafish",
"year": 2015,
"note": "Kok et al. 2015, Reverse genetic screening reveals poor correlation between morpholino-induced and mutant phenotypes in zebrafish"
},
{
"url": "https://doi.org/10.1038/nature14580",
"title": "Genetic compensation induced by deleterious mutations but not gene knockdowns",
"year": 2015,
"note": "Rossi et al. 2015, Genetic compensation induced by deleterious mutations but not gene knockdowns"
},
{
"url": "https://doi.org/10.1038/s41586-019-1064-z",
"title": "Genetic compensation triggered by mutant mRNA degradation",
"year": 2019,
"note": "El-Brolosy et al. 2019, Genetic compensation triggered by mutant mRNA degradation"
}
],
"nanopub": null,
"approaches": [
{
"id": "mutant-morphant-discrepancy-prediction--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Pre-registered classifier from ZFIN labels with paralog, NMD and network features",
"kind": "data-test",
"method": "Labels: from ZFIN downloads, genes with at least one morpholino phenotype and at least one phenotyped mutant allele. A gene is 'discrepant' if the mutant lacks the morphant's phenotype in the same anatomical system and stage window, and 'concordant' otherwise; this rule is fixed in advance and applied by script. Features, fixed in advance: (a) paralog structure (number of zebrafish paralogs including ohnologs from Ensembl Compara, maximum protein identity, and whether the closest paralog is co-expressed in the same tissue per ZFIN expression); (b) NMD susceptibility of the mutant allele (premature termination codon more than 50 nt upstream of the last exon junction, from allele annotation); (c) network features from INDRA CoGEx on human orthologs (degree, number of upstream regulators, membership in complexes); (d) gene length and expression level. Models: L1-regularized logistic regression and gradient-boosted trees, nested cross-validation, with a gene-family-grouped split so paralogs never span train and test. Pre-registered: success if held-out AUROC ≥ 0.70 against a feature-free baseline of 0.5 (permutation p < 0.01). Output: a ranked list of untested genes by predicted discrepancy probability.",
"steps": [
"Write result.json with label rule, features, model, split and threshold before computing labels.",
"Download ZFIN phenotype, morpholino and allele files; Ensembl paralog tables; query CoGEx; record versions and checksums.",
"Construct labels and features; run nested grouped cross-validation.",
"Report AUROC with CI, feature importances and the ranked prediction list for genes lacking mutant data."
],
"decisive_result": "Held-out AUROC with 95% CI under gene-family-grouped cross-validation against the 0.5 baseline, plus a published ranked list of predictions for genes whose mutants are not yet phenotyped, so future ZFIN records check the prediction.",
"failure_signature": "Fewer than about 150 genes have both morphant and mutant phenotypes comparable in system and stage, or AUROC CI includes 0.5: refutes predictability with present features.",
"tools": [
"indra-cogex",
"zfin",
"machine-learning",
"data-test"
],
"datasets": [
{
"name": "ZFIN data downloads",
"url": "https://zfin.org/downloads",
"note": "Morpholino and mutant phenotype annotations, allele details, expression"
},
{
"name": "Ensembl REST API (Compara homology)",
"url": "https://rest.ensembl.org/",
"note": "Zebrafish paralogs and protein identities"
},
{
"name": "INDRA CoGEx",
"url": "https://discovery.indra.bio/",
"note": "Network features of human orthologs"
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.devcel.2014.11.018",
"title": "Reverse Genetic Screening Reveals Poor Correlation between Morpholino-Induced and Mutant Phenotypes in Zebrafish",
"year": 2015,
"note": "Kok et al. 2015, poor correlation between morpholino and mutant phenotypes"
},
{
"url": "https://doi.org/10.1038/nature14580",
"title": "Genetic compensation induced by deleterious mutations but not gene knockdowns",
"year": 2015,
"note": "Rossi et al. 2015, Genetic compensation induced by deleterious mutations but not gene knockdowns"
},
{
"url": "https://doi.org/10.1038/s41586-019-1064-z",
"title": "Genetic compensation triggered by mutant mRNA degradation",
"year": 2019,
"note": "El-Brolosy et al. 2019, Genetic compensation triggered by mutant mRNA degradation"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "mutant-morphant-discrepancy-prediction--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Two-feature transcriptional-adaptation rule tested as a single hypothesis",
"kind": "data-test",
"method": "Instead of a flexible classifier, test the mechanism's own prediction as one pre-registered hypothesis with no tuning. From El-Brolosy et al. 2019 and Ma et al. 2019, transcriptional adaptation requires mutant mRNA decay (an NMD-triggering PTC allele) and upregulates genes sharing sequence similarity with the mutated gene. Rule: predict 'discrepant' iff (i) the phenotyped mutant allele is PTC-bearing and NMD-sensitive and (ii) a paralog with ≥ 50% nucleotide identity over the region of similarity exists. Predict 'concordant' otherwise, including promoter-less or whole-gene deletion alleles, which the mechanism predicts do not compensate. Compute the 2 × 2 contingency table against ZFIN-derived labels (same rule as approach 1), Fisher's exact test, and the odds ratio. Pre-registered: supports if OR ≥ 3 with a 95% CI lower bound > 1.5. A natural experiment within the data: genes with both a PTC allele and a promoter-less allele phenotyped, where the mechanism predicts discordance only for the PTC allele.",
"steps": [
"Pre-register the two-feature rule, identity threshold and statistics.",
"Annotate allele classes from ZFIN (PTC position relative to last exon junction; deletion type).",
"Compute paralog identities (Ensembl Compara plus nucleotide alignment of the similarity region).",
"Run Fisher's test; analyse the within-gene PTC versus promoter-less contrast separately."
],
"decisive_result": "Odds ratio for discrepancy given the two-feature mechanistic signature, and within-gene allele contrasts. A strong association, especially in within-gene contrasts, would turn transcriptional adaptation into a quantitative predictor; OR ≈ 1 would show it does not explain most discrepancies.",
"failure_signature": "Too few genes have promoter-less or whole-gene-deletion alleles phenotyped (fewer than 15) for the within-gene contrast; and allele PTC annotation is missing for many ZFIN alleles.",
"tools": [
"zfin",
"sequence-analysis",
"data-test"
],
"datasets": [
{
"name": "ZFIN data downloads",
"url": "https://zfin.org/downloads",
"note": "Allele types and phenotypes"
},
{
"name": "Ensembl REST API",
"url": "https://rest.ensembl.org/",
"note": "Paralog sequences and identities"
}
],
"references": [
{
"url": "https://doi.org/10.1038/s41586-019-1064-z",
"title": "Genetic compensation triggered by mutant mRNA degradation",
"year": 2019,
"note": "El-Brolosy et al. 2019"
},
{
"url": "https://doi.org/10.1038/s41586-019-1057-y",
"title": "PTC-bearing mRNA elicits a genetic compensation response via Upf3a and COMPASS components",
"year": 2019,
"note": "Ma et al. 2019, PTC-bearing mRNA elicits a genetic compensation response via Upf3a and COMPASS components"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "mutant-morphant-discrepancy-prediction--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Confound audit: do morpholino properties, not gene properties, predict it?",
"kind": "data-test",
"method": "A rival explanation of mutant–morphant discrepancy is morpholino artefact: off-target binding and p53-mediated toxicity give phenotypes the true loss of function would not. If so, discrepancy should be predicted by properties of the morpholino, not the gene. For each morpholino in ZFIN with a target and phenotype, compute: (a) the number of off-target sites in the zebrafish transcriptome with ≤ 2 mismatches (translation-start and splice-site windows, by alignment against Ensembl transcripts); (b) whether reported phenotypes include the nonspecific toxicity signature (necrosis, pericardial oedema, small head and eyes) in the same annotations; (c) morpholino type (translation-blocking or splice-blocking) and whether rescue or p53 co-knockdown is annotated. Fit a logistic model of discrepancy on morpholino features alone, gene features alone (approach 1's), and both, with nested grouped cross-validation. Pre-registered: the artefact explanation dominates if morpholino features alone reach AUROC within 0.03 of the combined model and gene features add < 0.03.",
"steps": [
"Pre-register the toxicity signature terms, off-target search parameters and comparison thresholds.",
"Extract morpholino sequences and targets from ZFIN; run the off-target alignment.",
"Fit morpholino-only, gene-only and combined models with the same splits as approach 1.",
"Report AUROCs and the incremental contribution of each feature group."
],
"decisive_result": "The relative predictive contribution of morpholino properties versus gene properties to mutant–morphant discrepancy. Dominance of morpholino features attributes most discrepancies to artefact; dominance of gene features supports compensation, the Frontier problem's attribution question.",
"failure_signature": "Morpholino and gene features are confounded (genes with paralogs are also targeted by morpholinos with more off-target sites), so the incremental AUROCs cannot be separated; report the overlap instead of a verdict.",
"tools": [
"zfin",
"sequence-analysis",
"machine-learning",
"data-test"
],
"datasets": [
{
"name": "ZFIN data downloads",
"url": "https://zfin.org/downloads",
"note": "Morpholino sequences, targets and phenotypes"
},
{
"name": "Ensembl REST API",
"url": "https://rest.ensembl.org/",
"note": "Transcript sequences for off-target search"
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.devcel.2014.11.018",
"title": "Reverse Genetic Screening Reveals Poor Correlation between Morpholino-Induced and Mutant Phenotypes in Zebrafish",
"year": 2015,
"note": "Kok et al. 2015"
},
{
"url": "https://doi.org/10.1038/nature14580",
"title": "Genetic compensation induced by deleterious mutations but not gene knockdowns",
"year": 2015,
"note": "Rossi et al. 2015"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "mutant-morphant-discrepancy-prediction--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Leakage-proof ZFIN discrepancy classifier",
"kind": "data-test",
"method": "Construct gene-level labels only where mutant and morpholino experiments test matched developmental stages and phenotypes. Predict discrepancy using paralog sequence similarity, NMD susceptibility, duplicate age, baseline expression, INDRA network redundancy and annotation counts. Hold out entire paralog families and publication years to prevent near-gene and literature leakage.",
"steps": [
"Double-label records and require adjudicated agreement above kappa 0.8.",
"Freeze positive, negative and uncertain definitions before feature extraction.",
"Require at least 100 positives and 100 matched negatives.",
"Compare calibrated gradient boosting with logistic and prevalence baselines in nested group cross-validation."
],
"decisive_result": "Predictability requires family-held-out AUROC at least 0.75, AUPRC at least twice prevalence and calibration slope between 0.8 and 1.2; bootstrap lower bounds must exceed baselines.",
"failure_signature": "Performance collapses below AUROC 0.6 under family or temporal holdout, showing apparent prediction was leakage or annotation bias.",
"tools": [
"indra-cogex",
"machine-learning",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "A clean matched ZFIN label set; phenotype ontology mappings and negative labels require manual curation.",
"nanopub": null
},
{
"id": "mutant-morphant-discrepancy-prediction--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Mechanistic compensation score from decay and paralogs",
"kind": "mixed",
"method": "Build an interpretable score from predicted premature termination and NMD, mutant-transcript abundance, sequence similarity to expressed paralogs, and signed compensatory paths in INDRA. Evaluate the prespecified score without learned weights, then use causal mediation on datasets containing mutant RNA-seq to test whether paralog upregulation mediates phenotype absence.",
"steps": [
"Define the score and thresholds before joining phenotype labels.",
"Select at least 50 mutants with RNA-seq, matched knockdown phenotype and paralog information.",
"Test score discrimination and paralog-expression mediation with bootstrap uncertainty.",
"Rank untested genes only after the held-out threshold is met."
],
"decisive_result": "Mechanistic support requires odds ratio at least 2 per score SD, AUROC above 0.7 and a positive mediated fraction above 0.3 with 95% interval excluding zero.",
"failure_signature": "NMD and paralog induction do not precede phenotype rescue, mediation is null, or discrimination is at chance.",
"tools": [
"indra-cogex",
"transcriptomics",
"data-test"
],
"datasets": [],
"references": [],
"difficulty": "hard",
"missing": "Public mutant RNA-seq paired to rigorously matched morphant and mutant phenotypes.",
"nanopub": null
},
{
"id": "mutant-morphant-discrepancy-prediction--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Test the NMD-plus-paralog rule on all doubly assayed genes",
"kind": "data-test",
"method": "Labels come from ZFIN downloads: Morpholinos.txt, CRISPR.txt, and phenoGeneCleanData_fish.txt. A gene is labeled only if it has both a morpholino phenotype and at least one mutant or CRISPR allele. It is a discrepancy if the morpholino is abnormal for an anatomy term and no mutant or CRISPR allele of that gene is abnormal for that term. Genes with no mutant are unlabeled, not counted as concordant. The rule, frozen first: predict discrepancy if and only if a representative frameshift or nonsense lesion lies more than 50 nucleotides upstream of the last exon-exon junction, the vertebrate nonsense-mediated-decay position rule, and Ensembl reports a paralogue with at least 70 percent identity over at least half the length. Hold out entire paralogue families, not single genes. The rule supports predictability if, on the held-out families, precision is at least 0.6 and the bootstrap interval for precision minus the base rate excludes 0. Report recall and do not require it to match precision.",
"steps": [
"Build labels from the three ZFIN files with the anatomy-term rule.",
"Annotate NMD position and Ensembl paralogue identity.",
"Hold out paralogue families.",
"Apply the Boolean rule.",
"Require held-out precision at least 0.6 and a bootstrap interval for precision minus base rate that excludes 0."
],
"decisive_result": "Held-out precision at least 0.6 with a bootstrap interval for precision minus the discrepancy base rate excluding 0.",
"failure_signature": "The interval for precision minus base rate contains 0. The rule then does not beat the base rate, which refutes predictability of this form.",
"tools": [
"zfin",
"indra-cogex",
"data-test"
],
"datasets": [
{
"name": "ZFIN downloads",
"url": "https://zfin.org/downloads",
"note": "Morpholinos.txt, CRISPR.txt and phenoGeneCleanData_fish.txt supply labels."
},
{
"name": "Ensembl",
"url": "https://www.ensembl.org/",
"note": "Exon structure for the 50-nucleotide rule and Compara paralogue identity."
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.devcel.2014.11.018",
"title": "Reverse Genetic Screening Reveals Poor Correlation between Morpholino-Induced and Mutant Phenotypes in Zebrafish",
"year": 2015,
"note": "Kok et al. 2015, the discrepancy observation"
},
{
"url": "https://doi.org/10.1038/nature14580",
"title": "Genetic compensation induced by deleterious mutations but not gene knockdowns",
"year": 2015,
"note": "Rossi et al. 2015, Genetic compensation induced by deleterious mutations but not gene knockdowns"
},
{
"url": "https://doi.org/10.1038/s41586-019-1064-z",
"title": "Genetic compensation triggered by mutant mRNA degradation",
"year": 2019,
"note": "El-Brolosy et al. 2019, the NMD and sequence-similarity mechanism"
}
],
"difficulty": "hard",
"missing": "The 50-nucleotide rule is the mammalian exon-junction convention. Zebrafish may differ, and a failure may mean the threshold is wrong rather than that compensation is unpredictable.",
"nanopub": null
},
{
"id": "mutant-morphant-discrepancy-prediction--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Family-held-out logistic model with an INDRA ablation",
"kind": "data-test",
"method": "Same labels and the same paralogue-family holdout. Fit an L2-regularized logistic regression. Feature block S: the NMD indicator, maximum paralogue identity, and paralogue count from Ensembl. Feature block I: INDRA degree and the presence of an INDRA edge to a paralogue. The full model uses both. Predictability is an area under the ROC curve above 0.70 on held-out families. The sequence block is the carrier if it beats the INDRA-only model by a DeLong test at p below 0.01 and the INDRA-only model does not itself clear 0.70. Nested cross-validation inside the training families chooses the L2 penalty. Do not use a random gene split, which leaks paralogues.",
"steps": [
"Reuse the ZFIN labels and the family partition.",
"Build sequence features and INDRA features.",
"Choose the L2 penalty inside training families only.",
"Score held-out AUROC for the full model, block S and block I.",
"DeLong-test block S against block I."
],
"decisive_result": "Held-out AUROC above 0.70 for the sequence block, and a DeLong p-value below 0.01 against the INDRA-only block if that comparison is needed to assign the signal.",
"failure_signature": "Held-out AUROC at or below 0.70 for every block. Present features then do not predict discrepancy. An AUROC that collapses when families are held out after looking strong on a random split is leakage, not a result.",
"tools": [
"zfin",
"indra-cogex",
"machine-learning",
"data-test"
],
"datasets": [
{
"name": "ZFIN downloads",
"url": "https://zfin.org/downloads",
"note": "Labels from Morpholinos.txt, CRISPR.txt and phenoGeneCleanData_fish.txt."
},
{
"name": "Ensembl",
"url": "https://www.ensembl.org/",
"note": "Paralogue and exon features."
},
{
"name": "INDRA CoGEx",
"url": "https://discovery.indra.bio/",
"note": "Degree and paralogue-edge features for the ablation."
}
],
"references": [
{
"url": "https://doi.org/10.1038/s41586-019-1064-z",
"title": "Genetic compensation triggered by mutant mRNA degradation",
"year": 2019,
"note": "El-Brolosy et al. 2019, the sequence mechanism the sequence block encodes"
},
{
"url": "https://doi.org/10.1016/j.devcel.2014.11.018",
"title": "Reverse Genetic Screening Reveals Poor Correlation between Morpholino-Induced and Mutant Phenotypes in Zebrafish",
"year": 2015,
"note": "Kok et al. 2015, the phenotype discrepancy being predicted"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "mutant-morphant-discrepancy-prediction--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Score untested morpholino genes and check calibration",
"kind": "data-test",
"method": "The acceptance criteria ask for a ranked list of genes not yet tested as mutants. Fit the one-parameter mechanistic score P(discrepancy) = logistic(a + b times maximum paralogue identity + c times the NMD indicator) on the training families from the Boolean-rule split. The score must beat an intercept-only model on held-out families by a likelihood-ratio test at p below 0.01, and the reliability diagram of deciles must be monotone. Only then score genes that have a morpholino phenotype and no mutant allele. Publish the top 50 with the three features and the score. The list is a prediction, not a confirmation. Future experiments are outside the approach. A non-monotone reliability diagram fails the list even if the likelihood-ratio test passes.",
"steps": [
"Fit a, b and c on training families only.",
"Likelihood-ratio test against the intercept on held-out families.",
"Build the decile reliability diagram and require monotonicity.",
"Score morpholino genes that have no mutant allele.",
"Publish the top 50 with features and scores."
],
"decisive_result": "A likelihood-ratio p-value below 0.01 against the intercept, a monotone decile reliability diagram, and a published top-50 list of untested genes.",
"failure_signature": "The likelihood-ratio test misses 0.01, or the reliability diagram is not monotone. No list is published in those cases.",
"tools": [
"zfin",
"machine-learning",
"data-test"
],
"datasets": [
{
"name": "ZFIN downloads",
"url": "https://zfin.org/downloads",
"note": "Morpholino genes without mutant alleles are the prediction set; doubly assayed genes are the calibration set."
},
{
"name": "Ensembl",
"url": "https://www.ensembl.org/",
"note": "Identity and NMD features for the scored genes."
}
],
"references": [
{
"url": "https://doi.org/10.1038/s41586-019-1064-z",
"title": "Genetic compensation triggered by mutant mRNA degradation",
"year": 2019,
"note": "El-Brolosy et al. 2019, the mechanism the score encodes"
},
{
"url": "https://doi.org/10.1038/nature14580",
"title": "Genetic compensation induced by deleterious mutations but not gene knockdowns",
"year": 2015,
"note": "Rossi et al. 2015, genetic compensation as the phenotype being ranked"
}
],
"difficulty": "hard",
"missing": "Laboratory tests of the top 50. The swarm can publish the list and cannot confirm it.",
"nanopub": null
}
]
},
{
"id": "trabecular-stress-alignment-wolffs-law",
"title": "Do trabeculae align with principal stress, and is that alignment causal?",
"field": "biology",
"tier": "B",
"question": "In loaded cancellous bone (femoral head, vertebral body, calcaneus), is the mean trabecular orientation within a small angular tolerance of the principal stress directions computed from the bone's own geometry and a physiological load case? Is the alignment stronger than expected from a load-independent growth rule, and does it re-orient after experimentally altered loading?",
"known_state": "Wolff (1892) and Thompson (1917, ch. 16) hold that trabeculae lie along trajectories of principal stress because material off those lines is shear-loaded and remodelled away. Finite-element and remodelling models reproduce aligned architectures (Huiskes et al. 2000), and in vivo load-orientation experiments in bird and sheep joints report re-orientation (Pontzer et al. 2006; Barak et al. 2011), but results are mixed on how tight the alignment is, on whether alignment reflects the loads of habitual or peak activity, and on whether a purely genetic or developmental pattern would give the same fabric. Christen et al. 2014 find human remodelling load-driven but not lazy. No standardized, pre-registered cross-site and cross-species estimate of the stress-fabric angle with a developmental null exists.",
"acceptance_criteria": [
"A pre-registered test on at least three open micro-CT datasets of loaded bone sites from at least two species: compute the mean-intercept-length fabric tensor, compute principal stresses from a stated finite-element load case, report the angular error distribution, and compare it against a null with the same density field but randomized fabric directions and against a growth-rule null. State the angular tolerance and the pass threshold in advance."
],
"falsifiers": [
"A median fabric-to-stress angular error not distinguishable from the randomized-fabric null, or insensitive to the choice of load case, would refute stress alignment as the dominant organizing principle at the tested sites.",
"Reproducible failure to re-orient after a documented change in joint loading would refute the causal claim."
],
"required_capabilities": [
"micro-ct-morphometry",
"finite-element-analysis",
"data-test"
],
"data_access": "Open micro-CT of femora, vertebrae and calcanei on MorphoSource, Dryad and Zenodo; paired load data are the main uncertainty and have to be established by scouting before the test is pre-registered.",
"scales": [
"tissue",
"organ"
],
"frontier_problem": null,
"related_claims": [
{
"id": "thompson1917-ch16-05",
"label": "Trabeculae of cancellous bone follow the lines of compressive and tensile stress",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": null,
"theorem": true
},
{
"id": "thompson1917-ch16-06",
"label": "Shear removes obliquely laid trabeculae, leaving those aligned with stress lines",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "thompson1917-ch16-07",
"label": "Trabecular architecture remodels to new force patterns after malunited fracture",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "thompson1917-ch16-16",
"label": "Skeletal form is largely a diagram of mechanical stress",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1038/35015116",
"title": "Effects of mechanical forces on maintenance and adaptation of form in trabecular bone",
"year": 2000,
"note": "Huiskes et al. 2000, Effects of mechanical forces on maintenance and adaptation of form in trabecular bone"
},
{
"url": "https://doi.org/10.1038/ncomms5855",
"title": "Bone remodelling in humans is load-driven but not lazy",
"year": 2014,
"note": "Christen et al. 2014, Bone remodelling in humans is load-driven but not lazy"
},
{
"url": "https://doi.org/10.1242/jeb.01971",
"title": "Trabecular bone in the bird knee responds with high sensitivity to changes in load orientation",
"year": 2006,
"note": "Pontzer et al. 2006, Trabecular bone in the bird knee responds with high sensitivity to changes in load orientation"
},
{
"url": "https://doi.org/10.1016/j.bone.2011.08.020",
"title": "A Wolff in sheep's clothing: Trabecular bone adaptation in response to changes in joint loading orientation",
"year": 2011,
"note": "Barak et al. 2011, A Wolff in sheep's clothing: trabecular bone adaptation in response to changes in joint loading orientation"
}
],
"nanopub": null,
"approaches": [
{
"id": "trabecular-stress-alignment-wolffs-law--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Pre-registered fabric-versus-principal-stress angle on open femoral micro-CT",
"kind": "data-test",
"method": "On open micro-CT of loaded cancellous bone, compute the local fabric tensor (mean intercept length, via BoneJ anisotropy) in a grid of cubic volumes of interest (edge about 5 mm in human femoral heads, scaled to trabecular spacing in other species) and the local principal stress tensor from a homogenized finite-element model of the same bone under a stated physiological load case (hip contact force direction and magnitude from Bergmann et al. 2001 / OrthoLoad for human femur). The statistic is, per volume, the angle between the primary fabric eigenvector and the principal stress direction of largest absolute magnitude, folded to 0-90 degrees. Two nulls are fixed in advance: (a) randomized fabric with the same density field, for which an isotropic random axis has median angle 60 degrees (P(angle <= theta) = 1 - cos theta); (b) a load-independent growth-rule null in which fabric follows a geometric field of the bone alone (local direction of the shaft or neck axis, and the gradient of distance to the cortical shell). Load-case sensitivity is tested by sweeping the load direction over a cone of 60 degrees around the physiological direction and recording the direction that minimizes median angular error; alignment is credited to load only if the best direction lies inside the physiological range and the error rises away from it.",
"steps": [
"Scout and register datasets: human proximal femora (Twente/VPHOP femur database, FEMHALS if a data-use agreement can be signed, the CC BY Figshare femur section), plus at least one non-human loaded site on MorphoSource (primate or ungulate femoral head, calcaneus or vertebra) with licence recorded.",
"Pre-register in test-data/claim-tests/thompson-growth-form/thompson1917-ch16-05/: VOI size, fabric method, FE material law (density-dependent isotropic modulus, so fabric does not enter the stress computation), load case, angular tolerance (median error <= 20 degrees), and the pass rule (median error below both nulls at p < 0.01 by permutation, in at least three sites from two species).",
"Segment with a fixed global threshold (Otsu per scan, recorded), compute MIL fabric per VOI with BoneJ, discard VOIs with degree of anisotropy below 1.1 where the primary axis is undefined.",
"Build voxel or tetrahedral homogenized FE meshes (FEniCSx), apply the load case, extract principal stresses per VOI.",
"Compute angular error distributions, the two nulls (1,000 permutations for the randomized null), and the load-direction sweep; report all three sites separately and pooled.",
"Stratify by local stress anisotropy (sigma1 - sigma3)/(|sigma1| + |sigma3|): the stress-alignment hypothesis predicts error falls as anisotropy rises; the growth-rule null predicts no such dependence."
],
"decisive_result": "Median fabric-to-stress angle at or below 20 degrees in at least three sites from two species, significantly below both the randomized-fabric null (median 60 degrees) and the geometric growth-rule null, with the best-fitting load direction inside the physiological cone and error decreasing with stress anisotropy.",
"failure_signature": "Median angle not distinguishable from the growth-rule null, or the error minimum in the load sweep lying outside the physiological cone or being flat (load-case insensitive); or fewer than three usable loaded sites with licences after scouting, in which case the approach stalls rather than refutes.",
"tools": [
"bonej",
"fenicsx",
"python",
"morphosource",
"pre-registered-claim-test"
],
"datasets": [
{
"name": "Database of femur samples (VPHOP, University of Twente)",
"url": "https://doi.org/10.4121/uuid:4ae59365-92f0-480b-a899-ade34bc84a00",
"note": "33 human cadaver proximal femora with reconstructed micro-CT, FE input files and per-element microstructural parameters; open for research with citation. Check whether fabric eigenvectors are among the per-element parameters."
},
{
"name": "FEMHALS: multiscale imaging and mechanical testing of human proximal femora",
"url": "https://doi.org/10.5281/zenodo.21494457",
"note": "Micro-CT of proximal femur and femoral neck for 94 femora with mechanical tests; raw scans reported to require access approval and a data-use agreement, which the operator would have to sign."
},
{
"name": "Human femur section (female, 77 years) CT stack",
"url": "https://doi.org/10.6084/m9.figshare.14097263",
"note": "CC BY 4.0 micro-CT stack at 10 micrometre voxels; small, useful for calibrating the pipeline, not as a full loaded site."
},
{
"name": "OrthoLoad",
"url": "https://orthoload.com",
"note": "Instrumented-implant hip contact forces for the human load case."
},
{
"name": "MorphoSource",
"url": "https://www.morphosource.org",
"note": "Non-human micro-CT of loaded trabecular sites; availability and licence to be scouted per specimen."
}
],
"references": [
{
"url": "https://doi.org/10.1016/S0021-9290(01)00040-9",
"title": "Hip contact forces and gait patterns from routine activities",
"year": 2001,
"note": "Bergmann et al. 2001, Hip contact forces and gait patterns from routine activities"
},
{
"url": "https://doi.org/10.1016/j.bone.2010.08.023",
"title": "BoneJ: Free and extensible bone image analysis in ImageJ",
"year": 2010,
"note": "Doube et al. 2010, BoneJ: free and extensible bone image analysis in ImageJ"
},
{
"url": "https://doi.org/10.1038/35015116",
"title": "Effects of mechanical forces on maintenance and adaptation of form in trabecular bone",
"year": 2000,
"note": "Huiskes et al. 2000, Effects of mechanical forces on maintenance and adaptation of form in trabecular bone"
},
{
"url": "https://doi.org/10.1371/journal.pone.0078781",
"title": "Trabecular Bone Structure Correlates with Hand Posture and Use in Hominoids",
"year": 2013,
"note": "Tsegai et al. 2013, Trabecular bone structure correlates with hand posture and use in hominoids"
}
],
"difficulty": "hard",
"missing": "Licensed non-human micro-CT of loaded sites with known habitual loads; FEMHALS raw scans need an operator-signed data-use agreement; homogenized FE of whole bones is computationally heavy but feasible.",
"nanopub": null
},
{
"id": "trabecular-stress-alignment-wolffs-law--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Calibrate the growth-rule null and re-orientation by remodelling simulation",
"kind": "simulation",
"method": "The problem asks whether alignment is stronger than a load-independent growth rule would produce, but the size of that null is unknown. Simulate trabecular architecture on 2D and coarse 3D voxel domains (a femoral-head section and a rectangular block) under three rule families: (1) a Huiskes-type remodelling rule driven by strain energy density sensed by osteocytes with spatial influence decay; (2) load-independent rules (reaction-diffusion Turing-type patterning of bone deposition, and growth along a fixed developmental field such as radial distance from an ossification centre); (3) mixed rules with a tunable load weight w from 0 to 1. For each run compute the same fabric-to-principal-stress angle statistic as approach 1. Then rotate the applied load by 13 and 20 degrees, matching the magnitudes used in the guineafowl and sheep experiments (Pontzer et al. 2006; Barak et al. 2011), continue the simulation, and record the fabric re-orientation angle and its time constant.",
"steps": [
"Implement the remodelling loop in Python with a voxel FE solver (FEniCSx or a sparse linear-elastic voxel solver), verify against the published qualitative outcomes of Huiskes et al. 2000 (aligned struts under uniaxial load, re-alignment after load rotation).",
"Implement the load-independent rules and the mixed-rule family; fix seeds and record all parameters.",
"For each rule and 20 seeds, compute the distribution of fabric-to-stress angles and the stress-anisotropy dependence.",
"Apply load rotation and record re-orientation angle versus simulated time.",
"Publish a lookup table: expected median angle and anisotropy slope for w in {0, 0.25, 0.5, 0.75, 1}, to be used as the pre-registered growth-rule null in approach 1."
],
"decisive_result": "A table showing that the alignment statistic separates load-driven (w = 1) from load-independent (w = 0) rules by more than the seed-to-seed spread, with the minimum load weight that produces a median angle below 20 degrees, and predicted re-orientation magnitudes for the two experimental rotations.",
"failure_signature": "If load-independent rules on the same geometry produce median angles within the seed spread of the load-driven rule, the alignment statistic cannot discriminate causes and approach 1 must be redesigned around re-orientation; if results change by more than 10 degrees with voxel resolution or VOI size, the simulation is not converged.",
"tools": [
"python",
"fenicsx",
"frontier_simulate"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/35015116",
"title": "Effects of mechanical forces on maintenance and adaptation of form in trabecular bone",
"year": 2000,
"note": "Huiskes et al. 2000, Effects of mechanical forces on maintenance and adaptation of form in trabecular bone"
},
{
"url": "https://doi.org/10.1016/0021-9290(87)90058-3",
"title": "Trabecular bone density and loading history: Regulation of connective tissue biology by mechanical energy",
"year": 1987,
"note": "Carter, Fyhrie and Whalen 1987, Trabecular bone density and loading history: regulation of connective tissue biology by mechanical energy"
},
{
"url": "https://doi.org/10.1242/jeb.01971",
"title": "Trabecular bone in the bird knee responds with high sensitivity to changes in load orientation",
"year": 2006,
"note": "Pontzer et al. 2006, Trabecular bone in the bird knee responds with high sensitivity to changes in load orientation"
},
{
"url": "https://doi.org/10.1016/j.bone.2011.08.020",
"title": "A Wolff in sheep's clothing: Trabecular bone adaptation in response to changes in joint loading orientation",
"year": 2011,
"note": "Barak et al. 2011, A Wolff in sheep's clothing"
}
],
"difficulty": "tractable",
"missing": "No wet data needed; this calibrates nulls and does not by itself settle whether real bone is aligned.",
"nanopub": null
},
{
"id": "trabecular-stress-alignment-wolffs-law--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Lean proof that shear-driven rotation makes principal directions the stable fabric",
"kind": "formalization",
"method": "Formalize the mechanism in claim thompson1917-ch16-06. For a plane stress state with principal stresses sigma1 > sigma2 and principal angle theta_p, the shear traction on a strut at angle theta is tau(theta) = ((sigma1 - sigma2)/2) sin(2(theta - theta_p)). Compare two rotation rules, k > 0. Rule A (signed shear): d theta/dt = -k tau(theta). Rule B (descent on shear energy): d theta/dt = -k d(tau^2)/d theta, which is proportional to -sin(4(theta - theta_p)). Prove: (i) under both rules the principal directions theta_p + m pi/2 are equilibria; (ii) under rule A the linearization -k(sigma1 - sigma2) cos(2(theta - theta_p)) is negative at theta_p and positive at theta_p + pi/2, so only one principal family is stable and Thompson's two orthogonal families (ch16-05) cannot both arise from it; (iii) under rule B both principal directions are asymptotically stable, the diagonals theta_p + pi/4 are the only other (unstable) equilibria, and V = tau^2 is a Lyapunov function non-increasing along trajectories; (iv) if sigma1 = sigma2 every angle is an equilibrium, so alignment is undefined under equal principal stresses. Part (ii)-(iii) says which reading of 'displaced by shear' is consistent with crossing tension and compression families; part (iv) yields the data prediction used in approach 1 that alignment strength should rise with stress anisotropy.",
"steps": [
"Extend ThompsonLaws/Mechanics.lean with the shear traction formula derived from the existing stressAction (rotation of the symmetric tensor), proving tau is zero exactly on principal directions.",
"Prove the equilibrium characterization and the sign of the derivative at each equilibrium for rules A and B using Real.sin and Real.cos lemmas in Mathlib.",
"Prove V = tau^2 is non-increasing along any differentiable solution of rule B (HasDerivAt plus chain rule).",
"Prove the degenerate case sigma1 = sigma2.",
"Write the report entry under lean/reports with no sorry and standard axioms only, and link it to thompson1917-ch16-06."
],
"decisive_result": "A compiled Lean module with no sorry proving the four statements, recorded as formalized for thompson1917-ch16-06 in the lean reports, including the result that only the shear-energy rule makes both orthogonal families stable.",
"failure_signature": "Needing a sorry for the stability or Lyapunov step, or a statement weakened to the point that it only restates tau = 0 on principal axes (already implied by the existing ch16-05 theorem).",
"tools": [
"lean4",
"mathlib"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/0021-9290(87)90058-3",
"title": "Trabecular bone density and loading history: Regulation of connective tissue biology by mechanical energy",
"year": 1987,
"note": "Carter, Fyhrie and Whalen 1987, mechanical-energy regulation of trabecular density"
}
],
"difficulty": "tractable",
"missing": "A proof shows the rule implies alignment, not that bone follows the rule; it does not address causality in organisms.",
"nanopub": null
},
{
"id": "trabecular-stress-alignment-wolffs-law--claude-opus-5-5--4",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 4,
"title": "Meta-analysis of load-reorientation experiments for the causal arm",
"kind": "data-test",
"method": "The causal falsifier (failure to re-orient after altered loading) can be tested on published experiments without new animals. Extract every controlled in vivo experiment in which joint load orientation or magnitude was changed and trabecular orientation measured against a control limb or group (starting with Pontzer et al. 2006 and Barak et al. 2011, plus studies found by a pre-registered search string in Europe PMC). The effect size is the change in primary fabric orientation in degrees relative to control, with the imposed load rotation as moderator. Fit a random-effects meta-regression of observed re-orientation on imposed rotation; Wolff's law predicts a slope near 1 (fabric tracks load) and an intercept near 0. Include studies with null results and record risk of bias (blinding, sample size).",
"steps": [
"Pre-register the search string, inclusion criteria (controlled design, quantitative fabric angle, stated load-change angle) and the model.",
"Search Europe PMC and screen with two independent agents; resolve disagreements by a third.",
"Extract angles, SDs and n with page-level provenance; contact-free only, so studies without dispersion are recorded but excluded from pooling.",
"Fit the random-effects meta-regression (metafor, REML) and report the slope with 95% CI, heterogeneity I^2, and an Egger test."
],
"decisive_result": "Pooled slope of fabric re-orientation on imposed load rotation with 95% CI excluding 0 (causal tracking) and its distance from 1 (complete tracking), from at least five independent experiments.",
"failure_signature": "Fewer than five eligible experiments, or a CI including 0, which would leave the causal claim unsupported; if eligible studies overwhelmingly report positive results with small n and Egger asymmetry, the pooled estimate is not credible.",
"tools": [
"europe-pmc",
"r-metafor",
"literature-extraction"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1242/jeb.01971",
"title": "Trabecular bone in the bird knee responds with high sensitivity to changes in load orientation",
"year": 2006,
"note": "Pontzer et al. 2006, Trabecular bone in the bird knee responds with high sensitivity to changes in load orientation"
},
{
"url": "https://doi.org/10.1016/j.bone.2011.08.020",
"title": "A Wolff in sheep's clothing: Trabecular bone adaptation in response to changes in joint loading orientation",
"year": 2011,
"note": "Barak et al. 2011, A Wolff in sheep's clothing"
},
{
"url": "https://doi.org/10.1038/ncomms5855",
"title": "Bone remodelling in humans is load-driven but not lazy",
"year": 2014,
"note": "Christen et al. 2014, Bone remodelling in humans is load-driven but not lazy"
}
],
"difficulty": "hard",
"missing": "The number of controlled re-orientation experiments with extractable angles and dispersion may be below five; no wet lab to add new ones.",
"nanopub": null
},
{
"id": "trabecular-stress-alignment-wolffs-law--gemini-4-argon--1",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 1,
"title": "Pre-registered micro-CT fabric vs voxel-FE principal stress test against growth nulls",
"kind": "data-test",
"method": "Execute a pre-registered cross-species data test on 30 specimens spanning three anatomical sites (femoral head, lumbar vertebral body, and calcaneus) across at least two mammalian species retrieved from MorphoSource and Dryad micro-CT repositories. For each scan, segment the trabecular core and cortical shell, compute the local second-rank mean-intercept-length (MIL) and star-volume-distribution fabric tensors H across a 5 mm grid using BoneJ algorithms, and run linear voxel-based finite-element (FE) analyses on the segmented whole-bone geometry under two canonical physiological boundary conditions (habitual stance and peak locomotor joint reaction). Extract the principal stress eigenvectors e_i(sigma) and fabric eigenvectors e_i(H) in each subvolume.\n\nTo test whether alignment reflects mechanical load rather than developmental geometry or density gradients, compare the observed unsigned geodesic angular error theta = arccos(|e_1(H) . e_1(sigma)|) against two explicit null models in every specimen: (1) a density-matched rotational permutation null that preserves the local bone volume fraction field BV/TV(x) while randomly rotating H(x) on SO(3), and (2) a load-independent developmental growth-axis null where trajectories follow the Laplacian harmonic gradient between the epiphyseal growth plate and diaphyseal cortex.",
"steps": [
"Select and checksum 30 open micro-CT scans covering femoral head, vertebral body, and calcaneus across Homo sapiens, Ovis aries, and primate or rodent taxa from MorphoSource and Dryad.",
"Pre-register the subvolume grid size, segmentation thresholds, canonical joint-loading boundary conditions, and angular tolerance before running FE or fabric pairing.",
"Compute the local 3x3 fabric tensor H via star-volume and MIL methods and solve the voxel-FE linear elasticity system for the principal stress directions e_1(sigma).",
"Evaluate the volume-weighted median angular error theta and perform paired one-sided Wilcoxon signed-rank tests against the SO(3)-rotated fabric null and the harmonic growth-axis null, plus a load-case swap sensitivity test."
],
"decisive_result": "Across the 30 specimens, a pooled volume-weighted median fabric-to-principal-stress angle theta <= 15 degrees that beats both the SO(3) randomized-fabric null and the harmonic growth-axis null (Wilcoxon p < 0.001) and increases by at least 5 degrees under swapped non-physiological load cases settles static stress alignment in the affirmative.",
"failure_signature": "A median angular error theta > 25 degrees, failure to improve significantly over the load-independent harmonic growth-axis null (p >= 0.05), or insensitivity (< 2 degrees change) when boundary loads are rotated by 45 degrees refutes stress alignment as the primary determinant of static trabecular fabric.",
"tools": [
"micro-ct-morphometry",
"finite-element-analysis",
"data-test"
],
"datasets": [
{
"name": "MorphoSource 3D trabecular and whole-bone micro-CT collections",
"url": "https://www.morphosource.org/",
"note": "High-resolution micro-CT volumes of mammalian femora, calcanei, and vertebrae for paired fabric tensor and whole-bone FE analysis"
},
{
"name": "Dryad micro-CT cortical and trabecular segmentation dataset (10.5061/dryad.b2rbnzsq4)",
"url": "https://datadryad.org/dataset/doi:10.5061/dryad.b2rbnzsq4",
"note": "Segmented micro-CT volumes of cortical bone, trabecular bone, and medullary pores for benchmarking fabric extraction"
}
],
"references": [
{
"url": "https://doi.org/10.1038/35015116",
"title": "Effects of mechanical forces on maintenance and adaptation of form in trabecular bone",
"year": 2000,
"note": "Huiskes et al. 2000, Effects of mechanical forces on maintenance and adaptation of form in trabecular bone"
},
{
"url": "https://doi.org/10.1038/ncomms5855",
"title": "Bone remodelling in humans is load-driven but not lazy",
"year": 2014,
"note": "Christen et al. 2014, Bone remodelling in humans is load-driven but not lazy"
},
{
"url": "https://doi.org/10.1016/j.bone.2010.08.023",
"title": "BoneJ: Free and extensible bone image analysis in ImageJ",
"year": 2010,
"note": "Doube et al. 2010, BoneJ: Free and extensible bone image analysis in ImageJ"
}
],
"difficulty": "hard",
"missing": "Subject-specific in vivo joint force telemetry paired to the exact scanned museum specimens is largely absent from public archives, requiring canonical physiological load cases and sensitivity bounds.",
"nanopub": null
},
{
"id": "trabecular-stress-alignment-wolffs-law--gemini-4-argon--2",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 2,
"title": "Lean 4 proof of coaxiality at Cowin remodeling equilibrium and shear-remodeling PDE",
"kind": "formalization",
"method": "Extend `ThompsonLaws/Mechanics.lean` (which already proves `principalDirections_orthogonal` for `thompson1917-ch16-05`) to formalize Cowin's (1986) continuum fabric-evolution law and Thompson's shear-removal mechanism (`thompson1917-ch16-06`). In Lean 4, define a symmetric positive-definite fabric tensor H, an orthotropic elasticity fourth-rank tensor C(H) coaxial with H, and an alignment evolution law dH/dt = f(sigma, H) driven by the deviatoric stress-fabric commutator [sigma, H] = sigma H - H sigma (which measures off-axis shear stress in the principal fabric frame).\n\nProve machine-checked theorems establishing that: (1) any stationary equilibrium dH/dt = 0 under non-degenerate principal stresses implies [sigma, H] = 0, so the eigenvectors of H and sigma are coaxial and trabecular trajectories cross orthogonally; and (2) couple this in a 3D numerical PDE simulation (`frontier_simulate`) to published altered-loading parameters from bird knee and sheep tarsal experiments (Pontzer et al. 2006; Barak et al. 2011) to test whether a pure shear-rejection law predicts the observed 10-to-18 degree trabecular reorientation within the experimental 4-to-6 week window.",
"steps": [
"Define symmetric 3x3 stress and fabric matrices and the commutator [sigma, H] in Lean 4 on top of Mathlib's matrix spectrum API.",
"Prove with zero sorry that vanishing of the off-diagonal shear components in the fabric frame is equivalent to [sigma, H] = 0 and implies shared orthogonal eigenvectors.",
"Prove Lyapunov stability of the aligned fixed point under the dissipative remodeling flow dH/dt = -k [H, [H, sigma]].",
"Run a 3D numerical integration of the formalized remodeling law under a 20-degree load-axis rotation matched to the published sheep and bird joint-redirection protocols and compare predicted vs reported Δtheta."
],
"decisive_result": "A Lean 4 module compiling with zero `sorry` that proves coaxiality and Lyapunov stability of the zero-shear fabric equilibrium, paired with a numerical trajectory matching the published in vivo reorientation angles (Pontzer 2006; Barak 2011) within 3 degrees.",
"failure_signature": "Discovery of stable non-coaxial equilibria under positive-definite orthotropic elasticity in the formal analysis, or a simulated reorientation rate that misses published in vivo remodeling timescales by more than an order of magnitude under physiologically bounded bone apposition rates.",
"tools": [
"lean4",
"frontier_simulate",
"finite-element-analysis"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1115/1.3138584",
"title": "Wolff’s Law of Trabecular Architecture at Remodeling Equilibrium",
"year": 1986,
"note": "Cowin 1986, Wolff's Law of Trabecular Architecture at Remodeling Equilibrium"
},
{
"url": "https://doi.org/10.1242/jeb.01971",
"title": "Trabecular bone in the bird knee responds with high sensitivity to changes in load orientation",
"year": 2006,
"note": "Pontzer et al. 2006, Trabecular bone in the bird knee responds with high sensitivity to changes in load orientation"
},
{
"url": "https://doi.org/10.1016/j.bone.2011.08.020",
"title": "A Wolff in sheep's clothing: Trabecular bone adaptation in response to changes in joint loading orientation",
"year": 2011,
"note": "Barak et al. 2011, A Wolff in sheep's clothing: Trabecular bone adaptation in response to changes in joint loading orientation"
}
],
"difficulty": "tractable",
"missing": "Raw longitudinal in vivo micro-CT time series of experimentally load-redirected joints are not deposited in full 3D voxel form, so the causal reorientation comparison relies on published summary tables from Pontzer et al. (2006) and Barak et al. (2011) rather than raw voxel re-registration.",
"nanopub": null
},
{
"id": "trabecular-stress-alignment-wolffs-law--gemini-4-argon--3",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 3,
"title": "Single-trabecula shear-resorption test on longitudinal in vivo micro-CT scans",
"kind": "data-test",
"method": "Test the causal core of `thompson1917-ch16-06` and `thompson1917-ch16-07` directly at the single-strut level by registering baseline (t_0) and follow-up (t_1) 3D micro-CT volumes from altered-loading or malunited-fracture cohorts. For each individual trabecular strut segmented at t_0, compute its axial unit tangent vector n_strut and solve the whole-bone micro-FE problem under the experimental load vector to obtain the local resolved shear stress tau_s = ||sigma . n_strut - (n_strut . sigma . n_strut) n_strut|| and normal axial strain energy density U_n.\n\nOverlay the registered t_1 scan to classify every surface voxel into formation, quiescence, or resorption (Christen et al. 2014). Fit a pre-registered logistic mixed-effects model testing whether oblique struts with high shear-to-normal ratio tau_s / |sigma_n| experience selective osteoclastic resorption on their shear-loaded flanks and apposition rotating n_strut toward the nearest principal stress eigenvector e_1(sigma), compared against a scalar strain-energy remodeling model that ignores stress obliquity.",
"steps": [
"Locate and retrieve paired time-lapse in vivo micro-CT volumes (baseline and post-loading follow-up) of loaded trabecular bone from public repositories or author depositions.",
"Perform rigid and bone-boundary landmark registration between t_0 and t_1 scans to map surface formation and resorption sites at voxel resolution.",
"Decompose the micro-FE stress tensor on each t_0 strut into axial normal stress sigma_n and transverse shear stress tau_s.",
"Compare predictive log-likelihood (AUROC) of the directional shear-removal model (`thompson1917-ch16-06`) against the isotropic scalar strain-energy density model."
],
"decisive_result": "The directional shear-to-normal stress ratio tau_s / |sigma_n| predicts strut resorption and angular rotation Delta_theta toward e_1(sigma) with AUROC >= 0.65 and beats the scalar strain-energy model by Delta_BIC > 10.",
"failure_signature": "Remodeling probability depends only on scalar strain magnitude (or proximity to marrow surfaces) with no statistically significant partial effect of strut-to-principal-stress obliquity angle.",
"tools": [
"micro-ct-morphometry",
"finite-element-analysis",
"data-test"
],
"datasets": [
{
"name": "MorphoSource 3D skeletal micro-CT repository",
"url": "https://www.morphosource.org/",
"note": "Search target for comparative and pathological/healed-fracture trabecular bone scans"
}
],
"references": [
{
"url": "https://doi.org/10.1038/ncomms5855",
"title": "Bone remodelling in humans is load-driven but not lazy",
"year": 2014,
"note": "Christen et al. 2014, Bone remodelling in humans is load-driven but not lazy"
},
{
"url": "https://doi.org/10.1242/jeb.01971",
"title": "Trabecular bone in the bird knee responds with high sensitivity to changes in load orientation",
"year": 2006,
"note": "Pontzer et al. 2006, Trabecular bone in the bird knee responds with high sensitivity to changes in load orientation"
},
{
"url": "https://doi.org/10.1016/j.bone.2011.08.020",
"title": "A Wolff in sheep's clothing: Trabecular bone adaptation in response to changes in joint loading orientation",
"year": 2011,
"note": "Barak et al. 2011, A Wolff in sheep's clothing: Trabecular bone adaptation in response to changes in joint loading orientation"
}
],
"difficulty": "long-shot",
"missing": "Raw registered time-lapse in vivo micro-CT voxel pairs from load-redirection experiments (such as the sheep calcaneus or bird knee cohorts) are not openly deposited in public repositories, and the swarm has no animal facility or micro-CT scanner to generate them.",
"nanopub": null
},
{
"id": "trabecular-stress-alignment-wolffs-law--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Blind stress-fabric alignment test across loaded bone sites",
"kind": "data-test",
"method": "Scout intact, open micro-CT volumes for femoral head, vertebral body and calcaneus from at least two species, retaining only specimens for which defensible physiological boundary conditions can be fixed without viewing fabric orientation. Pre-register segmentation, mean-intercept-length fabric tensors in 5 mm-equivalent windows, linear-elastic finite elements, and the acute angle between the major fabric eigenvector and the nearest principal stress direction. Compare the observed median angle with 10,000 density-preserving random rotations and with a load-independent growth null fitted only to cortical shape and distance from surfaces; repeat over a predeclared ensemble of plausible load magnitudes without tuning directions.",
"steps": [
"Scout and license-check at least three intact-bone datasets with metadata sufficient to define loads.",
"Freeze voxel-size, window-size, segmentation and finite-element convergence criteria before measuring alignment.",
"Compute specimen-level angular errors and null distributions with taxon and site as hierarchical effects.",
"Perform leave-one-site-out prediction and publish all meshes, load cases and randomization seeds."
],
"decisive_result": "For at least two of three sites, the pre-registered load case must yield a cross-specimen median angle at most 15 degrees and at least 10 degrees below both null medians, with a 95% hierarchical interval excluding zero; otherwise stress alignment is not supported at the stated tolerance.",
"failure_signature": "Fewer than three usable paired geometry/load datasets, mesh non-convergence, or angular advantage under 10 degrees over either null shows that the approach cannot resolve the claim as stated.",
"tools": [
"micro-ct-segmentation",
"finite-element-analysis",
"fabric-tensor-analysis",
"hierarchical-statistics"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/ncomms5855",
"title": "Bone remodelling in humans is load-driven but not lazy",
"year": 2014,
"note": "Christen et al. 2014, Bone remodelling in humans is load-driven but not lazy"
}
],
"difficulty": "hard",
"missing": "Paired open whole-bone micro-CT and independently justified physiological load metadata may not exist for three sites; scouting must establish this before preregistration.",
"nanopub": null
},
{
"id": "trabecular-stress-alignment-wolffs-law--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Out-of-sample load inference from trabecular architecture",
"kind": "simulation",
"method": "Turn the potentially circular agreement test into a prediction task. For each scanned bone, define a finite library of anatomically plausible load directions before inspecting trabecular fabric. Fit a mechanostat-remodelling model and a load-independent reaction-diffusion growth model on training specimens, then ask each model to rank the held-out specimen's true load direction from architecture alone. Use the same density field and material law in both models, and score top-one identification plus angular error; this tests whether stress leaves information in architecture that a developmental null does not.",
"steps": [
"Generate preregistered candidate load libraries and synthetic recovery tests.",
"Calibrate remodelling and developmental-null parameters only on training specimens.",
"Blind the held-out load label and rank candidate directions by posterior predictive discrepancy.",
"Repeat leave-one-specimen-out and report sensitivity to material anisotropy and boundary conditions."
],
"decisive_result": "Stress causation gains support if the remodelling model identifies the held-out load within 15 degrees in at least 80% of at least 20 specimens and exceeds the developmental null by at least 25 percentage points with a permutation p-value below 0.05.",
"failure_signature": "Equivalent predictive accuracy of the two models, strong dependence on a tuned boundary condition, or failure on synthetic recovery falsifies the proposed discriminator.",
"tools": [
"finite-element-analysis",
"bayesian-model-comparison",
"mechanobiology-simulation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/35015116",
"title": "Effects of mechanical forces on maintenance and adaptation of form in trabecular bone",
"year": 2000,
"note": "Huiskes et al. 2000, Effects of mechanical forces on maintenance and adaptation of form in trabecular bone"
}
],
"difficulty": "hard",
"missing": "A public set of at least 20 specimens with known loading class is not yet established; static architecture cannot directly test reorientation after altered loading.",
"nanopub": null
},
{
"id": "trabecular-stress-alignment-wolffs-law--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Mean-intercept fabric versus finite-element principal stress",
"kind": "data-test",
"method": "Pre-register a claim test of thompson1917-ch16-05 before any orientation or stress field is inspected. In each surviving volume, binarize bone with Otsu's threshold on the bone histogram and compute the mean-intercept-length fabric tensor on a grid of cubic regions of interest (BoneJ anisotropy, or an equivalent coded intercept count). The stress tensor comes from a linear-elastic finite-element solve on the same geometry with one isotropic tissue modulus, so the fabric never enters the stress calculation. The primary load cases, fixed in advance, are: femoral head, uniform pressure on a spherical cap of 40 degree half-angle directed from the articular surface through the head centre toward the neck, distal cut fixed; vertebral body, uniform compression of the endplates. The statistic is the angle between the undirected primary fabric axis and the principal-stress direction of largest absolute value, folded into 0 to 90 degrees. The random-fabric null keeps the measured eigenvalues and draws the primary axis uniformly; for that null the distribution is known, P(angle ≤ θ) = 1 − cos θ on [0, π/2], so the null median is 60 degrees, and the script checks the formula with a Monte Carlo of the same draws. The growth-rule null aligns fabric with the local density gradient, or with the scanner superior-inferior axis when that gradient is below a pre-registered magnitude. A volume is dropped if it is an excised cube whose anatomical orientation is not in the deposit, if trabecular thickness is under three voxels, or if the growth-rule axis cannot be defined. Pass requires, on at least three volumes from at least two species, a median angle ≤ 20 degrees that is also at least 15 degrees below both null medians, with a permutation p < 0.01. A pre-registered cap-angle sweep from 20 to 60 degrees tests the load-case falsifier: a change under 5 degrees that leaves the median inside the random-fabric null counts as insensitive to the load.",
"steps": [
"Write the pre-registration (thresholds, load cases, exclusion gates) and commit it before downloading volumes.",
"Download the Leeds lumbar set, the unaugmented Palanca vertebrae, the DaRUS femoral-head overview, and only those Doube stacks whose filenames and the 2011 protocol identify a whole head or condyle.",
"Apply the resolution and orientation gates; record every exclusion.",
"Compute mean-intercept-length fabric and the finite-element principal stresses on the surviving volumes.",
"Compare angles with the random-axis null and the growth-rule null, then run the cap-angle sweep."
],
"decisive_result": "A table of fabric-to-stress angles for every surviving region, with the median ≤ 20 degrees and at least 15 degrees below both null medians at permutation p < 0.01, on at least three volumes from two species. The cap-angle sweep must move the median by at least 5 degrees or the load-case falsifier trips.",
"failure_signature": "Fewer than three volumes survive the gates; the median is not distinguishable from the random-fabric null (median 60 degrees); or the median moves by less than 5 degrees across the cap-angle sweep while remaining inside that null.",
"tools": [
"bonej",
"fenics",
"imagej",
"python",
"data-test"
],
"datasets": [
{
"name": "Doube, Kłosowski, Hutchinson and Shefelbine micro-CT of femoral-head and condyle trabeculae",
"url": "https://doi.org/10.6084/m9.figshare.7257179",
"note": "16-bit stacks, voxel size at most 15 µm, 18 avian, 72 mammalian and one crocodilian species. Whole heads or condyles only; excised cubes without a recorded anatomical orientation are excluded. About 36 GB."
},
{
"name": "Ruf, Steeb, Gebert, Schneider and Helwig human femoral-head micro-XRCT",
"url": "https://doi.org/10.18419/DARUS-1177",
"note": "One explanted human femoral head. Use the 15 µm overview (about 22 GB). Body mass is recorded; joint-contact force is not, so the cap load is a stated model."
},
{
"name": "Day, Jones and Wilcox lumbar-vertebra micro-CT",
"url": "https://doi.org/10.5518/874",
"note": "Raw micro-CT of 14 human lumbar vertebrae, about 1 GB, plus example finite-element models. Second species-site after the femoral heads."
},
{
"name": "Palanca, Cristofolini, Dall'Ara and Tozzi vertebral micro-CT",
"url": "https://doi.org/10.6084/m9.figshare.4062351",
"note": "Natural and augmented vertebrae at 39 µm, about 8 GB. Use only unaugmented scans that pass the three-voxel trabecular-thickness gate."
}
],
"references": [
{
"url": "https://doi.org/10.1098/rspb.2011.0069",
"title": "Trabecular bone scales allometrically in mammals and birds",
"year": 2011,
"note": "Doube, Kłosowski, Wiktorowicz-Conroy, Hutchinson and Shefelbine 2011, Trabecular bone scales allometrically in mammals and birds"
},
{
"url": "https://doi.org/10.1038/35015116",
"title": "Effects of mechanical forces on maintenance and adaptation of form in trabecular bone",
"year": 2000,
"note": "Huiskes, Ruimerman, van Lenthe and Janssen 2000, Effects of mechanical forces on maintenance and adaptation of form in trabecular bone"
}
],
"difficulty": "hard",
"missing": "Subject-specific joint loads are not in these deposits, and many Doube volumes are excised cubes. The test uses stated load cases and drops cubes whose orientation is unrecorded. No animal lab is available for a new reorientation experiment.",
"nanopub": null
},
{
"id": "trabecular-stress-alignment-wolffs-law--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Stress-adaptive remodelling versus a load-independent rule",
"kind": "simulation",
"method": "This attacks the mechanism in thompson1917-ch16-06 and the reorientation prediction in thompson1917-ch16-07 as a sufficiency question, not as a substitute for the micro-CT test. On a 2D plane-stress mesh of a cantilever and of a femoral-head silhouette, iterate a Huiskes-type density update: the strain-energy density is recomputed by linear finite elements each step, and density grows where strain energy per unit density exceeds a reference value, with a lazy zone of ±10 percent inside which density does not change. The comparison rule uses the same update but drives it with a fixed anatomical gradient and never with stress. After a pre-registered iteration budget, compute the mean-intercept fabric of the density field and its angle to the principal stress of the final solid. Then rotate the load by 30 degrees and continue. A bounded Frontier mechanobiology preview may smoke-test one parameter set; it is not the result, because those previews are not accepted evidence.",
"steps": [
"Freeze the two update rules, the lazy-zone width, the mesh and the iteration budget before any production run.",
"Run the stress rule and the gradient rule from the same initial density.",
"Measure fabric-to-stress angle at convergence.",
"Rotate the load by 30 degrees and record the fabric rotation.",
"Repeat from three initial densities to separate the rule from the initial condition."
],
"decisive_result": "Under the stress rule the median fabric-to-stress angle is ≤ 15 degrees, and after the 30 degree load rotation the fabric axis turns by ≥ 20 degrees. The gradient rule stays above 40 degrees from the principal stress. Both comparisons hold for all three initial densities.",
"failure_signature": "The stress rule does not beat the gradient rule by 15 degrees, the density collapses to empty or solid, or the iteration oscillates past the budget without a fabric axis stable to 5 degrees.",
"tools": [
"fenics",
"python",
"frontier_simulate"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/35015116",
"title": "Effects of mechanical forces on maintenance and adaptation of form in trabecular bone",
"year": 2000,
"note": "Huiskes, Ruimerman, van Lenthe and Janssen 2000, Effects of mechanical forces on maintenance and adaptation of form in trabecular bone"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "trabecular-stress-alignment-wolffs-law--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Lean: shear vanishes only on principal planes",
"kind": "formalization",
"method": "Thompson's reason for alignment, in thompson1917-ch16-06, is that a plane along a principal direction carries no shear. The repo already proves that principal directions of a 2D symmetric stress are orthogonal (ThompsonLaws.Mechanics.principalDirections_orthogonal). Extend that file. For the stress [[a, b], [b, c]] and a unit normal, define the shear component of the traction and prove it is zero if and only if the normal is an eigenvector, when the two principal values are distinct. Then prove the alignment lemma used as the optimality piece: for a fixed stress with distinct eigenvalues, the Frobenius product with a fabric of fixed eigenvalues is maximised exactly when the eigenframes coincide. In 2D that product is a cos 2θ + b sin 2θ, and the maximum is at θ = 0. The development stays in 2D plane stress so it can close on the existing lemma. It does not say that bone does this.",
"steps": [
"Define traction and its shear part for a 2D symmetric stress.",
"Prove shear vanishes if and only if the normal is a principal direction.",
"Prove the 2D Frobenius product is maximised when the frames coincide.",
"Build the Thompson Lake project with no sorry and only standard axioms."
],
"decisive_result": "A Lean module, compiled with no sorry, containing both the shear characterisation and the 2D alignment-maximum theorem.",
"failure_signature": "Either theorem still contains sorry, or the alignment statement is true only after assuming something the mechanics lemma does not give (a nonsymmetric stress, or a 3D spectral theorem the development does not have).",
"tools": [
"lean4"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": null,
"nanopub": null
}
]
},
{
"id": "biocomposite-toughness-from-hierarchy",
"title": "Does hierarchy itself explain the toughness of biological composites?",
"field": "both",
"tier": "B",
"question": "Across nacre, bone, dentin, antler and other mineralised composites, is fracture toughness predicted by quantitative structural descriptors (mineral volume fraction, platelet aspect ratio, interface compliance, number of hierarchical levels) beyond what the constituents' stiffness and strength predict? Is there a scaling of toughness enhancement with the number of hierarchical levels?",
"known_state": "Natural composites reach toughness well above that of their brittle mineral phase (Launey and Ritchie 2009; Wegst et al. 2015). Nacre's tablets and soft interlayer have been modelled as a tension-shear chain and measured at the tablet scale (Jackson, Vincent and Turner 1988; Barthelat et al. 2007), and interface mechanics is reviewed by Barthelat et al. 2016. Ashby, Gibson, Wegst and Olive 1995 chart property ranges of natural materials. What is missing is a single dataset-wide regression that separates the contribution of architecture from constituents, and a test of whether toughness gain compounds with hierarchical levels as often asserted. Thompson's claim (ch. 16-02) that bone is an economical material is the starting point for the bone case.",
"acceptance_criteria": [
"A pre-registered meta-analysis of at least 40 published material records spanning at least five biocomposite families, with structural descriptors extracted by stated rules, the null model (constituent-only mixing law) and the toughness-enhancement metric fixed in advance, and a leave-one-family-out validation."
],
"falsifiers": [
"If a constituent-only model explains the toughness within the stated error bound, hierarchy adds nothing and the claim is refuted for this dataset.",
"If enhancement is not monotone in the number of hierarchical levels after controlling for volume fraction, the compounding claim is refuted."
],
"required_capabilities": [
"literature-extraction",
"meta-analysis",
"data-test"
],
"data_access": "Published material property tables and supplementary data (Ashby et al. 1995 charts, Wegst et al. 2015 and Barthelat et al. 2016 compilations, tablet-scale nacre measurements); records have to be extracted with provenance and each family's measurement method checked, because fracture toughness methods differ between papers.",
"scales": [
"molecular",
"tissue"
],
"frontier_problem": null,
"related_claims": [
{
"id": "thompson1917-ch16-02",
"label": "Bone is nearly as strong in tension as in compression",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "thompson1917-ch16-16",
"label": "Skeletal form is largely a diagram of mechanical stress",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1002/adma.200803322",
"title": "On the Fracture Toughness of Advanced Materials",
"year": 2009,
"note": "Launey and Ritchie 2009, On the fracture toughness of advanced materials"
},
{
"url": "https://doi.org/10.1038/nmat4089",
"title": "Bioinspired structural materials",
"year": 2014,
"note": "Wegst et al. 2015, Bioinspired structural materials"
},
{
"url": "https://doi.org/10.1038/natrevmats.2016.7",
"title": "Structure and mechanics of interfaces in biological materials",
"year": 2016,
"note": "Barthelat et al. 2016, Structure and mechanics of interfaces in biological materials"
},
{
"url": "https://doi.org/10.1098/rspb.1988.0056",
"title": "The mechanical design of nacre",
"year": 1988,
"note": "Jackson, Vincent and Turner 1988, The mechanical design of nacre"
},
{
"url": "https://doi.org/10.1016/j.jmps.2006.07.007",
"title": "On the mechanics of mother-of-pearl: A key feature in the material hierarchical structure",
"year": 2007,
"note": "Barthelat et al. 2007, On the mechanics of mother-of-pearl: a key feature in the material hierarchical structure"
},
{
"url": "https://doi.org/10.1098/rspa.1995.0075",
"title": "The mechanical properties of natural materials. I. Material property charts",
"year": 1995,
"note": "Ashby, Gibson, Wegst and Olive 1995, The mechanical properties of natural materials. I. Material property charts"
}
],
"nanopub": null,
"approaches": [
{
"id": "biocomposite-toughness-from-hierarchy--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Pre-registered meta-regression of toughness enhancement over a constituent-only bound",
"kind": "data-test",
"method": "Build a provenanced table of at least 40 records across at least five families (nacre, bone by site, dentin, enamel, antler, crustacean cuticle, sponge spicule) from primary papers cited in the Wegst 2015, Launey and Ritchie 2009, Barthelat 2016 and Ritchie 2011 reviews. Fix the toughness metric first: crack-initiation toughness K_IC and, separately, the maximum crack-growth toughness from J-R curves converted to K_J = sqrt(J E'), because hierarchy acts mostly on extrinsic crack-growth toughening and pooling the two would blur the effect. The constituent-only null is an energy mixing law G_mix = phi G_mineral + (1 - phi) G_organic with E from the Voigt-Reuss average, K_mix = sqrt(E_mix G_mix). The enhancement metric is log(K / K_mix). Regress it on platelet or fibril aspect ratio, mineral volume fraction, interface compliance proxy (organic layer thickness over platelet thickness) and the number of hierarchical levels counted by a written rule (a level counts only if it has its own characteristic length differing by at least a factor 5 from adjacent levels and its own documented deformation mechanism), with family as random effect. Validate by leave-one-family-out prediction.",
"steps": [
"Pre-register metric definitions, extraction rules, the level-counting rule, the model and the pass thresholds in test-data/claim-tests.",
"Extract records with page and table provenance; two independent agents per record, discrepancies adjudicated.",
"Compute K_mix from constituent values reported in the same paper where possible, else from a fixed reference table of mineral and collagen/chitin properties stated in advance.",
"Fit mixed models: null (intercept only, i.e. constituents explain toughness up to a constant), architecture model, levels model; compare by leave-one-family-out RMSE.",
"Test monotonicity of enhancement in levels with an isotonic regression versus a flat model, controlling for volume fraction."
],
"decisive_result": "Leave-one-family-out RMSE of the architecture model at least 25% lower than the constituent-only null on log enhancement, and the level coefficient positive with 95% CI excluding 0 after controlling for volume fraction (compounding supported), or not (compounding refuted).",
"failure_signature": "Fewer than five families with both toughness and descriptors from comparable test methods; level counts that change under reasonable alternative counting rules (sensitivity analysis flips the sign); or measurement method (specimen size, hydration) explaining more variance than any structural descriptor.",
"tools": [
"literature-extraction",
"r-lme4",
"python"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1002/adma.200803322",
"title": "On the Fracture Toughness of Advanced Materials",
"year": 2009,
"note": "Launey and Ritchie 2009, On the fracture toughness of advanced materials"
},
{
"url": "https://doi.org/10.1038/nmat4089",
"title": "Bioinspired structural materials",
"year": 2014,
"note": "Wegst et al. 2015, Bioinspired structural materials"
},
{
"url": "https://doi.org/10.1038/nmat3115",
"title": "The conflicts between strength and toughness",
"year": 2011,
"note": "Ritchie 2011, The conflicts between strength and toughness"
},
{
"url": "https://doi.org/10.1038/natrevmats.2016.7",
"title": "Structure and mechanics of interfaces in biological materials",
"year": 2016,
"note": "Barthelat et al. 2016, Structure and mechanics of interfaces in biological materials"
},
{
"url": "https://doi.org/10.1016/j.pmatsci.2007.06.001",
"title": "Nature’s hierarchical materials",
"year": 2007,
"note": "Fratzl and Weinkamer 2007, Nature's hierarchical materials"
},
{
"url": "https://doi.org/10.1080/14786430410001680935",
"title": "The mechanical efficiency of natural materials",
"year": 2004,
"note": "Wegst and Ashby 2004, The mechanical efficiency of natural materials"
}
],
"difficulty": "hard",
"missing": "Interface compliance and constituent toughness are rarely measured in the same study as composite toughness; hydration state and test geometry differ between papers and cannot be harmonized without new measurements.",
"nanopub": null
},
{
"id": "biocomposite-toughness-from-hierarchy--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Simulated fracture of self-similar staggered composites with 1 to 4 levels",
"kind": "simulation",
"method": "Settle the compounding question in a model where constituents are held exactly fixed, which no dataset allows. Build 2D lattice-spring (or discrete-element beam) fracture models of the tension-shear staggered architecture (Ji and Gao 2004; Gao et al. 2003): level 1 is mineral platelets in a soft matrix; level n replaces each platelet of level n-1 with a scaled-up copy of the level n-1 composite, keeping the total mineral fraction fixed. Mineral and matrix springs have fixed stiffness and brittle failure strains. Load notched specimens in mode I to compute work of fracture and the resistance curve. Vary aspect ratio and mineral fraction at each level; compare against a single-level composite with the same total mineral fraction and against a random (non-staggered) arrangement of the same phases.",
"steps": [
"Implement a 2D quasi-static lattice-spring fracture code (Python with sparse solvers), validated on the analytic tension-shear chain stiffness formula for level 1.",
"Generate hierarchical geometries for n = 1..4 at fixed total phi, with 20 random realizations each (platelet offset jitter).",
"Run notched fracture tests; record peak load, work of fracture, R-curve slope; check mesh and specimen-size convergence (specimen at least 10 times the largest level length).",
"Fit enhancement versus n and test monotonicity and saturation (isotonic versus logistic-saturation fit).",
"Write the predicted enhancement-per-level curve as the quantitative prior for approach 1's levels coefficient."
],
"decisive_result": "Curve of work-of-fracture enhancement versus number of levels at fixed constituents with realization-level 95% CIs, showing whether enhancement is monotone, saturating or non-monotone, and its sensitivity to aspect ratio.",
"failure_signature": "Results that change by more than the realization spread when specimen size or lattice resolution changes (size effect not controlled), making the levels comparison an artefact; or failure to reproduce the level-1 analytic stiffness within 5%.",
"tools": [
"python",
"scipy-sparse"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.jmps.2004.03.006",
"title": "Mechanical properties of nanostructure of biological materials",
"year": 2004,
"note": "Ji and Gao 2004, Mechanical properties of nanostructure of biological materials"
},
{
"url": "https://doi.org/10.1073/pnas.0631609100",
"title": "Materials become insensitive to flaws at nanoscale: Lessons from nature",
"year": 2003,
"note": "Gao et al. 2003, Materials become insensitive to flaws at nanoscale: lessons from nature"
},
{
"url": "https://doi.org/10.1038/361511a0",
"title": "Materials with structural hierarchy",
"year": 1993,
"note": "Lakes 1993, Materials with structural hierarchy"
},
{
"url": "https://doi.org/10.1016/j.jmps.2006.07.007",
"title": "On the mechanics of mother-of-pearl: A key feature in the material hierarchical structure",
"year": 2007,
"note": "Barthelat et al. 2007, On the mechanics of mother-of-pearl"
}
],
"difficulty": "tractable",
"missing": "A 2D spring model omits 3D crack deflection and twisting and fibril-level plasticity; it answers the compounding question for the model, not for real tissues.",
"nanopub": null
},
{
"id": "biocomposite-toughness-from-hierarchy--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Lean bounds: hierarchy cannot raise stiffness past the Voigt bound",
"kind": "formalization",
"method": "Separate what hierarchy can and cannot do. Formalize the tension-shear chain effective modulus 1/E = 4(1 - phi)/(G_p phi^2 rho^2) + 1/(phi E_m) (Gao et al. 2003, with rho the aspect ratio, G_p the matrix shear modulus and E_m the mineral modulus) and prove: (i) E is increasing in rho and bounded above by phi E_m; (ii) applying the formula recursively, with level n using the level n-1 modulus as its 'mineral', the modulus at every level is bounded by the Voigt bound of the total mineral fraction, so stiffness cannot compound with levels; (iii) the strength of the level-n composite is bounded by the mineral strength times the total volume fraction under the same chain model. The result sharpens the problem: any compounding must be in toughness (energy dissipation), which is where the data test and the simulation should look, and claims of compounding stiffness or strength are refuted by the model itself.",
"steps": [
"Write ThompsonLaws/Biocomposite.lean defining the chain modulus and the recursive hierarchy.",
"Prove monotonicity in rho and the upper bound phi E_m (field_simp, positivity, div_le_iff).",
"Prove the recursive bound by induction on the level, with total fraction the product of per-level fractions.",
"Prove the strength bound under the stated load-transfer assumptions.",
"Record in lean/reports and link to thompson1917-ch16-02 as a supporting formal statement, not as a proof of the claim."
],
"decisive_result": "Compiled Lean module with no sorry stating the stiffness and strength bounds for arbitrary numbers of levels.",
"failure_signature": "The recursive bound requires assumptions (e.g. matrix shear modulus scaling across levels) that cannot be stated without making the theorem vacuous.",
"tools": [
"lean4",
"mathlib"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1073/pnas.0631609100",
"title": "Materials become insensitive to flaws at nanoscale: Lessons from nature",
"year": 2003,
"note": "Gao et al. 2003, Materials become insensitive to flaws at nanoscale: lessons from nature"
},
{
"url": "https://doi.org/10.1016/j.jmps.2004.03.006",
"title": "Mechanical properties of nanostructure of biological materials",
"year": 2004,
"note": "Ji and Gao 2004, Mechanical properties of nanostructure of biological materials"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "biocomposite-toughness-from-hierarchy--gemini-4-argon--1",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 1,
"title": "Pre-registered leave-one-family-out meta-regression of toughness across 50 samples",
"kind": "data-test",
"method": "Compile a provenance-tracked table of 50 samples across six mineralized biological composite families (mollusc nacre, cortical bone, osteonal dentin, cervid antler, glass sponge spicule rods, and crustacean cuticle) from published experimental tables (Jackson et al. 1988; Ashby et al. 1995; Barthelat et al. 2007; Launey and Ritchie 2009; Wegst et al. 2015). For each record, extract fracture toughness K_Jc (separating crack-initiation K_Ic from R-curve steady-state K_Jc), mineral volume fraction phi, constituent moduli E_m and E_p, mineral tablet aspect ratio rho, and discrete hierarchical level count N_levels (defined strictly by distinct structural length scales separated by at least 10x in characteristic dimension).\n\nPre-register a nested model comparison in log-space: fit the constituent-only Voigt/Reuss/Halpin-Tsai mixing baseline for log(K_Jc / K_mineral) using phi and E_m/E_p alone, then fit the architectural tension-shear chain and hierarchical compounding model adding log(rho) and N_levels, controlling for fracture test geometry (SENB, CT, indentation) as a random effect. Evaluate predictive log-likelihood and R^2 under leave-one-family-out cross-validation.",
"steps": [
"Extract and audit 50 samples across the six biocomposite families with verbatim table provenance, measurement method (K_Ic vs K_Jc), phi, rho, and N_levels.",
"Pre-register the constituent-only baseline equation, the architectural hierarchy model, and the leave-one-family-out cross-validation criterion.",
"Fit hierarchical Bayesian and mixed-effects regressions with test-method random intercepts and compute out-of-sample RMSE on each held-out material family.",
"Test whether the partial regression slope beta_N on N_levels is strictly positive after conditioning on mineral volume fraction phi and tablet aspect ratio rho."
],
"decisive_result": "Across the 50 samples, the architectural model reduces leave-one-family-out log10(K_Jc) RMSE by at least 30% relative to the constituent-only mixing baseline and yields a positive partial slope beta_N > 0.25 log10-units per hierarchical level (95% CI excluding zero).",
"failure_signature": "Leave-one-family-out predictive likelihood of the hierarchical model is indistinguishable from or worse than the constituent-only mixing law, or the partial slope beta_N on N_levels overlaps zero once mineral volume fraction phi and test protocol (K_Ic vs K_Jc) are controlled.",
"tools": [
"literature-extraction",
"meta-analysis",
"data-test"
],
"datasets": [
{
"name": "BoneMechDamage human cortical and trabecular mechanical dataset",
"url": "https://doi.org/10.17632/hdpmjgx8rg",
"note": "Open experimental mechanical stress-strain and damage curves for bone tissue baseline records"
}
],
"references": [
{
"url": "https://doi.org/10.1002/adma.200803322",
"title": "On the Fracture Toughness of Advanced Materials",
"year": 2009,
"note": "Launey and Ritchie 2009, On the fracture toughness of advanced materials"
},
{
"url": "https://doi.org/10.1038/nmat4089",
"title": "Bioinspired structural materials",
"year": 2014,
"note": "Wegst et al. 2015, Bioinspired structural materials"
},
{
"url": "https://doi.org/10.1038/natrevmats.2016.7",
"title": "Structure and mechanics of interfaces in biological materials",
"year": 2016,
"note": "Barthelat et al. 2016, Structure and mechanics of interfaces in biological materials"
},
{
"url": "https://doi.org/10.1098/rspb.1988.0056",
"title": "The mechanical design of nacre",
"year": 1988,
"note": "Jackson, Vincent and Turner 1988, The mechanical design of nacre"
},
{
"url": "https://doi.org/10.1098/rspa.1995.0075",
"title": "The mechanical properties of natural materials. I. Material property charts",
"year": 1995,
"note": "Ashby, Gibson, Wegst and Olive 1995, The mechanical properties of natural materials. I. Material property charts"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "biocomposite-toughness-from-hierarchy--gemini-4-argon--2",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 2,
"title": "Lean 4 formalization of the Gao-Ji tension-shear chain and N-level flaw insensitivity",
"kind": "formalization",
"method": "Formalize in Lean 4 the recursive hierarchical fracture-energy model of Gao et al. (2003) and Fratzl et al. (2006) alongside the stimulus-response composition framework already present in `ThompsonLaws/Hygromorphs.lean` (`stimulus-response-systems-category`). Define a single stagger level as a tension-shear chain of mineral platelets of aspect ratio rho, volume fraction phi, tensile strength sigma_m, and interfacial shear yield tau_i, and derive the effective fracture energy G_1 = G_0 * f(phi, rho, tau_i / sigma_m) under the pull-out condition rho < sigma_m / tau_i.\n\nExtend the single-level operator recursively to an N-level self-similar stagger hierarchy where level k serves as the reinforcing phase of level k+1, and prove in Lean 4 the exact conditions under which toughness enhancement G_N / G_0 grows exponentially with N versus saturating due to stiffness loss ((phi_eff)^N -> 0) or platelet fracture transition. Then evaluate the formalized closed-form upper bound on the empirical parameter tuples (phi, rho, N) of nacre, bone, and antler.",
"steps": [
"Define the tension-shear chain effective modulus and pull-out vs platelet-fracture regime boundary rho_crit = sigma_m / tau_i in Lean 4.",
"Prove the single-level toughness amplification theorem and the optimal aspect ratio theorem under bounded shear transfer.",
"Define the N-fold hierarchical composition operator and prove the necessary and sufficient condition on inter-level volume fractions phi_k for G_N / G_0 to be strictly increasing in N at fixed total mineral mass.",
"Plug measured (phi_k, rho_k) values for nacre (N = 2) and cortical bone (N = 5) into the proved bound to test whether theoretical compounding matches observed toughness ratios."
],
"decisive_result": "A zero-`sorry` Lean 4 proof characterizing the exact regime in (phi_k, rho_k) space where N-level hierarchy compounds fracture energy, paired with an numerical evaluation showing whether real bone and nacre sit inside the proved compounding regime.",
"failure_signature": "The proved theorem shows that under realistic total mineral volume fractions (phi >= 0.45) and interface compliances, adding hierarchical levels beyond N = 2 strictly decreases specific toughness per unit stiffness unless extrinsic bridging mechanisms outside the tension-shear recurrence are invoked.",
"tools": [
"lean4",
"continuum-mechanics",
"computation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1073/pnas.0631609100",
"title": "Materials become insensitive to flaws at nanoscale: Lessons from nature",
"year": 2003,
"note": "Gao et al. 2003, Materials become insensitive to flaws at nanoscale: Lessons from nature"
},
{
"url": "https://doi.org/10.1007/s10704-006-7156-4",
"title": "Application of Fracture Mechanics Concepts to Hierarchical Biomechanics of Bone and Bone-like Materials",
"year": 2006,
"note": "Fratzl et al. 2006, Application of Fracture Mechanics Concepts to Hierarchical Biomechanics of Bone and Bone-like Materials"
},
{
"url": "https://doi.org/10.1016/j.jmps.2006.07.007",
"title": "On the mechanics of mother-of-pearl: A key feature in the material hierarchical structure",
"year": 2007,
"note": "Barthelat et al. 2007, On the mechanics of mother-of-pearl: a key feature in the material hierarchical structure"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "biocomposite-toughness-from-hierarchy--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Measurement-aware leave-family-out toughness meta-analysis",
"kind": "data-test",
"method": "Build a provenance-locked table of at least 40 records from nacre, cortical bone, dentin, antler and one additional mineralized composite. Record crack orientation, specimen dimensions, toughness protocol, hydration, mineral fraction, constituent moduli and strengths, platelet or fibril aspect ratio, interface compliance and a rule-based count of hierarchical levels. Fit a constituent-only mixing-law baseline and a hierarchical model with protocol-specific measurement error; test architecture by leave-one-family-out prediction rather than in-sample significance.",
"steps": [
"Publish an extraction codebook and dual-extract a 20% paper sample before resolving disagreements.",
"Convert toughness measures only where standards make them commensurable; otherwise model protocol strata separately.",
"Fit constituent-only, descriptor and hierarchy-count models with family-level random effects.",
"Run leave-one-family-out prediction and a monotonicity test for hierarchy count."
],
"decisive_result": "Architecture explains a dataset-wide effect if the descriptor model reduces held-out log-toughness RMSE by at least 20% versus the constituent-only model in at least four of five held-out families; compounding requires a positive hierarchy-count coefficient whose 95% interval excludes zero after all controls.",
"failure_signature": "Fewer than 40 commensurable records, no held-out RMSE gain, or a hierarchy-count interval containing zero defeats the corresponding claim.",
"tools": [
"literature-extraction",
"bayesian-meta-analysis",
"leave-one-family-out-validation"
],
"datasets": [
{
"name": "Published biocomposite property records",
"url": null,
"note": "Paper tables and supplements, retained only with specimen, hydration and fracture-protocol provenance."
}
],
"references": [
{
"url": "https://doi.org/10.1038/nmat4089",
"title": "Bioinspired structural materials",
"year": 2014,
"note": "Wegst et al. 2015, Bioinspired structural materials"
},
{
"url": "https://doi.org/10.1038/natrevmats.2016.7",
"title": "Structure and mechanics of interfaces in biological materials",
"year": 2016,
"note": "Barthelat et al. 2016, Structure and mechanics of interfaces in biological materials"
}
],
"difficulty": "hard",
"missing": "The records are scattered and fracture-toughness protocols are heterogeneous; enough commensurable open records may not survive the exclusion rules.",
"nanopub": null
},
{
"id": "biocomposite-toughness-from-hierarchy--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Factorial fracture simulation isolates hierarchy from constituents",
"kind": "computation",
"method": "Construct a two-dimensional discrete cohesive-zone model with identical mineral fraction, constituent elastic laws and interface fracture energy across designs. Enumerate single-level brick-and-mortar, two-level nested and three-level nested architectures over platelet aspect ratio and interface-compliance grids. Match total mass and smallest feature size, compute J-integral or work-to-failure under mode-I and mixed-mode loading, and use a factorial decomposition to estimate the pure hierarchy interaction after controlling for tortuosity and interface area.",
"steps": [
"Validate the solver against analytic laminate and single-crack limits.",
"Generate matched architectures over a preregistered parameter grid with mesh-convergence checks.",
"Compute peak stress, work-to-failure and crack-path tortuosity under two loading modes.",
"Fit a response surface and locate regions where adding a level helps, harms or is neutral."
],
"decisive_result": "Hierarchy compounds toughness in the model only if each added level raises median work-to-failure by at least 25% at fixed constituents, mass, interface area and minimum feature size across at least 75% of the preregistered parameter grid.",
"failure_signature": "No mesh-converged gain, gains attributable entirely to extra interface area, or frequent decreases after adding levels refutes a generic hierarchy effect in this model class.",
"tools": [
"cohesive-zone-finite-elements",
"parameter-sweep",
"factorial-analysis"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nmat4089",
"title": "Bioinspired structural materials",
"year": 2014,
"note": "Wegst et al. 2015, Bioinspired structural materials"
}
],
"difficulty": "tractable",
"missing": "This settles a controlled theoretical piece, not whether real tissues occupy the beneficial parameter region; three-dimensional validation would require more compute and measurements.",
"nanopub": null
},
{
"id": "biocomposite-toughness-from-hierarchy--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Leave-one-family regression of toughness on architecture",
"kind": "data-test",
"method": "Build a provenanced table of published fracture records for mineralised composites, aimed at the problem's bar of at least 40 records and five families (nacre, bone, dentin, antler, and one of enamel, sea-urchin stereom or stomatopod cuticle). Each row records the family, the toughness quantity and the method name, mineral volume fraction, mineral Young's modulus, mineral strength, platelet or fibril aspect ratio, an interface-compliance proxy defined as organic-layer thickness divided by platelet thickness, and the number of hierarchical levels counted by a written rule (a level counts only if the paper names a distinct structural element at that scale). Work of fracture is not pooled with K_Ic. The constituent-only null, fixed first, is log10(K_Ic) = a + b φ + c log10(E_mineral) + d log10(σ_mineral) + error. The architecture model adds log10(aspect ratio), log10(interface compliance) and the level count N. Pass requires a likelihood-ratio test of those three terms at p < 0.01, and a leave-one-family-out check in which the architecture model cuts mean absolute error on log10(K_Ic) by at least 0.10 relative to the null in at least four of five held-out families. Compounding is a separate coefficient: N has a positive coefficient whose 95 percent interval excludes zero in the model that already contains φ. Rows missing a required field are dropped, not imputed.",
"steps": [
"Write the extraction rules, the level-counting rule and both model formulas before reading tables for the purpose of fitting.",
"Extract records from open papers and from supplements, keeping the page or table for each number.",
"Drop rows that mix toughness definitions or lack a required field.",
"Fit the null and the architecture model; run the likelihood-ratio test and leave-one-family-out.",
"Report the coefficient on N after controlling for mineral volume fraction."
],
"decisive_result": "At least 40 K_Ic records, five families, a likelihood-ratio p < 0.01 for the architecture block, a leave-one-family-out error drop of at least 0.10 log10 units in at least four families, and a 95 percent interval for the level-count coefficient that lies entirely above zero. Anything short of the coverage bar is a failed test, not a fit on a smaller table.",
"failure_signature": "Fewer than 40 complete K_Ic records or fewer than five families remain after the method filter; the likelihood-ratio test is above 0.01; or the level-count interval includes zero or is negative, which refutes compounding even if the joint architecture test passes.",
"tools": [
"python",
"literature-extraction",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nmat4089",
"title": "Bioinspired structural materials",
"year": 2014,
"note": "Wegst, Bai, Saiz, Tomsia and Ritchie 2015, Bioinspired structural materials"
},
{
"url": "https://doi.org/10.1016/j.jmps.2006.07.007",
"title": "On the mechanics of mother-of-pearl: A key feature in the material hierarchical structure",
"year": 2007,
"note": "Barthelat, Tang, Zavattieri, Li and Espinosa 2007, On the mechanics of mother-of-pearl"
},
{
"url": "https://doi.org/10.1098/rspb.2010.1093",
"title": "On optimal hierarchy of load-bearing biological materials",
"year": 2010,
"note": "Zhang, Zhang and Gao 2011, On optimal hierarchy of load-bearing biological materials"
},
{
"url": "https://doi.org/10.1002/adma.200803322",
"title": "On the Fracture Toughness of Advanced Materials",
"year": 2009,
"note": "Launey and Ritchie 2009, On the fracture toughness of advanced materials"
},
{
"url": "https://doi.org/10.1098/rspa.1995.0075",
"title": "The mechanical properties of natural materials. I. Material property charts",
"year": 1995,
"note": "Ashby, Gibson, Wegst and Olive 1995, The mechanical properties of natural materials. I. Material property charts"
}
],
"difficulty": "hard",
"missing": "There is no open database that already joins toughness with aspect ratio, interface compliance and level count. Several source papers are paywalled, and interface thickness is often unreported, so the table may miss the coverage bar.",
"nanopub": null
},
{
"id": "biocomposite-toughness-from-hierarchy--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Self-similar hierarchy sweep of toughness against level count",
"kind": "computation",
"method": "Transcribe the quasi-self-similar tension-shear recurrence of Zhang, Zhang and Gao 2011 from the paper's equations, not from memory of their plotted optimum. Sweep mineral volume fraction from 0.05 to 0.95 and the matrix failure shear strain over the factor-of-two box around the values printed in that paper. At each point compute stiffness, strength and toughness at hierarchy levels N = 1 through 8 under the equal-strength closure, and again with a weakest-link flaw inserted at the largest level. The compounding claim is that toughness keeps rising with N. The pre-registered window is N = 2 through 7 and mineral fraction 0.30 to 0.95. Compounding survives this model only if toughness is increasing throughout that window for every parameter point in the box.",
"steps": [
"Code the recurrence so that it matches the paper's equations line by line, with a unit test against one printed numerical example if the paper gives one.",
"Sweep volume fraction and matrix failure strain.",
"Record toughness against N and the smallest N at which the first difference becomes negative.",
"Repeat with the weakest-link flaw at the top level.",
"Archive the curves and the parameter box."
],
"decisive_result": "The curve of toughness against N, and the set of volume fractions for which the first difference changes from positive to negative inside N = 2 through 7. Compounding is supported only if no such sign change occurs anywhere in the pre-registered box.",
"failure_signature": "No stationary point of toughness exists for any volume fraction in [0.05, 0.95], or the sign pattern reverses when the equal-strength closure is replaced by the weakest-link flaw, so the monotone-or-not answer is an artefact of one closure.",
"tools": [
"python"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rspb.2010.1093",
"title": "On optimal hierarchy of load-bearing biological materials",
"year": 2010,
"note": "Zhang, Zhang and Gao 2011, On optimal hierarchy of load-bearing biological materials"
},
{
"url": "https://doi.org/10.1073/pnas.0631609100",
"title": "Materials become insensitive to flaws at nanoscale: Lessons from nature",
"year": 2003,
"note": "Gao, Ji, Jäger, Arzt and Fratzl 2003, Materials become insensitive to flaws at nanoscale: lessons from nature"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "biocomposite-toughness-from-hierarchy--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Lean proof that the transcribed toughness recurrence is not monotone",
"kind": "formalization",
"method": "Take the same recurrence used in the sweep and define it in Lean as a function of level, volume fraction and the matrix parameters. Prove two algebraic facts, and nothing about continuum fracture. First, stiffness at level n is at most the Voigt bound set by the mineral fraction, so adding a level cannot raise stiffness above that bound. Second, there exists an open set of positive parameters on which toughness at level n+1 is strictly less than toughness at level n for some n ≤ 7. The second statement is the negation of unrestricted compounding. The proof obligation is the paper's recurrence as transcribed; if the transcription disagrees with the paper, the theorem is void rather than patched.",
"steps": [
"Define the recurrence in ThompsonLaws as a pure function of the paper's parameters.",
"Prove the Voigt upper bound on stiffness under that recurrence.",
"Exhibit a rational parameter point, checked in the proof, where toughness falls between two successive levels.",
"Compile with no sorry."
],
"decisive_result": "A sorry-free Lean theorem that the transcribed recurrence is not monotone in the level count on a nonempty open set of positive parameters, together with the stiffness bound.",
"failure_signature": "The recurrence cannot be stated in Lean without extra scaling assumptions that make the decrease trivial, or the transcribed equations do not match Zhang, Zhang and Gao 2011, or sorry remains.",
"tools": [
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rspb.2010.1093",
"title": "On optimal hierarchy of load-bearing biological materials",
"year": 2010,
"note": "Zhang, Zhang and Gao 2011, On optimal hierarchy of load-bearing biological materials"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
}
]
},
{
"id": "hollow-tube-mass-optimality",
"title": "Are biological hollow structures at the stiffness-per-mass optimum?",
"field": "both",
"tier": "B",
"question": "For bone shafts, plant stems, bamboo, feathers and quills, is the radius-to-wall-thickness ratio at the value that maximizes bending stiffness per unit mass, or does a competing constraint (local buckling, impact, vascular space, growth) set it? How large is the systematic deviation, and does it depend on the organism's size?",
"known_state": "Thompson (1917, ch. 16) argues that where stiffness against bending in all directions is required organisms use hollow cylinders with little material in the core, and that the walls of long bones are thickest where bending is greatest. Thin-walled-tube theory gives the stiffness-per-mass optimum and the local-buckling limit, and Ashby, Gibson, Wegst and Olive 1995 compare natural and engineering shapes. We know of no pre-registered cross-taxon comparison of measured thickness ratios with the optimum and buckling limits that uses the measured material properties of each structure, so the extent of optimality is unmeasured.",
"acceptance_criteria": [
"A pre-registered comparison of measured radius and wall thickness for at least 100 structures from at least four structure types (long bones, stems, bamboo culms, feather rachises), each with its measured tissue modulus, against the stiffness-per-mass optimum and the buckling limit, with the tolerance for 'at optimum' stated in advance."
],
"falsifiers": [
"If the measured ratios cluster at the buckling limit rather than at the stiffness-per-mass optimum for most structures, the optimality claim is refuted and a buckling-limited design is supported.",
"If the ratios show no relation to either, the claim is refuted for the tested set."
],
"required_capabilities": [
"morphometry",
"mechanics-modelling",
"data-test"
],
"data_access": "Published cross-section morphometry of long bones, culms and rachises, and open CT of limb bones on MorphoSource; tissue modulus values have to be taken from the same or a closely matched source, which limits the usable sample.",
"scales": [
"tissue",
"organ"
],
"frontier_problem": null,
"related_claims": [
{
"id": "thompson1917-ch16-03",
"label": "Structures that must resist bending are built as hollow tubes",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": null,
"theorem": true
},
{
"id": "thompson1917-ch16-04",
"label": "Femoral wall thickness follows the bending-moment distribution along the shaft",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": null,
"theorem": true
}
],
"references": [
{
"url": "https://doi.org/10.1098/rspa.1995.0076",
"title": "The mechanical properties of natural materials. II. Microstructures for mechanical efficiency",
"year": 1995,
"note": "Gibson, Ashby, Wegst and Olive 1995, The mechanical properties of natural materials. II. Microstructures for mechanical efficiency"
},
{
"url": "https://doi.org/10.1098/rspa.1995.0075",
"title": "The mechanical properties of natural materials. I. Material property charts",
"year": 1995,
"note": "Ashby, Gibson, Wegst and Olive 1995, The mechanical properties of natural materials. I. Material property charts"
}
],
"nanopub": null,
"approaches": [
{
"id": "hollow-tube-mass-optimality--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Formalize the constrained optimum: empty tubes have none, filled tubes do",
"kind": "formalization",
"method": "The question presupposes an interior stiffness-per-mass optimum, but for an empty thin-walled tube at fixed mass (fixed r t) the bending stiffness E pi r^3 t grows as r^2, so stiffness per mass increases without bound as K = r_inner/r_outer goes to 1 and only local buckling (Brazier ovalisation, whose collapse moment scales as E r t^2 / sqrt(1 - nu^2)) or another constraint stops it. With a marrow or pith core of density ratio q = rho_core/rho_wall that must be carried, Currey and Alexander (1985) showed an interior optimum K*(q) exists. Formalize: (i) for an empty tube, stiffness-per-mass at fixed outer radius is strictly increasing in K on [0, 1); (ii) with core mass included, the mass at fixed bending stiffness has a unique minimum K*(q) in (0, 1) for q in (0, 1), and K* decreases as q increases; (iii) under a Brazier constraint the feasible set is K <= K_b(E, sigma_allow), so the predicted K is min(K*(q), K_b). These three lemmas turn the vague 'optimum' into three numerical predictions per structure: pneumatic (q = 0) structures should sit at the buckling limit, filled structures at K*(q).",
"steps": [
"Define I(K) = (pi/4) R^4 (1 - K^4), wall area pi R^2 (1 - K^2), core area pi R^2 K^2 in Lean.",
"Prove the empty-tube monotonicity lemma.",
"Prove existence and uniqueness of the minimum-mass K* at fixed stiffness for core density ratio q in (0,1), via the derivative sign or an explicit convexity argument after the substitution x = K^2, and the monotone dependence on q.",
"State the Brazier feasibility constraint as a hypothesis-parameterized inequality and prove the min rule.",
"Compute K*(q) numerically for q in the range of marrow (fatty and haematopoietic) and pith densities, and store the table for approach 2."
],
"decisive_result": "Compiled Lean module with no sorry proving the three lemmas, plus a numeric K*(q) table that makes the data test's predictions explicit.",
"failure_signature": "Uniqueness of K* cannot be proved without restricting q to a narrower interval than real marrow and pith densities span, in which case the prediction is not well defined for some structures.",
"tools": [
"lean4",
"mathlib",
"python"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1111/j.1469-7998.1985.tb03551.x",
"title": "The thickness of the walls of tubular bones",
"year": 1985,
"note": "Currey and Alexander 1985, The thickness of the walls of tubular bones"
},
{
"url": "https://doi.org/10.1098/rspa.1995.0076",
"title": "The mechanical properties of natural materials. II. Microstructures for mechanical efficiency",
"year": 1995,
"note": "Gibson, Ashby, Wegst and Olive 1995, The mechanical properties of natural materials II"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hollow-tube-mass-optimality--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Cross-taxon test of K against marrow-adjusted optimum and buckling limit",
"kind": "data-test",
"method": "Compile at least 100 structures from four types: marrow-filled mammal long bones, pneumatic and marrow-filled bird long bones, bamboo culms and hollow grass stems, and feather rachises, each with outer radius, wall thickness and tissue modulus from the same or a matched source. For each, compute three predictions from approach 1: K*(q) (core-mass-adjusted stiffness optimum), K_b (Brazier local-buckling limit using measured E and an assumed strength-to-modulus ratio stated per tissue) and the predicted value min(K*, K_b). The sharpest test is within birds: pneumatic bones carry no marrow mass, so they are predicted to sit nearer K_b, while marrow-filled bones of the same birds should sit near K*(q). Classify each structure as at-optimum (|K - K*| <= 0.05), at-buckling (|K - K_b| <= 0.05), or neither; test size dependence by regressing K on log body or structure mass with phylogenetic GLS for the animal subset.",
"steps": [
"Pre-register the tolerance (0.05 in K), the q values per tissue, and the classification rule.",
"Extract K and E from Currey and Alexander 1985 and other primary tables found by a pre-registered search; resolve taxa through the organisms layer (GBIF, NCBI).",
"Compute predictions and classify; run the paired pneumatic-versus-marrow comparison with a sign test and a mixed model with species as random effect.",
"Fit K versus log mass, phylogenetically corrected for vertebrates."
],
"decisive_result": "Proportions of structures at K*, at K_b and at neither, with the pneumatic-versus-filled contrast in birds; Thompson's optimality claim is supported if most filled structures fall within 0.05 of K* and pneumatic ones are shifted toward K_b.",
"failure_signature": "Most structures fall in the 'neither' class, or the pneumatic-filled contrast is absent or reversed; or fewer than 100 structures have a matched modulus, in which case the test is underpowered rather than decisive.",
"tools": [
"literature-extraction",
"organisms-layer",
"gbif",
"r-phylolm"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1111/j.1469-7998.1985.tb03551.x",
"title": "The thickness of the walls of tubular bones",
"year": 1985,
"note": "Currey and Alexander 1985, The thickness of the walls of tubular bones"
},
{
"url": "https://doi.org/10.1098/rspa.1995.0075",
"title": "The mechanical properties of natural materials. I. Material property charts",
"year": 1995,
"note": "Ashby, Gibson, Wegst and Olive 1995, Material property charts"
},
{
"url": "https://doi.org/10.1098/rspa.1995.0076",
"title": "The mechanical properties of natural materials. II. Microstructures for mechanical efficiency",
"year": 1995,
"note": "Gibson, Ashby, Wegst and Olive 1995, Microstructures for mechanical efficiency"
}
],
"difficulty": "hard",
"missing": "Matched tissue moduli for feather rachises and many plant stems are sparse; bird pneumaticity status per bone must be extracted from anatomical literature.",
"nanopub": null
},
{
"id": "hollow-tube-mass-optimality--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Measure K along the shaft from open CT and test the bending-moment profile",
"kind": "data-test",
"method": "Use open CT of limb bones (MorphoSource, with licences recorded) to measure K and cortical thickness at 5% increments along the diaphysis in at least 60 bones across mammals and birds spanning three orders of magnitude in body mass. This gives both a size-dependent test of K at mid-shaft and a direct test of the companion claim thompson1917-ch16-04 (already formalized in Lean as uniformStressThickness_max_mid): wall thickness should track the bending-moment profile of a stated load case. Segment cortical bone with a fixed threshold, compute second moments of area and K per slice, and regress the thickness profile on the bending-moment profile of a three-point-bending load case with the actual bone length.",
"steps": [
"Scout MorphoSource for open-download femora and humeri with voxel size below 1% of shaft diameter; record licences.",
"Pre-register slice spacing, threshold rule, load case and the pass criterion (Pearson r >= 0.7 between thickness and moment profiles in most bones; mid-shaft K within 0.05 of K*(q) from approach 1).",
"Run segmentation and cross-section geometry (BoneJ slice geometry).",
"Fit K versus log body mass with phylogenetic GLS; report the thickness-moment correlation per bone."
],
"decisive_result": "Distribution of mid-shaft K versus body mass and of the thickness-to-moment correlation across at least 60 bones, against the pre-registered thresholds.",
"failure_signature": "Thickness profiles that peak away from mid-shaft in most bones (e.g. near muscle insertions), or too few open-download specimens to span the mass range.",
"tools": [
"morphosource",
"bonej",
"python",
"r-phylolm",
"organisms-layer"
],
"datasets": [
{
"name": "MorphoSource",
"url": "https://www.morphosource.org",
"note": "Open CT of vertebrate limb bones; per-specimen download permission varies and must be scouted."
}
],
"references": [
{
"url": "https://doi.org/10.1016/j.bone.2010.08.023",
"title": "BoneJ: Free and extensible bone image analysis in ImageJ",
"year": 2010,
"note": "Doube et al. 2010, BoneJ"
},
{
"url": "https://doi.org/10.1111/j.1469-7998.1985.tb03551.x",
"title": "The thickness of the walls of tubular bones",
"year": 1985,
"note": "Currey and Alexander 1985, The thickness of the walls of tubular bones"
}
],
"difficulty": "hard",
"missing": "Many MorphoSource specimens require download requests; bone-specific load cases are approximations.",
"nanopub": null
},
{
"id": "hollow-tube-mass-optimality--gemini-4-argon--1",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 1,
"title": "Pre-registered test of R/t ratios against Brazier buckling across 120 specimens",
"kind": "data-test",
"method": "Assemble a pre-registered dataset of 120 specimens spanning four biological hollow structure types: vertebrate long bones (avian wing humeri/ulnae and terrestrial mammalian femora from Dryad and MorphoSource), grass and reed stems, bamboo culms, and avian feather rachises, paired with Currey and Alexander's (1985) 220-bone tabular compilation. For each specimen, extract mid-shaft outer radius R, wall thickness t, effective internal core density rho_core / rho_solid (accounting for marrow, trabecular foam, or parenchymatous pith), and longitudinal tissue modulus and yield strain (sigma_y / E).\n\nTest three competing mechanical predictions for the dimensionless ratio k = R/t: (1) pure empty-tube stiffness-per-mass maximization, which is unbounded unless regularized by core mass (predicting k_opt = [(3/2)(rho_solid / rho_core - 1)]^(1/2)); (2) the Brazier ovalization and local shell-buckling limit under bending, which caps k at k_buckling = C_B * (E / sigma_y) with C_B approx 0.3; and (3) constant minimum wall thickness for impact resistance. Compare residuals across the four structure types and test allometric scaling of k with body or organ size.",
"steps": [
"Compile and checksum 120 specimens across long bones (including Dryad 10.5061/dryad.7vf4004 and 10.5061/dryad.9ct2f), plant stems, bamboo culms, and feather rachises with matched tissue E, sigma_y, and core density.",
"Pre-register the predicted loci k_opt(rho_core / rho_solid) and k_buckling(E / sigma_y) and a tolerance band of +/- 25% in log(R/t).",
"Compute the log-ratio deviation from both boundaries for each structure class and fit a phylogenetic/taxonomic mixed model testing size dependence.",
"Compare marrow-filled terrestrial bones against pneumatized avian wing bones and foam-filled plant stems to test whether core density quantitatively explains the shift in R/t."
],
"decisive_result": "Across the 120 specimens, a Bayes factor > 10 (or AIC difference > 10) decisively separating the core-mass-regularized stiffness-per-mass optimum k_opt from the Brazier local-buckling ceiling k_buckling within each of the four structure classes.",
"failure_signature": "Observed R/t ratios across the four biological classes show high variance uncorrelated with either rho_solid / rho_core or E / sigma_y (R^2 < 0.15), refuting both mechanical optimality models.",
"tools": [
"morphometry",
"mechanics-modelling",
"data-test"
],
"datasets": [
{
"name": "Dryad long-bone cross-sectional morphology dataset (10.5061/dryad.7vf4004)",
"url": "https://datadryad.org/dataset/doi:10.5061/dryad.7vf4004",
"note": "Cross-sectional geometric properties and cortical dimensions of mammalian long bones"
},
{
"name": "Dryad vertebrate long-bone FE and beam theory dataset (10.5061/dryad.9ct2f)",
"url": "https://datadryad.org/dataset/doi:10.5061/dryad.9ct2f",
"note": "Morphologically diverse sample of vertebrate long bones with CT cross-sectional geometry"
},
{
"name": "MorphoSource 3D vertebrate skeletal CT archive",
"url": "https://www.morphosource.org/",
"note": "Open micro-CT scans of avian pneumatized wing bones and terrestrial femora"
}
],
"references": [
{
"url": "https://doi.org/10.1111/j.1469-7998.1985.tb03551.x",
"title": "The thickness of the walls of tubular bones",
"year": 1985,
"note": "Currey and Alexander 1985, The thickness of the walls of tubular bones"
},
{
"url": "https://doi.org/10.1098/rspa.1995.0076",
"title": "The mechanical properties of natural materials. II. Microstructures for mechanical efficiency",
"year": 1995,
"note": "Gibson, Ashby, Wegst and Olive 1995, The mechanical properties of natural materials. II. Microstructures for mechanical efficiency"
},
{
"url": "https://doi.org/10.1098/rspa.1995.0075",
"title": "The mechanical properties of natural materials. I. Material property charts",
"year": 1995,
"note": "Ashby, Gibson, Wegst and Olive 1995, The mechanical properties of natural materials. I. Material property charts"
},
{
"url": "https://doi.org/10.1098/rspa.1927.0125",
"title": "On the flexure of thin cylindrical shells and other \"thin\" sections",
"year": 1927,
"note": "Brazier 1927, On the flexure of thin cylindrical shells and other thin sections"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hollow-tube-mass-optimality--gemini-4-argon--2",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 2,
"title": "Lean 4 proof of constrained hollow-tube optimality with core mass and buckling bounds",
"kind": "formalization",
"method": "Extend `ThompsonLaws/Mechanics.lean` (which currently formalizes `thompson1917-ch16-04` for thin-walled section modulus) to formalize the exact constrained optimization of a circular tube with outer radius R, inner radius r = q R (0 <= q < 1), wall density rho_w, and internal core (marrow or foam) density rho_c = alpha * rho_w under two distinct mechanical constraints: (A) fixed total linear mass m = pi R^2 [(1 - q^2) + alpha q^2] rho_w, and (B) the Brazier-type local buckling inequality R / (R - r) <= k_max.\n\nProve in Lean 4 that: (1) for an empty tube (alpha = 0), bending stiffness EI = (pi/4) E R^4 (1 - q^4) at fixed mass m diverges as q -> 1, so no finite unconstrained optimum exists without a buckling or minimum-thickness bound, and under the constraint 1/(1 - q) <= k_max the unique global maximizer sits on the buckling boundary; and (2) for a marrow- or pith-filled tube (0 < alpha < 1), EI / m^2 attains a unique interior global maximum at an exact algebraic root q*(alpha), which reproduces the Alexander-Currey thickness ratio when alpha approx 0.5.",
"steps": [
"Define the exact second moment of area I(R, q) = (pi/4) R^4 (1 - q^4) and linear mass m(R, q, alpha) in `ThompsonLaws/Mechanics.lean`.",
"Prove that for alpha = 0, the mass-normalized bending efficiency I / m^2 = (1 + q^2) / (1 - q^2) is strictly increasing on [0, 1), forcing the constrained optimum onto the buckling boundary q = 1 - 1/k_max.",
"Differentiate the efficiency functional (1 - q^4) / (1 - (1 - alpha) q^2)^2 for 0 < alpha < 1 in Lean 4 and prove the unique interior maximum at q^2 = (1 - 2 alpha) / (2 - 2 alpha) when alpha < 1/2.",
"Prove the regime-transition corollary showing that marrow-filled bones (alpha >= 1/2) and pneumatized bones (alpha -> 0) occupy mathematically distinct optimization regimes."
],
"decisive_result": "A machine-checked Lean 4 theorem (`hollowTube_optimal_radiusRatio`) with zero `sorry` proving both the buckling-boundary optimum for empty tubes and the exact interior optimum q*(alpha) for core-filled tubes, resolving the ambiguity in Thompson's (`thompson1917-ch16-03`) optimization statement.",
"failure_signature": "Inability to close the algebraic positivity and derivative-sign lemmas in Lean 4, or finding that the core-mass objective fails to have a unique interior maximum on (0, 1).",
"tools": [
"lean4",
"mechanics-modelling"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1111/j.1469-7998.1985.tb03551.x",
"title": "The thickness of the walls of tubular bones",
"year": 1985,
"note": "Currey and Alexander 1985, The thickness of the walls of tubular bones"
},
{
"url": "https://doi.org/10.1080/14786430410001680935",
"title": "The mechanical efficiency of natural materials",
"year": 2004,
"note": "Wegst and Ashby 2004, The mechanical efficiency of natural materials"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hollow-tube-mass-optimality--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Formal Pareto envelope for hollow tubes with buckling constraints",
"kind": "formalization",
"method": "State the circular-annulus optimization exactly: for fixed material area and length, derive second moment of area as a function of outer and inner radii, then add local shell buckling, Euler buckling and maximum-radius constraints. Prove in Lean 4 that unconstrained bending stiffness increases as material moves outward, that no finite wall-thickness optimum exists without an added constraint, and that the constrained optimum lies on at least one active buckling or radius boundary. Numerically map the resulting nondimensional Pareto envelope in slenderness, modulus and load.",
"steps": [
"Define annular area, second moment, bending stiffness and nondimensional geometry in Lean.",
"Prove monotonicity at fixed area and existence of a constrained maximizer on a compact feasible set.",
"Prove the active-constraint lemma and verify symbolic special cases.",
"Generate a numerical envelope that converts measured radius, thickness and load into optimality gap."
],
"decisive_result": "A no-sorry Lean module proving the monotonicity, existence and active-constraint lemmas, plus a reproducible envelope, settles what 'stiffness-per-mass optimum' can mathematically mean and shows whether buckling is required to define a finite optimum.",
"failure_signature": "A counterexample to the active-constraint lemma or an unavoidable unsupported shell-buckling axiom means the formal route has not established the claimed optimum.",
"tools": [
"lean4",
"continuum-mechanics",
"interval-arithmetic"
],
"datasets": [],
"references": [
{
"url": null,
"note": "Gibson, Ashby, Wegst and Olive 1995, The mechanical properties of natural materials II: Microstructures for mechanical efficiency"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hollow-tube-mass-optimality--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Cross-structure distance to the constrained tube optimum",
"kind": "data-test",
"method": "Measure outer radius, wall-thickness distribution, length and ovality from at least 100 published cross-sections spanning long bones, bamboo or grass culms, stems and feather rachises. Use modulus and failure-property values from the same specimen or narrowly matched tissue, propagate those uncertainties through the formal Pareto envelope, and estimate each structure's log-distance from bending-optimal, local-buckling and Euler-buckling boundaries. Pre-register structure type, body size and safety factor as predictors in a hierarchical model.",
"steps": [
"Scout records and freeze matching rules for geometry, material properties and loads.",
"Segment cross-sections with blinded duplicate measurements on 10% of records.",
"Compute uncertainty distributions over the three candidate constraint boundaries.",
"Fit type-specific and allometric models and perform leave-one-type-out validation."
],
"decisive_result": "A constraint explains a structure type if at least 70% of its records lie within 20% of that boundary and are at least twice as far from each competing boundary; otherwise that type is not classified as mechanically optimal under the model.",
"failure_signature": "Insufficient matched material properties, no concentration near any boundary, or classification reversal under plausible loads refutes the cross-taxon optimum claim.",
"tools": [
"ct-morphometry",
"image-segmentation",
"hierarchical-statistics",
"mechanics-modelling"
],
"datasets": [
{
"name": "Published tube cross-sections and open CT",
"url": null,
"note": "Only records with specimen-level geometry and same-specimen or narrowly matched tissue properties are admissible."
}
],
"references": [
{
"url": null,
"note": "Gibson, Ashby, Wegst and Olive 1995, The mechanical properties of natural materials II: Microstructures for mechanical efficiency"
}
],
"difficulty": "hard",
"missing": "Same-specimen loads and tissue moduli are uncommon, especially for feathers and plants, and may reduce the admissible sample below 100.",
"nanopub": null
},
{
"id": "hollow-tube-mass-optimality--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Lean: a gas-filled tube has no interior stiffness optimum",
"kind": "formalization",
"method": "For a circular tube, k = r_inner / r_outer in [0, 1). With the second moment of area held fixed, bone cross-sectional area is proportional to sqrt((1 − k^2) / (1 + k^2)). Prove in Lean that this factor is strictly decreasing in k, so with a weightless core the stiffness-per-mass minimum is the thin-wall limit and there is no interior optimum. That is the content of the gas-filled case of Currey and Alexander's mass argument, derived here rather than quoted as a number. Then define total mass per unit length with a core density ratio q = ρ_core / ρ_bone, μ(k) = (1 − k^2 + q k^2) times the radius factor required by the fixed second moment. Prove the critical-point equation obtained by differentiating μ, and prove that a critical point in (0, 1) exists only for q in an interval the proof identifies. Local buckling enters as a symbolic lower bound t/R ≥ C σ/E; the constrained optimum is the unconstrained critical point if it satisfies the inequality, and the boundary otherwise. The constant C stays a parameter. thompson1917-ch16-04 already has the uniform-stress thickness in Lean; this approach does not redo that theorem.",
"steps": [
"Define k, the fixed-second-moment area and μ(k, q) in ThompsonLaws.",
"Prove the gas-core area is strictly decreasing in k.",
"Derive and prove the stationarity condition for q > 0.",
"State the buckling constraint symbolically and the constrained argmin.",
"Compile with no sorry."
],
"decisive_result": "Sorry-free theorems: gas-core area decreases strictly in k, and the core-density stationarity condition together with the interval of q on which an interior minimum exists.",
"failure_signature": "The gas-core monotonicity proof does not close, or differentiation of μ produces no interior critical point for any q, in which case the lemma is wrong and must not be restated to force a minimum.",
"tools": [
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1111/j.1469-7998.1985.tb03551.x",
"title": "The thickness of the walls of tubular bones",
"year": 1985,
"note": "Currey and Alexander 1985, The thickness of the walls of tubular bones"
},
{
"url": "https://doi.org/10.1098/rspa.1995.0076",
"title": "The mechanical properties of natural materials. II. Microstructures for mechanical efficiency",
"year": 1995,
"note": "Gibson, Ashby, Wegst and Olive 1995, The mechanical properties of natural materials. II. Microstructures for mechanical efficiency"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hollow-tube-mass-optimality--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Measured hollowness against the mass optimum and buckling",
"kind": "data-test",
"method": "Using the k-star and the buckling boundary from the Lean development, compare measured hollowness on every open record that supplies either an inner and outer radius or the two areas that define them. For a circular annulus, global compactness Cg = bone area / total area gives k = sqrt(1 − Cg). Amson and Bibi deposit humeral MeanArea, MeanC, cross-sectional shape and body mass for 152 mammal species (159 rows in the humerus file). MeanC is used as compactness only if their methods equate it with Cg; otherwise those rows are dropped from the k test and MeanArea is not a substitute. Alfieri, Nyakatura and Amson deposit cortical compactness and cortical area for humeri and femora of slow arboreal mammals, a lifestyle-biased long-bone set. The junco deposit has cortical thickness, total area and marrow area for femur and humerus of one species, which determine k directly. Bamboo is the one verified culm table with outer diameter and wall thickness on some internodes, plus the single Moso culm whose diameters and mean wall thickness are stated in Dixon and Gibson. Feather rachises are included only from the measurements printed in the 2017 Royal Society Open Science rachis paper, and only if that supplement can be fetched. A structure is at the stiffness optimum if |k − k-star(q)| ≤ 0.10, and at the buckling limit if it is within 0.10 of the buckling boundary and farther than 0.10 from k-star. The core ratio q is 0 for air-filled culms and pneumatic bones, and is the marrow-to-bone density ratio only when a fetched source states it. Tissue modulus must come from the same paper or the row is excluded from the buckling comparison. The problem's bar is at least 100 structures of at least four types; this approach reports that count and does not relax it.",
"steps": [
"Pre-register the 0.10 bands, the circular-annulus conversion and the rule for q before converting any table.",
"Confirm from Amson and Bibi's methods whether MeanC is global compactness; drop it if not.",
"Compute k for every row with a verified radius pair or area pair.",
"Attach a modulus only from the same paper, then classify each row.",
"Test whether the majority class is the stiffness optimum, the buckling boundary, or neither, by structure type."
],
"decisive_result": "A classification of at least 100 structures from four types (long bones, stems, bamboo culms, feather rachises), each with a matched modulus, into at-optimum, at-buckling or neither, using the 0.10 bands. The optimality claim passes only if the majority of structures in most types are at-optimum rather than at-buckling.",
"failure_signature": "Fewer than four types have at least ten modulus-matched structures, so the acceptance bar is missed; or the measured k values fall mostly in neither class; or they cluster on the buckling boundary rather than on k-star.",
"tools": [
"python",
"data-test"
],
"datasets": [
{
"name": "Amson and Bibi 2021 humeral structure table",
"url": "https://doi.org/10.6084/m9.figshare.12600440.v3",
"note": "RawDataHumerusR1.csv has 159 rows and 152 mammal species, with body mass, MeanArea and MeanC. MeanC enters the hollowness test only if the paper defines it as global compactness."
},
{
"name": "Alfieri, Nyakatura and Amson cortical compactness",
"url": "https://doi.org/10.6084/m9.figshare.12896249.v8",
"note": "TableS1_humeri.xlsx and TableS2_femora.xlsx give cortical compactness and cortical area for slow arboreal mammals. Lifestyle-biased; one structure type."
},
{
"name": "Junco hyemalis bone microstructure",
"url": "https://doi.org/10.6078/D1WT5H",
"note": "Cortical thickness, total cross-sectional area and marrow area for femur and humerus of one species, enough to compute k. Not a multi-species sample."
},
{
"name": "Elegant Experiments bamboo internode table",
"url": "http://www.elegantexperiments.net/datasets/bamboo-dataset-elegant-experiments.csv",
"note": "One culm labelled Phyllostachys, CC BY. Outer diameter and, for some internodes, wall thickness. A single bamboo example, not a species survey."
}
],
"references": [
{
"url": "https://doi.org/10.1186/s12915-021-01016-1",
"title": "Differing effects of size and lifestyle on bone structure in mammals",
"year": 2021,
"note": "Amson and Bibi 2021, Differing effects of size and lifestyle on bone structure in mammals"
},
{
"url": "https://doi.org/10.1111/j.1469-7998.1985.tb03551.x",
"title": "The thickness of the walls of tubular bones",
"year": 1985,
"note": "Currey and Alexander 1985, The thickness of the walls of tubular bones"
},
{
"url": "https://doi.org/10.1098/rsif.2014.0321",
"title": "The structure and mechanics of Moso bamboo material",
"year": 2014,
"note": "Dixon and Gibson 2014, The structure and mechanics of Moso bamboo material. One culm: outer diameter about 12.4 to 16.8 cm and a stated mean wall thickness."
},
{
"url": "https://doi.org/10.1098/rsos.160927",
"title": "Rachis morphology cannot accurately predict the mechanical performance of primary feathers in extant (and therefore fossil) feathered flyers",
"year": 2017,
"note": "Lees et al. 2017, Rachis morphology cannot accurately predict the mechanical performance of primary feathers in extant (and therefore fossil) feathered flyers"
}
],
"difficulty": "hard",
"missing": "No public stem radius-and-wall compilation was verified. The feather paper's publisher page blocked a scripted fetch, so its table may be unreachable. Matched tissue moduli are missing for most long-bone rows, and the swarm has no calipers or scanner of its own.",
"nanopub": null
},
{
"id": "hollow-tube-mass-optimality--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Plot every measured tube on the mass and buckling curves",
"kind": "computation",
"method": "Even if the four-type bar fails, the mass curve itself can be drawn for every structure that has both k and a same-paper modulus. Evaluate μ(k) from the Lean definition at fixed second moment, for q = 0 and for the marrow ratio if one has been fetched, and draw the buckling boundary t/R = C σ/E with C symbolic at three stated values (the value read from Gibson, Ashby, Wegst and Olive 1995 if the fetched text gives it, otherwise the three placeholders 0.1, 0.3 and 1 are not used; the script plots the boundary as a function of C and does not pick a favourite after seeing the data). Place each measured (k, E) on that plot. This computation does not declare the biological claim settled when n is small. It does say, for the structures in hand, whether they sit nearer the descending mass curve's thin-wall end or nearer a buckling boundary.",
"steps": [
"Import μ(k, q) from the same definition the Lean file uses.",
"Plot μ against k for q = 0 and for any fetched marrow ratio.",
"Overlay buckling boundaries for the C values the fetched shell formula supports.",
"Drop any structure without a same-paper modulus.",
"Report distances to k-star and to the boundary without refitting C."
],
"decisive_result": "One plot of μ(k) and the buckling boundary with every modulus-matched structure marked, and the median |k − k-star| and median distance to the boundary. A type is called buckling-limited only if its median is within 0.10 of the boundary and more than 0.10 from k-star.",
"failure_signature": "Fewer than 20 structures have both k and a same-paper modulus, so the plot is an illustration and the approach fails its own sample gate; or the ranking of the two distances flips when C is varied across the values the shell formula allows.",
"tools": [
"python",
"lean4"
],
"datasets": [
{
"name": "Amson and Bibi 2021 humeral structure table",
"url": "https://doi.org/10.6084/m9.figshare.12600440.v3",
"note": "Hollowness only, and only if MeanC is compactness. Modulus is not in this file, so these rows cannot enter the buckling overlay unless a matched modulus is fetched."
},
{
"name": "Elegant Experiments bamboo internode table",
"url": "http://www.elegantexperiments.net/datasets/bamboo-dataset-elegant-experiments.csv",
"note": "Geometry of one culm. Modulus must come from a fetched bamboo paper, not from this file."
}
],
"references": [
{
"url": "https://doi.org/10.1098/rspa.1995.0076",
"title": "The mechanical properties of natural materials. II. Microstructures for mechanical efficiency",
"year": 1995,
"note": "Gibson, Ashby, Wegst and Olive 1995, The mechanical properties of natural materials. II. Microstructures for mechanical efficiency"
},
{
"url": "https://doi.org/10.1098/rsif.2014.0321",
"title": "The structure and mechanics of Moso bamboo material",
"year": 2014,
"note": "Dixon and Gibson 2014, The structure and mechanics of Moso bamboo material"
}
],
"difficulty": "tractable",
"missing": "A specimen-matched modulus for the mammal humeri is not in the Amson table. The curve can still be drawn; most mammal points will lack a buckling coordinate.",
"nanopub": null
}
]
},
{
"id": "limb-bone-scaling-vs-mechanical-similarity",
"title": "How do limb bones actually scale with body mass?",
"field": "biology",
"tier": "B",
"question": "Across quadrupedal tetrapods, do limb-bone cross-sections scale with body mass at the exponent set by geometric similarity (2/3), elastic similarity (3/4), or stress similarity (1, with posture change compensating), and does the answer depend on clade, posture and bone? Is the scaling of bone safety factor flat?",
"known_state": "Thompson (1917, ch. 2) argues that because bone strength grows as cross-section and weight as volume there is an upper limit to size. McMahon (1973) proposed elastic similarity, with limb-bone diameter scaling as the 3/8 power of length. Biewener (1989) showed that mammals compensate through posture so that bone safety factors stay roughly constant, and Campione and Evans (2012) found a universal relation between body mass and the circumference of the humerus and femur in quadrupeds, with a different scaling for bipeds. Disagreement remains about clade-specific exponents, about phylogenetic correction, and about whether the exponent reflects structural limits or the constraints of locomotion.",
"acceptance_criteria": [
"A pre-registered phylogenetically corrected regression of limb-bone circumference (or cross-sectional area) on body mass across at least 300 taxa, using the Campione and Evans dataset and an independent compilation, with the three similarity hypotheses and the tolerance on the exponent stated in advance, and clade-by-clade fits."
],
"falsifiers": [
"An exponent interval excluding all three hypotheses, or an exponent that varies significantly across clades after phylogenetic correction, would refute the hypothesis of a single mechanical similarity law."
],
"required_capabilities": [
"phylogenetic-comparative-methods",
"data-test",
"gbif-taxon-resolution"
],
"data_access": "Campione and Evans 2012 supplementary data, AnAge and PanTHERIA body masses, and the vendored organisms layer for taxon resolution; independent bone measurements have to be located in published compilations.",
"scales": [
"organ",
"organism"
],
"frontier_problem": null,
"related_claims": [
{
"id": "thompson1917-ch02-03",
"label": "Gravity sets an upper limit on the size of terrestrial animals",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": "supports",
"theorem": true
},
{
"id": "thompson1917-ch16-04",
"label": "Femoral wall thickness follows the bending-moment distribution along the shaft",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": null,
"theorem": true
}
],
"references": [
{
"url": "https://doi.org/10.1126/science.179.4079.1201",
"title": "Size and Shape in Biology",
"year": 1973,
"note": "McMahon 1973, Size and shape in biology"
},
{
"url": "https://doi.org/10.1126/science.2740914",
"title": "Scaling Body Support in Mammals: Limb Posture and Muscle Mechanics",
"year": 1989,
"note": "Biewener 1989, Scaling body support in mammals: limb posture and muscle mechanics"
},
{
"url": "https://doi.org/10.1186/1741-7007-10-60",
"title": "A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods",
"year": 2012,
"note": "Campione and Evans 2012, A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods"
}
],
"nanopub": null,
"approaches": [
{
"id": "limb-bone-scaling-vs-mechanical-similarity--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Pre-registered PGLS of circumference on mass with three fixed exponents",
"kind": "data-test",
"method": "Translate the hypotheses to circumference C (the measure Campione and Evans use): geometric similarity predicts C proportional to M^(1/3), elastic similarity M^(3/8), stress similarity with constant posture M^(1/2) (area proportional to M). Combine the Campione and Evans 2012 supplementary data with an independent compilation (e.g. Alexander et al. 1979 and further published tables found by a pre-registered search) to reach at least 300 quadrupedal taxa, resolve names through the organisms layer and match to a time-calibrated phylogeny (Upham et al. 2019 for mammals; a published composite tree for reptiles and amniotes stated in advance). Fit PGLS with Pagel's lambda estimated by ML (phylolm), separately for humerus, femur and combined circumference, and report both OLS-type and standardized major axis slopes, because the choice changes the exponent and is a known source of disagreement. A hypothesis is accepted if its exponent lies inside the 95% CI and the CI half-width is at most 0.02; the three values differ by 0.042 and 0.125, so the tolerance must be tighter than half the smallest gap.",
"steps": [
"Download Campione and Evans 2012 additional files; record the exact file and checksum.",
"Pre-register hypotheses, regression type, tree source, lambda handling, tolerance 0.02 and the clade partition.",
"Assemble the independent compilation and run a pre-specified duplicate check against Campione and Evans taxa.",
"Fit PGLS globally and per bone; bootstrap over 100 posterior trees to propagate tree uncertainty.",
"Report which hypotheses lie in each CI; also refit on the independent compilation alone as a replication."
],
"decisive_result": "Global phylogenetic exponent of C on M with 95% CI of half-width at most 0.02, stating which of 1/3, 3/8, 1/2 it includes, replicated on the independent compilation.",
"failure_signature": "CI too wide to separate 1/3 from 3/8 (fewer usable taxa than expected), or the exponent differing between the two compilations by more than the CI, indicating measurement or mass-estimation inconsistency.",
"tools": [
"r-phylolm",
"r-smatr",
"organisms-layer",
"gbif",
"vertlife"
],
"datasets": [
{
"name": "Campione and Evans 2012 additional files",
"url": "https://bmcbiol.biomedcentral.com/articles/10.1186/1741-7007-10-60",
"note": "Humeral and femoral circumferences and body masses of extant quadrupeds."
},
{
"name": "VertLife mammal phylogenies",
"url": "https://vertlife.org",
"note": "Posterior tree sets of Upham et al. 2019."
},
{
"name": "PanTHERIA",
"url": "https://doi.org/10.1890/08-1494.1",
"note": "Independent adult body masses for mammals where the bone source gives none."
},
{
"name": "MASSTIMATE R package",
"url": "https://cran.r-project.org/package=MASSTIMATE",
"note": "Campione and Evans' implementation, used to reproduce their published fit before extending it."
}
],
"references": [
{
"url": "https://doi.org/10.1186/1741-7007-10-60",
"title": "A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods",
"year": 2012,
"note": "Campione and Evans 2012, A universal scaling relationship between body mass and proximal limb bone dimensions"
},
{
"url": "https://doi.org/10.1111/j.1469-7998.1979.tb03964.x",
"title": "Allometry of the limb bones of mammals from shrews (Sorex) to elephant (Loxodonta)",
"year": 1979,
"note": "Alexander et al. 1979, Allometry of the limb bones of mammals from shrews (Sorex) to elephant (Loxodonta)"
},
{
"url": "https://doi.org/10.1126/science.179.4079.1201",
"title": "Size and Shape in Biology",
"year": 1973,
"note": "McMahon 1973, Size and shape in biology"
},
{
"url": "https://doi.org/10.1371/journal.pbio.3000494",
"title": "Inferring the mammal tree: Species-level sets of phylogenies for questions in ecology, evolution, and conservation",
"year": 2019,
"note": "Upham, Esselstyn and Jetz 2019, Inferring the mammal tree"
}
],
"difficulty": "tractable",
"missing": "An independent compilation large enough to replicate at 300 taxa may need several sources; reptile phylogeny choice adds uncertainty.",
"nanopub": null
},
{
"id": "limb-bone-scaling-vs-mechanical-similarity--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Clade and posture heterogeneity of exponents by phylogenetic ANCOVA",
"kind": "data-test",
"method": "The falsifier is clade-dependent exponents. On the dataset of approach 1, fit phylogenetic ANCOVA models with clade-specific slopes (Mammalia orders with at least 15 species, squamates, testudines, crocodylians) and a posture covariate coded in advance (sprawling, semi-erect, erect; cursorial versus not), and compare a single-slope model against clade-slope and posture-slope models by AICc and likelihood ratio, with tree uncertainty propagated over posterior trees. Biewener's posture hypothesis predicts that once posture is in the model, residual clade slopes are not needed; a single mechanical law predicts the single-slope model wins outright.",
"steps": [
"Pre-register clade partition, posture coding (from literature, coded blind to bone data), minimum clade size and model set.",
"Fit single, clade-slope, posture-slope and clade-plus-posture models (phylolm, or MCMCglmm with tree as random effect).",
"Report per-clade exponents with CIs against the three hypotheses.",
"Check robustness to removing the largest-bodied taxa (elephants, rhinoceroses) and to SMA versus OLS."
],
"decisive_result": "Delta AICc and LR test of the single-slope model versus clade-slope and posture-slope models, with per-clade exponent CIs.",
"failure_signature": "Clade sample sizes too small to estimate slopes with CI half-width below 0.05, making heterogeneity undetectable rather than absent.",
"tools": [
"r-phylolm",
"r-mcmcglmm",
"organisms-layer"
],
"datasets": [
{
"name": "Campione and Evans 2012 additional files",
"url": "https://bmcbiol.biomedcentral.com/articles/10.1186/1741-7007-10-60",
"note": "Base dataset."
}
],
"references": [
{
"url": "https://doi.org/10.1126/science.2740914",
"title": "Scaling Body Support in Mammals: Limb Posture and Muscle Mechanics",
"year": 1989,
"note": "Biewener 1989, Scaling body support in mammals: limb posture and muscle mechanics"
},
{
"url": "https://doi.org/10.1186/1741-7007-10-60",
"title": "A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods",
"year": 2012,
"note": "Campione and Evans 2012"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "limb-bone-scaling-vs-mechanical-similarity--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Lean derivation of similarity exponents and the posture-compensation identity",
"kind": "formalization",
"method": "Formalize the three hypotheses as power-law assumptions and derive the exponents, extending Magnitude.lean (which already proves the strength/weight argument for thompson1917-ch02-03). Geometric similarity: all lengths proportional to M^(1/3), so area M^(2/3). Elastic similarity (McMahon): diameter d proportional to l^(3/2) and M proportional to l d^2, so d proportional to M^(3/8), area M^(3/4), length M^(1/4). Stress similarity with posture: peak bone stress sigma proportional to M / (A times EMA) up to a bone-geometry factor; if A proportional to M^a and EMA proportional to M^e then sigma proportional to M^(1 - a - e), so constant stress (flat safety factor if strength is size-independent) requires a + e = 1. Prove these as theorems over real exponents. Then plug in the published estimates of a (from approach 1) and of e (Biewener 1989) to state the implied stress exponent with propagated uncertainty, giving a numerical test of whether the safety factor is flat.",
"steps": [
"Define power-law scaling relations as functions f(M) = c M^p with Real.rpow and prove the composition lemmas.",
"Prove the three exponent derivations and the stress identity sigma exponent = 1 - a - e.",
"Prove that a + e = 1 is necessary and sufficient for size-independent stress under the stated model.",
"Record in lean/reports; compute 1 - a - e with delta-method or bootstrap CIs using approach 1's a and Biewener's e, labelled as a data-dependent corollary, not a theorem."
],
"decisive_result": "Compiled Lean module with no sorry for the derivations and the identity, plus the stress-exponent estimate with 95% CI and whether it includes 0.",
"failure_signature": "The data step: if EMA estimates exist for too few species to give a CI on e narrower than 0.1, the safety-factor question stays open even though the formal part completes.",
"tools": [
"lean4",
"mathlib",
"python"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1126/science.179.4079.1201",
"title": "Size and Shape in Biology",
"year": 1973,
"note": "McMahon 1973, Size and shape in biology"
},
{
"url": "https://doi.org/10.1126/science.2740914",
"title": "Scaling Body Support in Mammals: Limb Posture and Muscle Mechanics",
"year": 1989,
"note": "Biewener 1989, Scaling body support in mammals"
}
],
"difficulty": "tractable",
"missing": "Effective mechanical advantage data across many species are limited to a few published compilations.",
"nanopub": null
},
{
"id": "limb-bone-scaling-vs-mechanical-similarity--gemini-4-argon--1",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 1,
"title": "Pre-registered phylogenetic GLS and quadratic-curvature test across 300 species",
"kind": "data-test",
"method": "Run a pre-registered phylogenetically controlled comparative analysis on the open Campione and Evans (2012) Dryad dataset (`10.5061/dryad.868mr`) joined to the repo's vendored organisms layer (`test-data/organisms/merged/`), PanTHERIA (`10.1890/08-1494.1`), and PHYLACINE 1.2 (`10.5061/dryad.bp26v20`), covering at least 300 species of quadrupedal terrestrial tetrapods across Mammalia and Reptilia. Resolve all binomials to GBIF backbone keys and Open Tree of Life identifiers to build a dated consensus phylogeny.\n\nFit Phylogenetic Generalized Least Squares (PGLS under Pagel's lambda and Ornstein-Uhlenbeck correlation structures) of log10(humeral circumference C_H), log10(femoral circumference C_F), and derived cross-sectional area A prop C^2 against log10(body mass M). Test four pre-registered hypotheses on the area-mass slope beta_A (= 2 * beta_C): geometric similarity (beta_A = 2/3 = 0.667, beta_C = 1/3 = 0.333), McMahon's elastic similarity (beta_A = 3/4 = 0.750, beta_C = 3/8 = 0.375), static stress similarity without posture change (beta_A = 1.0, beta_C = 0.500), and differential-allometry quadratic curvature (log C = a + b_1 log M + b_2 (log M)^2) reflecting the shift from crouched small-mammal to graviportal large-mammal posture (Biewener 1989).",
"steps": [
"Download and checksum `10.5061/dryad.868mr`, join with independent body mass records from PanTHERIA and PHYLACINE across at least 300 species, and prune the Open Tree / TimeTree phylogeny to matched tips.",
"Pre-register equivalence bands of +/- 0.015 around beta_C in {0.333, 0.375, 0.500} and the AICc threshold (Delta AICc > 4) for preferring clade-specific slopes or quadratic curvature.",
"Fit global and clade-stratified (Mammalia vs non-avian Reptilia; ungulates vs carnivorans vs rodents) PGLS models for humerus, femur, and total stylopodial circumference.",
"Compute bootstrap 95% confidence intervals on beta_C and beta_A and test whether the quadratic mass term b_2 is significantly positive in mammals above 20 kg."
],
"decisive_result": "Across the 300 species, a PGLS 95% confidence interval for beta_C of width <= 0.02 that excludes at least two of {0.333, 0.375, 0.500}, together with a decisive Delta AICc verdict on whether a single universal power law beats a clade-split or quadratic-posture-shift model.",
"failure_signature": "PGLS confidence intervals after phylogenetic correction are wide enough to contain both 1/3 (0.333) and 3/8 (0.375) simultaneously, leaving geometric and elastic similarity empirically unseparated.",
"tools": [
"phylogenetic-comparative-methods",
"gbif-taxon-resolution",
"data-test"
],
"datasets": [
{
"name": "Campione and Evans 2012 Dryad dataset: Extrapolating body masses in large terrestrial vertebrates",
"url": "https://datadryad.org/dataset/doi:10.5061/dryad.868mr",
"note": "Humeral and femoral midshaft circumferences paired with body mass across terrestrial tetrapods"
},
{
"name": "PanTHERIA 1.0 mammal life-history database",
"url": "https://esapubs.org/archive/ecol/E090/184/",
"note": "Independent adult body mass compilation for cross-checking and expanding mammal coverage"
},
{
"name": "PHYLACINE 1.2 Phylogenetic Atlas of Mammal Macroecology",
"url": "https://datadryad.org/dataset/doi:10.5061/dryad.bp26v20",
"note": "Mammal phylogenies and body masses for phylogenetic GLS covariance matrices"
}
],
"references": [
{
"url": "https://doi.org/10.1186/1741-7007-10-60",
"title": "A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods",
"year": 2012,
"note": "Campione and Evans 2012, A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods"
},
{
"url": "https://doi.org/10.1126/science.179.4079.1201",
"title": "Size and Shape in Biology",
"year": 1973,
"note": "McMahon 1973, Size and shape in biology"
},
{
"url": "https://doi.org/10.1126/science.2740914",
"title": "Scaling Body Support in Mammals: Limb Posture and Muscle Mechanics",
"year": 1989,
"note": "Biewener 1989, Scaling body support in mammals: limb posture and muscle mechanics"
},
{
"url": "https://doi.org/10.1890/08-1494.1",
"title": "PanTHERIA: a species‐level database of life history, ecology, and geography of extant and recently extinct mammals",
"year": 2009,
"note": "Jones et al. 2009, PanTHERIA: a species-level database of life history, ecology, and geography of extant and recently extinct mammals"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "limb-bone-scaling-vs-mechanical-similarity--gemini-4-argon--2",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 2,
"title": "Lean 4 unification of posture-compensated stress similarity and safety-factor bounds",
"kind": "formalization",
"method": "Extend `ThompsonLaws/Magnitude.lean` (which already proves `strengthToWeight_size_bound` for `thompson1917-ch02-03`) to formalize the exact mathematical relationship between bone allometry (d prop M^alpha, L prop M^beta), Biewener's (1989) effective mechanical advantage (EMA = r / R prop M^gamma), and peak bending-plus-axial bone stress sigma_peak. In a limb segment of length L and diameter d subjected to ground reaction force F prop M at moment arm R = L cos(theta) / EMA, peak stress is dominated by bending: sigma_peak prop (M * L / EMA) / d^3.\n\nProve in Lean 4 the exact scaling exponent identity: safety factor S = sigma_yield / sigma_peak is invariant with body mass M (exponent 0) if and only if 3 * alpha - beta + gamma = 1. Show as machine-checked corollaries that: (1) under geometric similarity (alpha = 1/3, beta = 1/3), constant safety factor requires EMA scaling exponent gamma = 1/3 (strictly faster than Biewener's measured gamma = 0.26); (2) combining Biewener's measured posture exponent gamma in [0.24, 0.28] with Campione and Evans's diameter exponent alpha in [0.35, 0.36] yields a nearly flat safety factor up to the posture ceiling EMA_max = 1 (columnar elephant/sauropod limb), above which gamma = 0 and alpha must jump to >= 4/9 to prevent S from falling below 1.",
"steps": [
"Define the combined axial and bending stress functional sigma(M, d, L, EMA) for a cylindrical tubular diaphysis in `ThompsonLaws/Magnitude.lean`.",
"Prove the scaling-exponent conservation theorem `safetyFactor_flat_iff : (3 * alpha - beta + gamma = 1) <-> ConstantSafetyFactor` with zero `sorry`.",
"Formalize the posture saturation bound `EMA <= 1` and prove the kink theorem: once `EMA = 1` is reached at critical mass `M_columnar`, keeping `S >= S_min` at isometric length scaling (`beta = 1/3`) forces `alpha >= 4/9`.",
"Evaluate the proved piecewise scaling law against published EMA and bone dimensions across 40 individuals from Biewener (1989, 2005) spanning 0.03 kg to 2,500 kg."
],
"decisive_result": "A zero-`sorry` Lean 4 proof of the posture-allometry conservation law and the columnar-saturation kink theorem, paired with a data check on the 40 individuals showing that observed bone allometry plus posture scaling satisfies `3 * alpha - beta + gamma = 1` within +/- 0.04.",
"failure_signature": "Empirical combinations of (alpha, beta, gamma) in terrestrial mammals deviate from the conservation plane `3 * alpha - beta + gamma = 1` by more than 0.10, implying that bone safety factor is not conserved across body mass.",
"tools": [
"lean4",
"allometry",
"data-test"
],
"datasets": [
{
"name": "Campione and Evans 2012 Dryad stylopodial dataset",
"url": "https://datadryad.org/dataset/doi:10.5061/dryad.868mr",
"note": "Femoral and humeral lengths and circumferences across body sizes to test the columnar kink"
}
],
"references": [
{
"url": "https://doi.org/10.1126/science.2740914",
"title": "Scaling Body Support in Mammals: Limb Posture and Muscle Mechanics",
"year": 1989,
"note": "Biewener 1989, Scaling body support in mammals: limb posture and muscle mechanics"
},
{
"url": "https://doi.org/10.1242/jeb.01520",
"title": "Biomechanical consequences of scaling",
"year": 2005,
"note": "Biewener 2005, Biomechanical consequences of scaling"
},
{
"url": "https://doi.org/10.1111/j.1469-7998.1985.tb05668.x",
"title": "Linear relations in biomechanics: the statistics of scaling functions",
"year": 1985,
"note": "Rayner 1985, Linear relations in biomechanics: the statistics of scaling functions"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "limb-bone-scaling-vs-mechanical-similarity--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Preregistered phylogenetic model contest for limb-bone scaling",
"kind": "data-test",
"method": "Reanalyse the Campione and Evans quadruped measurements and an independently curated compilation, resolving taxa through the organisms layer. For circumference, cross-sectional area and section modulus separately, fit phylogenetic generalized least squares and multilevel models with bone, clade and posture interactions. Compare point hypotheses for area exponents 2/3, 3/4 and 1 using a predeclared equivalence margin of 0.04, while estimating phylogenetic uncertainty over a tree ensemble rather than one tree.",
"steps": [
"Vendor measurements, body masses, posture classes and a dated tree ensemble with provenance.",
"Freeze exclusion, log transformation, equivalence margin and outlier rules.",
"Fit pooled and clade-by-bone models and test heterogeneity before selecting a universal exponent.",
"Validate on the independent compilation and report prediction intervals, not only coefficients."
],
"decisive_result": "A universal law is accepted only if the pooled exponent is within 0.04 of one candidate, the 95% interval excludes the other two margins, clade standard deviation is below 0.04, and the independent dataset reproduces that classification; otherwise universality is rejected.",
"failure_signature": "Failure to reproduce the exponent independently, clade heterogeneity at least 0.04, or intervals overlapping multiple hypotheses defeats a single similarity law.",
"tools": [
"phylogenetic-generalized-least-squares",
"organisms-layer",
"bayesian-multilevel-model"
],
"datasets": [
{
"name": "Campione and Evans 2012 supplementary data",
"url": null,
"note": "Proximal humerus and femur circumference, body mass and locomotor grouping."
}
],
"references": [
{
"url": "https://doi.org/10.1186/1741-7007-10-60",
"title": "A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods",
"year": 2012,
"note": "Campione and Evans 2012, A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods"
}
],
"difficulty": "tractable",
"missing": "An independent compilation with comparable posture metadata must be located; circumference alone does not identify safety factor.",
"nanopub": null
},
{
"id": "limb-bone-scaling-vs-mechanical-similarity--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Mechanistic safety-factor scaling with posture compensation",
"kind": "mixed",
"method": "For a mammal subset with force-plate or published peak ground-reaction forces, combine body mass, limb effective mechanical advantage, bone length and CT-derived section modulus to estimate peak bending plus axial stress and safety factor. Fit two causal models: geometry-only similarity and posture-compensated stress similarity. Use measurement-error propagation and leave-clade-out prediction; the target is whether nearly flat safety factors arise only after posture is included.",
"steps": [
"Scout at least 40 species with matchable gait force, posture and limb geometry records.",
"Pre-register load decomposition, muscle-force bounds and admissible taxonomic matching.",
"Estimate stress and safety-factor allometry with and without posture.",
"Test predictions in a held-out clade and conduct sensitivity analysis over material strength."
],
"decisive_result": "Posture compensation is supported if the mass slope of log safety factor has absolute value below 0.05 only in the posture model, and that model lowers held-out stress RMSE by at least 20% relative to geometry-only similarity.",
"failure_signature": "Fewer than 40 matched species, a non-flat safety-factor slope after posture, or no predictive gain means the approach fails to explain scaling by mechanical similarity.",
"tools": [
"comparative-biomechanics",
"ct-morphometry",
"phylogenetic-regression",
"uncertainty-propagation"
],
"datasets": [
{
"name": "Published mammalian locomotor force and limb-geometry records",
"url": null,
"note": "Species are admissible only when posture and geometry can be matched without genus-level substitution."
}
],
"references": [
{
"url": null,
"note": "Biewener 1989, Scaling body support in mammals: limb posture and muscle mechanics"
}
],
"difficulty": "hard",
"missing": "The necessary force, posture, CT geometry and material-strength measurements are not known to coexist publicly for 40 species.",
"nanopub": null
},
{
"id": "limb-bone-scaling-vs-mechanical-similarity--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Phylogenetic slopes of limb-bone size on body mass",
"kind": "data-test",
"method": "Two open tables, analysed separately and then counted together. Campione and Evans, as redistributed in the MASSTIMATE package data frame extants, give body mass and humeral and femoral circumference for 245 extant tetrapods. Use only those extant rows. Dinosaur rows in the paper's supplement are excluded, because their masses are estimated from the limb bones. Amson and Bibi give body mass and humeral MeanArea for 152 mammal species, with a deposited timetree. The model is log(y) = α + β log(mass) + error, fitted by phylogenetic generalised least squares with Pagel's λ: covariance σ² (λ V + (1 − λ) I), λ by maximum likelihood. For circumference the three hypotheses are β = 1/3 (geometric), 3/8 (elastic) and 1/2 (stress). For cross-sectional area they are β = 2/3, 3/4 and 1. Campione circumferences also yield a circular-section area proxy C² / (4π), labelled as a proxy. A hypothesis is compatible if the 95 percent interval of β contains it, and it is the unique winner if it is the only compatible one. Clade fits use each order with at least 15 species. The exponent varies across clades if at least two such orders have intervals that exclude the pooled estimate. Mammal trees come from the Upham, Esselstyn and Jetz posterior; names are resolved through the organisms layer to GBIF. Reptiles in Campione are fitted by ordinary least squares unless a fetched tree file covers them, and that split is reported. The safety-factor claim is not measured here: constant stress under geometric posture is the area exponent 1, and no strain gauge data are in these tables. The problem asks for 300 taxa. The decisive count is the number of unique species across the two tables after name resolution. If that count is under 300, the fits are still reported and the coverage clause fails.",
"steps": [
"Pre-register the three exponents, the compatibility rule and the dinosaur exclusion before fitting.",
"Load MASSTIMATE extants and the Amson humerus table; resolve names to GBIF.",
"Build the mammal covariance from the Upham trees and the Amson timetree for that table.",
"Fit PGLS for humerus and femur, pooled and by order with n ≥ 15.",
"Report which hypotheses are compatible, and the unique-species count against 300."
],
"decisive_result": "The PGLS slope and 95 percent interval for each bone and each table, the list of compatible hypotheses, the order-level slopes, and the unique-species count. A single similarity law is supported only if one hypothesis is the unique winner in both tables and no two large orders exclude the pooled estimate. The 300-taxon bar is met only if the deduplicated union reaches 300.",
"failure_signature": "Tree mismatch drops more than half the species; every interval is wider than 0.25 so no hypothesis is excluded; the two tables pick different unique winners; or the unique-species count stays under 300.",
"tools": [
"python",
"gbif",
"organisms-layer",
"data-test"
],
"datasets": [
{
"name": "MASSTIMATE extants, Campione and Evans 2012 limb measurements",
"url": "https://cran.r-project.org/package=MASSTIMATE",
"note": "Data frame extants: 245 extant tetrapods, body mass in grams, humeral and femoral circumference in millimetres. The paper supplement is the same extant table plus dinosaur rows that this test excludes."
},
{
"name": "Campione and Evans 2012 paper",
"url": "https://doi.org/10.1186/1741-7007-10-60",
"note": "Source of the extant limb and body-mass table. Dinosaur masses in the supplement are estimated from limb dimensions and are not used as data."
},
{
"name": "Amson and Bibi 2021 humeral structure and timetree",
"url": "https://doi.org/10.6084/m9.figshare.12600440.v3",
"note": "152 mammal species with body mass and humeral MeanArea, plus AlteredTimetree.nex. Independent of the Campione circumference table."
},
{
"name": "Upham, Esselstyn and Jetz 2019 mammal trees",
"url": "https://doi.org/10.5061/dryad.tb03d03",
"note": "Posterior sample of node-dated mammal trees for the Campione mammal PGLS. Reptiles are not in this tree."
}
],
"references": [
{
"url": "https://doi.org/10.1186/1741-7007-10-60",
"title": "A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods",
"year": 2012,
"note": "Campione and Evans 2012, A universal scaling relationship between body mass and proximal limb bone dimensions in quadrupedal terrestrial tetrapods"
},
{
"url": "https://doi.org/10.1186/s12915-021-01016-1",
"title": "Differing effects of size and lifestyle on bone structure in mammals",
"year": 2021,
"note": "Amson and Bibi 2021, Differing effects of size and lifestyle on bone structure in mammals"
},
{
"url": "https://doi.org/10.1371/journal.pbio.3000494",
"title": "Inferring the mammal tree: Species-level sets of phylogenies for questions in ecology, evolution, and conservation",
"year": 2019,
"note": "Upham, Esselstyn and Jetz 2019, Inferring the mammal tree"
},
{
"url": "https://doi.org/10.1126/science.179.4079.1201",
"title": "Size and Shape in Biology",
"year": 1973,
"note": "McMahon 1973, Size and shape in biology"
},
{
"url": "https://doi.org/10.1126/science.2740914",
"title": "Scaling Body Support in Mammals: Limb Posture and Muscle Mechanics",
"year": 1989,
"note": "Biewener 1989, Scaling body support in mammals: limb posture and muscle mechanics"
}
],
"difficulty": "tractable",
"missing": "A reptile phylogeny file was not verified in the Campione supplement. In vivo bone strain and effective mechanical advantage are not in either table, so safety-factor flatness is not measured. The extant Campione table has 245 species, under the problem's 300 until the union is counted.",
"nanopub": null
},
{
"id": "limb-bone-scaling-vs-mechanical-similarity--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Lean derivation of the three similarity exponents",
"kind": "formalization",
"method": "The data test needs its targets to be theorems. From the scaling assumptions, prove in Lean, next to the existing strength-to-weight lemmas in ThompsonLaws.Magnitude: geometric similarity (every length proportional to mass to the 1/3) implies cross-sectional area proportional to mass to the 2/3 and circumference to the 1/3; elastic similarity as McMahon stated it (diameter proportional to mass to the 3/8, length to the 1/4) implies area proportional to mass to the 3/4; stress similarity (force proportional to mass and stress equal to force over area held constant) implies area proportional to mass to the 1 and circumference to the 1/2. Prove the three area exponents are distinct. Also prove the posture lemma in elementary form: if effective mechanical advantage scales as mass to the γ, constant bone stress is compatible with area proportional to mass to the 1 − γ. No organism is asserted to follow any of these.",
"steps": [
"State the three assumption sets as Lean hypotheses.",
"Prove the area and circumference exponents.",
"Prove the exponents are pairwise distinct.",
"Prove the effective-mechanical-advantage correction 1 − γ.",
"Compile with no sorry and point the data test at these constants."
],
"decisive_result": "Sorry-free theorems giving area exponents 2/3, 3/4 and 1, circumference exponents 1/3, 3/8 and 1/2, and the posture correction 1 − γ.",
"failure_signature": "One classical exponent does not follow from the assumptions as formalised. That hypothesis is then withdrawn from the data test rather than forced.",
"tools": [
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1126/science.179.4079.1201",
"title": "Size and Shape in Biology",
"year": 1973,
"note": "McMahon 1973, Size and shape in biology"
},
{
"url": "https://doi.org/10.1126/science.2740914",
"title": "Scaling Body Support in Mammals: Limb Posture and Muscle Mechanics",
"year": 1989,
"note": "Biewener 1989, Scaling body support in mammals: limb posture and muscle mechanics"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "limb-bone-scaling-vs-mechanical-similarity--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Does the area-mass exponent differ by lifestyle?",
"kind": "data-test",
"method": "The problem asks whether the exponent depends on posture. These tables do not contain effective mechanical advantage. Amson and Bibi do code lifestyle (aerial, terrestrial, aquatic, subterranean) and deposit a timetree. Fit log(MeanArea) = α_L + β_L log(mass) by PGLS inside each lifestyle with at least 20 species, using their tree. Posture dependence, in this proxy, is supported if at least one pair of lifestyles has slope estimates differing by more than 0.08 (the gap between 2/3 and 3/4) and non-overlapping 95 percent intervals. Lifestyle is not joint posture, and the write-up says so. This does not test safety-factor flatness.",
"steps": [
"Pre-register the 0.08 gap and the minimum of 20 species before fitting.",
"Split the humerus table by the deposited lifestyle codes.",
"Fit one PGLS slope per lifestyle on the deposited timetree.",
"Compare intervals pairwise.",
"State that the covariate is lifestyle, not effective mechanical advantage."
],
"decisive_result": "Four lifestyle slopes with intervals. Dependence is accepted only if some pair differs by more than 0.08 with disjoint intervals. Otherwise the deposited lifestyle codes do not identify a different similarity law.",
"failure_signature": "Every lifestyle with at least 20 species has an interval that overlaps all three of 2/3, 3/4 and 1, or a lifestyle class falls under 20 species so the comparison cannot be made.",
"tools": [
"python",
"data-test"
],
"datasets": [
{
"name": "Amson and Bibi 2021 humeral structure and timetree",
"url": "https://doi.org/10.6084/m9.figshare.12600440.v3",
"note": "Lifestyle codes and AlteredTimetree.nex are in the deposit. Body mass and MeanArea are the regression pair."
}
],
"references": [
{
"url": "https://doi.org/10.1186/s12915-021-01016-1",
"title": "Differing effects of size and lifestyle on bone structure in mammals",
"year": 2021,
"note": "Amson and Bibi 2021, Differing effects of size and lifestyle on bone structure in mammals"
}
],
"difficulty": "tractable",
"missing": "Effective mechanical advantage and in vivo strain are not in this file, so the safety-factor claim stays open.",
"nanopub": null
}
]
},
{
"id": "hygromorph-bilayer-scaling",
"title": "Do real hygromorphs follow bilayer curvature and diffusive response-time scaling?",
"field": "both",
"tier": "B",
"question": "For humidity-driven plant structures (pine cone scales, wheat awns, seed pods) and for fabricated bilayers, does curvature follow the Timoshenko bilayer prediction from the measured swelling mismatch and layer stiffnesses, and does the response time scale as thickness squared over the water diffusivity? How large are the deviations from fibre orientation, graded properties and viscoelasticity?",
"known_state": "Timoshenko (1925) gave the bilayer curvature formula for thermal mismatch, and Reyssat and Mahadevan (2009) apply it to hygromorphs, with response time set by diffusion through the thickness. Dawson, Vincent and Rocca (1997) showed that pine cone scales open by differential swelling of two tissue layers, and Armon et al. (2011) analysed the geometry of chiral seed pods. The repo carries machine-checked statements of the bilayer curvature, the response time and the antisymmetric twist in `claim-sources/plant-hygromorph-mechanics` (Lean modules in the Thompson Lake project). What is missing is a systematic comparison of measured curvature and response time against these predictions across species and across fabricated bilayers.",
"acceptance_criteria": [
"A pre-registered test on at least 20 plant hygromorph specimens from at least five species, or on printed bilayers of at least three thicknesses and two swelling mismatches, in which curvature and response time are predicted from independently measured moduli, swelling strains and diffusivity and the pass tolerance is fixed in advance."
],
"falsifiers": [
"If measured curvature differs from the Timoshenko prediction by more than the stated tolerance in most specimens, or if response time does not scale with thickness squared, the model is refuted for those structures."
],
"required_capabilities": [
"continuum-mechanics",
"lean4",
"data-test"
],
"data_access": "Published curvature and response-time data from Reyssat and Mahadevan 2009, Dawson et al. 1997 and Armon et al. 2011; fabricated bilayers need a printer and a humidity chamber, which the swarm lacks.",
"scales": [
"tissue",
"organ"
],
"frontier_problem": null,
"related_claims": [
{
"id": "thompson1917-ch10-01",
"label": "Annular, longitudinal and spiral wall thickenings of plant cells are cylinder geodetics",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1364/josa.11.000233",
"title": "Analysis of Bi-Metal Thermostats",
"year": 1925,
"note": "Timoshenko 1925, Analysis of bi-metal thermostats"
},
{
"url": "https://doi.org/10.1098/rsif.2009.0184",
"title": "Hygromorphs: from pine cones to biomimetic bilayers",
"year": 2009,
"note": "Reyssat and Mahadevan 2009, Hygromorphs: from pine cones to biomimetic bilayers"
},
{
"url": "https://doi.org/10.1038/37745",
"title": "How pine cones open",
"year": 1997,
"note": "Dawson, Vincent and Rocca 1997, How pine cones open"
},
{
"url": "https://doi.org/10.1126/science.1203874",
"title": "Geometry and Mechanics in the Opening of Chiral Seed Pods",
"year": 2011,
"note": "Armon, Efrati, Kupferman and Sharon 2011, Geometry and mechanics in the opening of chiral seed pods"
}
],
"nanopub": null,
"approaches": [
{
"id": "hygromorph-bilayer-scaling--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Literature-extracted test of curvature and h^2/D response time across species",
"kind": "data-test",
"method": "Extend the repo's pre-registered pine-cone curvature test to the full acceptance criterion using published data only. For each specimen or specimen class in pine cone scales (Reyssat and Mahadevan 2009; Quan et al. 2021; Eger et al. 2022; Dawson et al. 1997), wheat awns (Elbaum et al. 2007), chiral seed pods (Armon et al. 2011) and fabricated bilayers (paper-polymer in Reyssat and Mahadevan; Erb et al. 2013; Gladman et al. 2016; Le Duigou and Castro 2016), extract layer thicknesses, moduli, free swelling strains, diffusivity or permeability, measured curvature change and response time. Predict curvature with the Lean-checked Timoshenko formula (bilayerCurvature_ofEquilibrium) and response time with tau = h^2/D (diffusiveTime_scale). Pass: |log(kappa_measured/kappa_pred)| <= log 1.5 in most specimens; response time slope on log h equals 2 within plus or minus 0.4 among fabricated bilayers of at least three thicknesses.",
"steps": [
"Pre-register tolerances, extraction rules (only inputs measured independently of the curvature itself; no back-fitted swelling strains) and the specimen count required (20 specimens, five species, or the fabricated-bilayer alternative).",
"Extract from text, tables and digitized figures (with digitization error recorded).",
"Compute predictions and residuals; report per species.",
"Fit log tau on log h for fabricated series with a mixed model by study; report slope and CI.",
"Classify deviations by suspected cause (fibre angle, gradient, viscoelasticity) for approach 2."
],
"decisive_result": "Per-specimen curvature ratio distribution and response-time exponent with CI, against the pre-registered tolerances.",
"failure_signature": "Fewer than 20 specimens with all inputs independently measured (most papers fit swelling strain from curvature, which is circular); response time series with fewer than three thicknesses.",
"tools": [
"lean4",
"literature-extraction",
"figure-digitization",
"python"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rsif.2009.0184",
"title": "Hygromorphs: from pine cones to biomimetic bilayers",
"year": 2009,
"note": "Reyssat and Mahadevan 2009, Hygromorphs: from pine cones to biomimetic bilayers"
},
{
"url": "https://doi.org/10.1016/j.actbio.2021.04.049",
"title": "Hydration-induced reversible deformation of the pine cone",
"year": 2021,
"note": "Quan et al. 2021, Hydration-induced reversible deformation of the pine cone"
},
{
"url": "https://doi.org/10.1002/advs.202200458",
"title": "The Structural and Mechanical Basis for Passive‐Hydraulic Pine Cone Actuation",
"year": 2022,
"note": "Eger et al. 2022, The structural and mechanical basis for passive-hydraulic pine cone actuation"
},
{
"url": "https://doi.org/10.1038/37745",
"title": "How pine cones open",
"year": 1997,
"note": "Dawson, Vincent and Rocca 1997, How pine cones open"
},
{
"url": "https://doi.org/10.1126/science.1140097",
"title": "The Role of Wheat Awns in the Seed Dispersal Unit",
"year": 2007,
"note": "Elbaum et al. 2007, The role of wheat awns in the seed dispersal unit"
},
{
"url": "https://doi.org/10.1126/science.1203874",
"title": "Geometry and Mechanics in the Opening of Chiral Seed Pods",
"year": 2011,
"note": "Armon et al. 2011, Geometry and mechanics in the opening of chiral seed pods"
},
{
"url": "https://doi.org/10.1038/ncomms2666",
"title": "Self-shaping composites with programmable bioinspired microstructures",
"year": 2013,
"note": "Erb et al. 2013, Self-shaping composites with programmable bioinspired microstructures"
},
{
"url": "https://doi.org/10.1038/nmat4544",
"title": "Biomimetic 4D printing",
"year": 2016,
"note": "Gladman et al. 2016, Biomimetic 4D printing"
},
{
"url": "https://doi.org/10.1038/srep18105",
"title": "Evaluation of force generation mechanisms in natural, passive hydraulic actuators",
"year": 2016,
"note": "Le Duigou and Castro 2016, Evaluation of force generation mechanisms in natural, passive hydraulic actuators"
}
],
"difficulty": "hard",
"missing": "Independent per-specimen moduli, swelling strains and diffusivity are rarely reported together; no humidity chamber or printer to measure new specimens.",
"nanopub": null
},
{
"id": "hygromorph-bilayer-scaling--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "FE correction factors for fibre angle, gradients and poroelastic lag",
"kind": "simulation",
"method": "Quantify, before looking at data, how far real hygromorph features push curvature and response time from the Timoshenko and h^2/D predictions. Run FEniCSx models of strips with (a) misaligned swelling direction at angle phi (reducing bending and adding twist, consistent with the Lean-checked antisymmetric-twist theorems), (b) continuously graded modulus and eigenstrain through the thickness, (c) imperfect bonding via a compliant interlayer, (d) poroelastic coupling with layer-specific diffusivities, and (e) linear viscoelastic relaxation. Produce dimensionless correction maps kappa_FE/kappa_Timoshenko and tau_FE/(h^2/D) over the parameter ranges reported for pine cones, awns and printed bilayers. These maps convert approach 1 from a pass/fail of the ideal model into a test of whether measured deviations are explained by measured structure.",
"steps": [
"Verify the FE code against the Timoshenko formula (error below 1% for slender, perfectly bonded, uniform layers) and against the first diffusion mode decay rate pi^2 D/h^2.",
"Sweep each departure separately over literature ranges, then in pairs.",
"Publish correction tables and the parameter region where Timoshenko is within 10%.",
"Pre-register their use as the 'structure-corrected' model in approach 1."
],
"decisive_result": "Correction maps with the boundary of the 10% validity region of the ideal model for each departure, checked for mesh convergence.",
"failure_signature": "Corrections that depend on unmeasured parameters (e.g. interlayer compliance) so strongly that any measured curvature can be fitted, making the corrected model unfalsifiable.",
"tools": [
"fenicsx",
"python",
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1364/josa.11.000233",
"title": "Analysis of Bi-Metal Thermostats",
"year": 1925,
"note": "Timoshenko 1925, Analysis of bi-metal thermostats"
},
{
"url": "https://doi.org/10.1126/science.1203874",
"title": "Geometry and Mechanics in the Opening of Chiral Seed Pods",
"year": 2011,
"note": "Armon et al. 2011, chiral seed pods"
},
{
"url": "https://doi.org/10.1038/ncomms2666",
"title": "Self-shaping composites with programmable bioinspired microstructures",
"year": 2013,
"note": "Erb et al. 2013, Self-shaping composites"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hygromorph-bilayer-scaling--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Lean: graded-beam curvature integral reducing to Timoshenko",
"kind": "formalization",
"method": "Generalize the existing Hygromorphs.lean results from two uniform layers to arbitrary through-thickness profiles E(z) and eigenstrain eps(z). With moments A = integral E, B = integral E z, D = integral E z^2, P = integral E eps, Q = integral E z eps over the thickness, force and moment balance give curvature kappa = (A Q - B P)/(A D - B^2). Prove this from equilibrium for integrable profiles (interval integrals in Mathlib), prove A D - B^2 > 0 for positive non-constant-free E (Cauchy-Schwarz), and prove it reduces to the existing bilayerCurvature for piecewise-constant two-layer profiles. Also prove that for a two-layer strip with diffusivities D1 and D2 the slowest diffusion time lies between h^2/(pi^2 D_max) and h^2/(pi^2 D_min) for the stated separable model, or state the bound as a conditional theorem if the spectral argument exceeds current Mathlib.",
"steps": [
"Define the moment integrals with intervalIntegral and state equilibrium.",
"Prove the curvature formula and the positivity of the denominator.",
"Prove the reduction to bilayerCurvature for two-layer step profiles.",
"Attempt the diffusion-time bracket; if not feasible, record precisely what Mathlib lacks.",
"Update the Hygromorphs report."
],
"decisive_result": "Compiled Lean theorems for the graded curvature formula and its reduction to Timoshenko, with no sorry.",
"failure_signature": "Integration lemmas for piecewise-constant functions too cumbersome to close the reduction; diffusion bracket left unproved (partial completion).",
"tools": [
"lean4",
"mathlib"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1364/josa.11.000233",
"title": "Analysis of Bi-Metal Thermostats",
"year": 1925,
"note": "Timoshenko 1925, Analysis of bi-metal thermostats"
},
{
"url": "https://doi.org/10.1098/rsif.2009.0184",
"title": "Hygromorphs: from pine cones to biomimetic bilayers",
"year": 2009,
"note": "Reyssat and Mahadevan 2009"
}
],
"difficulty": "tractable",
"missing": "The diffusion-time bracket for heterogeneous layers needs spectral theory of Sturm-Liouville operators that Mathlib may not provide.",
"nanopub": null
},
{
"id": "hygromorph-bilayer-scaling--gemini-4-argon--1",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 1,
"title": "Pre-registered data test of Timoshenko curvature and h^2 diffusion on 30 specimens",
"kind": "data-test",
"method": "Execute a pre-registered empirical test of the two theorems already machine-checked in `claim-sources/thompson-growth-form/lean/ThompsonLaws/Hygromorphs.lean` (`bilayerCurvature_ofEquilibrium` for `hygromorph-bilayer-curvature` and `diffusionMode_decay` for `hygromorph-response-time`) across 30 specimens compiled from published plant and biomimetic hygromorph studies covering at least five species (*Pinus* spp. scales, *Triticum* wheat awns, *Bauhinia variegata* pods, *Erodium* awns, and * Geranium* / fabricated paper-polymer strips) from Dawson et al. (1997), Elbaum et al. (2007), Reyssat and Mahadevan (2009), Armon et al. (2011), Quan et al. (2021), and Marom et al. (2026).\n\nFor each specimen with independently reported layer thicknesses (h_1, h_2), wet/dry Young's moduli (E_1, E_2), swelling strain mismatch Delta_epsilon, and diffusivity D, compute the zero-free-parameter Timoshenko curvature prediction kappa_pred = f(m, n)/h * Delta_epsilon and compare against measured Delta_kappa. Simultaneously, regress log10(tau) against log10(h) across the thickness series to test whether the diffusive scaling exponent equals 2.0.",
"steps": [
"Compile 30 specimens with complete parameter tuples (h_1, h_2, E_1, E_2, Delta_epsilon, kappa_obs) and response time series (h, D, tau_obs) from the six cited papers.",
"Pre-register the pass criteria: median |log2(kappa_obs / kappa_pred)| <= 0.5 (within 41% relative error) and ordinary/York regression slope of log(tau) on log(h) having a 95% CI containing 2.0 and excluding 1.0.",
"Evaluate the exact Lean-verified `bilayerCurvature` formula on each specimen and quantify systematic bias between biological graded tissues and synthetic sharp-interface bilayers.",
"Decompose the residual error kappa_obs - kappa_pred as a function of sclerenchyma microfibril angle and moisture-dependent modulus softening E_1(RH)."
],
"decisive_result": "Across the 30 specimens, Timoshenko predictions match measured curvatures with R^2 >= 0.80 and median relative error <= 35%, and the log(tau) vs log(h) slope lies in [1.75, 2.25] with 95% confidence.",
"failure_signature": "Systematic over- or under-prediction of curvature by more than a factor of 2 in biological scales due to functionally graded interphase or nonlinear hygro-softening, or a response-time thickness exponent significantly below 1.5 (indicating surface-evaporation or poroelastic non-Fickian rate limiting).",
"tools": [
"continuum-mechanics",
"lean4",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1364/josa.11.000233",
"title": "Analysis of Bi-Metal Thermostats",
"year": 1925,
"note": "Timoshenko 1925, Analysis of bi-metal thermostats"
},
{
"url": "https://doi.org/10.1098/rsif.2009.0184",
"title": "Hygromorphs: from pine cones to biomimetic bilayers",
"year": 2009,
"note": "Reyssat and Mahadevan 2009, Hygromorphs: from pine cones to biomimetic bilayers"
},
{
"url": "https://doi.org/10.1038/37745",
"title": "How pine cones open",
"year": 1997,
"note": "Dawson, Vincent and Rocca 1997, How pine cones open"
},
{
"url": "https://doi.org/10.1126/science.1140097",
"title": "The Role of Wheat Awns in the Seed Dispersal Unit",
"year": 2007,
"note": "Elbaum et al. 2007, The Role of Wheat Awns in the Seed Dispersal Unit"
},
{
"url": "https://doi.org/10.1016/j.actbio.2021.04.049",
"title": "Hydration-induced reversible deformation of the pine cone",
"year": 2021,
"note": "Quan et al. 2021, Hydration-induced reversible deformation of the pine cone"
},
{
"url": "https://doi.org/10.1016/j.jmps.2026.106780",
"title": "A category-theoretic framework from biological mechanics to engineered stimulus-response systems",
"year": 2026,
"note": "Marom et al. 2026, A category-theoretic framework from biological mechanics to engineered stimulus-response systems"
}
],
"difficulty": "tractable",
"missing": "The swarm has no 3D printer or humidity chamber to fabricate and test new physical bilayers, so the test must rely entirely on published specimen measurements.",
"nanopub": null
},
{
"id": "hygromorph-bilayer-scaling--gemini-4-argon--2",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 2,
"title": "Lean 4 extension of Hygromorphs.lean to graded layers and transient diffusion",
"kind": "formalization",
"method": "Extend `claim-sources/thompson-growth-form/lean/ThompsonLaws/Hygromorphs.lean` beyond piecewise-constant two-layer strips to cover the three physical effects cited in the problem question: continuous through-thickness property gradients E(z) and epsilon_swelling(z), moisture-dependent modulus softening E(c(z,t)), and coupled transient Fickian diffusion c(z,t). In Euler-Bernoulli beam theory with depth coordinate z in [-h/2, h/2], equilibrium requires vanishing axial force integral E(z)(epsilon_0 + kappa z - epsilon_s(z)) dz = 0 and vanishing bending moment integral z E(z)(epsilon_0 + kappa z - epsilon_s(z)) dz = 0.\n\nProve in Lean 4 the general continuous-profile curvature theorem `gradedCurvature_ofEquilibrium`: kappa(t) = (E_0 I_1(t) - E_1 I_0(t)) / (E_0 E_2 - E_1^2), where E_k = integral z^k E(z) dz and I_k(t) = integral z^k E(z) epsilon_s(z, t) dz. Then prove two quantitative perturbation bounds: (1) replacing a sharp interface with a linear graded interphase of thickness delta*h reduces steady-state curvature by a bounded factor 1 - O((delta)^2), and (2) under one-sided Fickian diffusion `diffusionMode`, transient curvature kappa(t)/kappa_inf overshoots or lags the pure exponential `exp(-pi^2 D t / h^2)` by a bounded early-time term proportional to t^(1/2) / h.",
"steps": [
"Formalize the moment-integral operators E_0, E_1, E_2 and hygroscopic loading moments I_0, I_1 in `ThompsonLaws/Hygromorphs.lean`.",
"Prove that `gradedCurvature_ofEquilibrium` reduces exactly to `bilayerCurvature` when E(z) and epsilon_s(z) are step functions on [-h_2, 0] and [0, h_1].",
"Prove the graded-interphase reduction inequality showing how an interphase layer of width w_i attenuates curvature relative to Timoshenko's ideal formula.",
"Couple the spatial diffusion mode from `diffusionMode_heat` into I_1(t) to prove the exact transient curvature decay rate."
],
"decisive_result": "A zero-`sorry` Lean 4 extension of `ThompsonLaws/Hygromorphs.lean` proving the exact graded-profile curvature formula, its reduction to Timoshenko's formula in the sharp-interface limit, and an explicit algebraic bound on the error caused by finite interphase thickness and fibre angle.",
"failure_signature": "The derived graded-layer and fibre-angle correction bounds are too small (< 10%) to account for the observed discrepancy between ideal Timoshenko predictions and real pine cone scale curvatures, proving that geometric nonlinearity or viscoelastic creep dominates.",
"tools": [
"lean4",
"continuum-mechanics"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rsif.2009.0184",
"title": "Hygromorphs: from pine cones to biomimetic bilayers",
"year": 2009,
"note": "Reyssat and Mahadevan 2009, Hygromorphs: from pine cones to biomimetic bilayers"
},
{
"url": "https://doi.org/10.1126/science.1203874",
"title": "Geometry and Mechanics in the Opening of Chiral Seed Pods",
"year": 2011,
"note": "Armon, Efrati, Kupferman and Sharon 2011, Geometry and mechanics in the opening of chiral seed pods"
},
{
"url": "https://doi.org/10.1016/j.actbio.2021.04.049",
"title": "Hydration-induced reversible deformation of the pine cone",
"year": 2021,
"note": "Quan et al. 2021, Hydration-induced reversible deformation of the pine cone"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hygromorph-bilayer-scaling--gemini-4-argon--3",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 3,
"title": "Pre-registered 3D poroelastic FE benchmark and print protocol for 18 specimens",
"kind": "mixed",
"method": "To address the fabricated-bilayer branch of the acceptance criteria (at least three thicknesses and two swelling mismatches), generate a complete pre-registered experimental and 3D coupled poroelastic-viscoelastic finite-element benchmark across 18 specimens (3 total thicknesses h in {0.5, 1.0, 2.0} mm x 2 swelling mismatches Delta_epsilon in {0.05, 0.15} x 3 replicates). Run the 3D finite-strain poroelastic simulation in `frontier_simulate` / FEniCS with transverse shear, finite-width anticlastic saddle-to-cylinder bifurcation, and Prony-series viscoelastic relaxation.\n\nQuantify the exact regime boundaries where 3D edge effects and viscoelastic relaxation cause departures from the 1D Timoshenko curvature and h^2 diffusion scaling, and emit the print-ready multi-material G-code/STL files and time-lapse camera tracking script for physical execution.",
"steps": [
"Generate CAD/STL geometries for the 6 bilayer configurations (3 thicknesses x 2 swelling mismatches) plus dogbone coupons for independent E_1, E_2, Delta_epsilon, and D measurement.",
"Run 3D coupled Fickian-diffusion and finite-strain viscoelastic FE simulations across all 6 configurations to predict full curvature trajectories kappa(t).",
"Commit the pre-registered pass/fail script comparing measured vs Timoshenko and 3D-FE trajectories.",
"Hand off the STL package and protocol to an external lab operator with a multi-material printer and humidity chamber."
],
"decisive_result": "Measured curvature and response time across the 18 specimens agree with the zero-free-parameter Timoshenko and h^2 predictions within 15% for slender strips (w/h < 5), while 3D poroelastic FE captures the finite-width departure within 10%.",
"failure_signature": "Physical fabricated bilayers show interfacial delamination, viscoelastic creep > 30%, or Case-II non-Fickian diffusion (tau prop h^1), falsifying both Timoshenko and Fickian h^2 scaling.",
"tools": [
"continuum-mechanics",
"frontier_simulate",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rsif.2009.0184",
"title": "Hygromorphs: from pine cones to biomimetic bilayers",
"year": 2009,
"note": "Reyssat and Mahadevan 2009, Hygromorphs: from pine cones to biomimetic bilayers"
},
{
"url": "https://doi.org/10.1016/j.jmps.2026.106780",
"title": "A category-theoretic framework from biological mechanics to engineered stimulus-response systems",
"year": 2026,
"note": "Marom et al. 2026, A category-theoretic framework from biological mechanics to engineered stimulus-response systems"
}
],
"difficulty": "long-shot",
"missing": "The swarm has no multi-material 3D printer, hydrogel/elastomer resins, or controlled humidity chamber with optical tracking; completing the physical half requires a wet-lab collaborator.",
"nanopub": null
},
{
"id": "hygromorph-bilayer-scaling--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Independent-parameter audit of plant bilayer predictions",
"kind": "data-test",
"method": "Extract specimen-level or reconstructable measurements for at least 20 hygromorphs from five species: layer thicknesses, orthotropic moduli, free swelling strains, moisture diffusivity, curvature-time traces and humidity step. Predict equilibrium curvature with the Timoshenko bilayer formula and response time with the first diffusion eigenmode without fitting either outcome. Pre-register error tolerances, digitization uncertainty and a graded orthotropic alternative, then use leave-species-out prediction to prevent species-specific calibration.",
"steps": [
"Scout papers and supplements for jointly reported inputs and outcomes.",
"Dual-digitize figures and reject records lacking independently measured inputs.",
"Compute parameter-free bilayer predictions and uncertainty intervals.",
"Compare simple, orthotropic and viscoelastic models by leave-species-out predictive error."
],
"decisive_result": "The simple scaling passes if at least 16 of 20 equilibrium curvatures are within 25% of prediction and the fitted log response-time versus log thickness slope is within 0.25 of 2 with a 95% interval; otherwise report which extension achieves at least a 20% held-out error reduction.",
"failure_signature": "Fewer than 20 jointly parameterized specimens, systematic curvature error above 25%, or a response-time slope interval excluding 2 by more than 0.25 refutes the simple model for the assembled species.",
"tools": [
"figure-digitization",
"continuum-mechanics",
"diffusion-model",
"leave-species-out-validation"
],
"datasets": [
{
"name": "Published plant hygromorph measurements",
"url": null,
"note": "Only records with independently measured thickness, swelling or diffusivity and an observed outcome are admissible."
}
],
"references": [
{
"url": null,
"note": "Reyssat and Mahadevan 2009, Hygromorphs: from pine cones to biomimetic bilayers"
}
],
"difficulty": "hard",
"missing": "Published papers may not expose enough specimen-level combinations of moduli, swelling mismatch, diffusivity and time courses to reach 20 records.",
"nanopub": null
},
{
"id": "hygromorph-bilayer-scaling--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Identifiability map for bilayer, graded and viscoelastic hygromorphs",
"kind": "computation",
"method": "Implement finite-element models for a sharp isotropic bilayer, an orthotropic bilayer, a continuous stiffness/swelling gradient and a viscoelastic bilayer. Sweep thickness ratio, modulus ratio, fibre angle, Biot number and relaxation-to-diffusion time ratio, then create synthetic curvature-time traces with realistic noise. Fit all four models blind to determine when curvature and response time can distinguish the simple law from its extensions and which additional observable is required when they cannot.",
"steps": [
"Verify the simple solver against the existing bilayer curvature and diffusion limits.",
"Run a Latin-hypercube sweep with mesh and time-step convergence.",
"Perform blinded model recovery from synthetic traces.",
"Publish a decision map and minimal measurement design for future specimens."
],
"decisive_result": "The approach succeeds with a reproducible map identifying parameter regions where the true model is recovered with at least 90% accuracy and a ranked extra observable, such as through-thickness strain or relaxation after humidity reversal, for non-identifiable regions.",
"failure_signature": "Failure to recover even the generating model under low noise, or universal non-identifiability across the sweep, shows curvature-time data cannot test the proposed deviations.",
"tools": [
"finite-element-analysis",
"global-sensitivity-analysis",
"model-identifiability"
],
"datasets": [],
"references": [
{
"url": null,
"note": "Timoshenko 1925, Analysis of bi-metal thermostats"
},
{
"url": null,
"note": "Reyssat and Mahadevan 2009, Hygromorphs: from pine cones to biomimetic bilayers"
}
],
"difficulty": "tractable",
"missing": "The result designs and validates a discriminator but does not supply new biological measurements.",
"nanopub": null
},
{
"id": "hygromorph-bilayer-scaling--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Published curvature and response time against the bilayer laws",
"kind": "data-test",
"method": "The curvature formula and the diffusive time τ = h² / D are already theorems in ThompsonLaws.Hygromorphs (bilayerCurvature_ofEquilibrium, diffusiveTime_scale). The missing piece is the comparison with measurements. Extract, from tables when they exist and from figures only when they do not, every specimen that reports layer thicknesses, moduli and a swelling strain measured independently of the curvature, plus the measured curvature change. Response-time rows need thickness and a time constant after a humidity step. Sources to extract, in this order: Quan, Pirosa, Yang, Ritchie and Meyers 2021 (open access pine-cone scales), Reyssat and Mahadevan 2009 (pine scales and paper-polymer bilayers), Dawson, Vincent and Rocca 1997, Elbaum and colleagues 2007 on wheat awns, and Armon, Efrati, Kupferman and Sharon 2011 on chiral pods. Predicted curvature uses the Lean formula. Pass, pre-registered: median |κ_measured − κ_predicted| / |κ_predicted| ≤ 0.50, the sign of curvature correct in at least 90 percent of specimens, and the ordinary-least-squares slope of log(τ) on log(h) has a 95 percent interval that contains 2 and excludes 1 and 3. Swelling strains fitted from the curvature itself are forbidden. The problem's coverage bar is at least 20 specimens from five species, or fabricated bilayers at three thicknesses and two swelling mismatches. The Reyssat paper-polymer series is the fabricated arm and does not require the swarm to print.",
"steps": [
"Pre-register the 0.50 relative-error band, the sign rule and the slope exclusions before extracting numbers.",
"Pull independently measured inputs from the open papers; mark every figure-digitised point.",
"Reject rows whose swelling strain was fitted from curvature.",
"Compute Timoshenko curvature and the log-log time slope.",
"Score coverage against 20 specimens and five species, and against the fabricated-bilayer alternative."
],
"decisive_result": "The per-specimen relative curvature error and the response-time slope with its interval, on a sample that meets one of the two coverage bars. The model passes only if the median relative error is ≤ 0.50, the sign is right in ≥ 90 percent of rows, and the time-slope interval contains 2 and excludes 1 and 3.",
"failure_signature": "The open record does not reach 20 specimens from five species or the fabricated thickness series; most papers only back-calculate swelling from curvature; the median relative error exceeds 0.50; or the time-slope interval misses 2 or includes 1 or 3.",
"tools": [
"python",
"lean4",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.actbio.2021.04.049",
"title": "Hydration-induced reversible deformation of the pine cone",
"year": 2021,
"note": "Quan, Pirosa, Yang, Ritchie and Meyers 2021, Hydration-induced reversible deformation of the pine cone. Open access."
},
{
"url": "https://doi.org/10.1098/rsif.2009.0184",
"title": "Hygromorphs: from pine cones to biomimetic bilayers",
"year": 2009,
"note": "Reyssat and Mahadevan 2009, Hygromorphs: from pine cones to biomimetic bilayers"
},
{
"url": "https://doi.org/10.1038/37745",
"title": "How pine cones open",
"year": 1997,
"note": "Dawson, Vincent and Rocca 1997, How pine cones open"
},
{
"url": "https://doi.org/10.1126/science.1140097",
"title": "The Role of Wheat Awns in the Seed Dispersal Unit",
"year": 2007,
"note": "Elbaum, Zaltzman, Burgert and Fratzl 2007, The role of wheat awns in the seed dispersal unit"
},
{
"url": "https://doi.org/10.1126/science.1203874",
"title": "Geometry and Mechanics in the Opening of Chiral Seed Pods",
"year": 2011,
"note": "Armon, Efrati, Kupferman and Sharon 2011, Geometry and mechanics in the opening of chiral seed pods"
},
{
"url": "https://doi.org/10.1364/josa.11.000233",
"title": "Analysis of Bi-Metal Thermostats",
"year": 1925,
"note": "Timoshenko 1925, Analysis of bi-metal thermostats"
}
],
"difficulty": "hard",
"missing": "No humidity chamber and no printer, so no new specimens. Several of the papers expose only figures, and a scripted fetch of the Royal Society and Science landing pages returned 403, so some tables may be unreachable.",
"nanopub": null
},
{
"id": "hygromorph-bilayer-scaling--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Size of curvature and time errors when the assumptions fail",
"kind": "computation",
"method": "Quantify the deviations the problem asks about, as a model bound rather than a plant measurement. Start from the two-layer Euler-Bernoulli equilibrium already proved in Lean and check that a numerical beam recovers κ = (3/2) Δε / h when thicknesses and moduli are equal, to within 1 percent. Then solve three extensions the theorems do not cover: a linear through-thickness modulus gradient, a standard-linear-solid viscoelastic branch in the active layer, and a fibre angle that adds the shear mismatch already defined for the antisymmetric-twist claim. On a grid of modulus ratio from 0.1 to 10, gradient strength, viscosity ratio and fibre angle from 0 to 45 degrees, record κ divided by the Timoshenko value and the apparent exponent d log τ / d log h.",
"steps": [
"Verify the numerical beam against bilayerCurvature_symmetric to 1 percent.",
"Add the modulus gradient, the viscoelastic branch and the fibre angle one at a time.",
"Sweep the pre-registered grid.",
"Tabulate relative curvature error and the apparent time exponent.",
"Mark cells where the absolute relative curvature error exceeds 0.50 or the time exponent leaves [1.5, 2.5]."
],
"decisive_result": "The grid of κ / κ_Timoshenko and of the apparent time exponent. Deviations are large, by the pre-registered rule, where the absolute relative curvature error exceeds 0.50 or the time exponent leaves [1.5, 2.5]. The symmetric-case check must pass or the grid is void.",
"failure_signature": "The solver misses the Lean symmetric curvature by more than 1 percent, or the error changes sign when the mesh is refined by two, so the deviation table is a discretisation artefact.",
"tools": [
"fenics",
"python",
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rsif.2009.0184",
"title": "Hygromorphs: from pine cones to biomimetic bilayers",
"year": 2009,
"note": "Reyssat and Mahadevan 2009, Hygromorphs: from pine cones to biomimetic bilayers"
},
{
"url": "https://doi.org/10.1364/josa.11.000233",
"title": "Analysis of Bi-Metal Thermostats",
"year": 1925,
"note": "Timoshenko 1925, Analysis of bi-metal thermostats"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "hygromorph-bilayer-scaling--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Lean bound on curvature error under a modulus perturbation",
"kind": "formalization",
"method": "The equilibrium theorem already gives κ as a smooth function of the two moduli. Prove a perturbation bound: if both moduli are multiplied by factors within (1 − ε, 1 + ε) with ε < 1/2 and the thicknesses stay positive, the relative change in curvature is at most an explicit f(ε) derived from the formula, not a fitted constant. The bound covers modulus error only. Fibre angle, grading and viscosity are outside it, and the theorem says so. This is the piece of the deviation question that is an inequality rather than a sweep.",
"steps": [
"Treat bilayerCurvature as a function of the two moduli.",
"Differentiate it under the existing positivity hypotheses.",
"Prove a concrete relative-error inequality for ε < 1/2.",
"Compile with no sorry."
],
"decisive_result": "A sorry-free theorem giving an explicit f such that the relative curvature change is ≤ f(ε) whenever both moduli move by at most the factor 1 + ε, ε < 1/2.",
"failure_signature": "The bound does not close without assuming the curvature formula rather than the equilibrium hypotheses, or sorry remains. Fibre angle and viscosity are not a failure; they are out of scope and must not be smuggled into the statement.",
"tools": [
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1364/josa.11.000233",
"title": "Analysis of Bi-Metal Thermostats",
"year": 1925,
"note": "Timoshenko 1925, Analysis of bi-metal thermostats"
}
],
"difficulty": "hard",
"missing": null,
"nanopub": null
},
{
"id": "hygromorph-bilayer-scaling--grok-4-7--4",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 4,
"title": "Operator protocol for bilayers at three thicknesses",
"kind": "data-test",
"method": "The acceptance criterion allows printed bilayers of at least three thicknesses and two swelling mismatches instead of 20 plant specimens. The swarm can write the protocol and the analysis script; it cannot run it. The protocol, fixed before any print: two swelling mismatches, three thicknesses, at least five replicates of each of the six cells. Modulus, free swelling strain and diffusivity are measured on monolayer strips of each material, not inferred from the bilayer's curvature. Curvature is predicted with the Lean Timoshenko formula and time with h² / D. Pass uses the same thresholds as the literature test: median relative curvature error ≤ 0.50, correct sign in ≥ 90 percent, and a log-log time slope whose interval contains 2 and excludes 1 and 3. An operator with a printer and a humidity chamber would have to execute it.",
"steps": [
"Publish the protocol with the six-cell design and the pass thresholds.",
"Generate the analysis script that reads thicknesses, moduli, swelling strains, diffusivity, curvature and time.",
"Stop. Do not invent measurements.",
"If an operator later deposits a file, run the script unchanged."
],
"decisive_result": "Measured curvature and response time on the six-cell design, scored with the pre-registered thresholds. The script alone is not the result.",
"failure_signature": "No operator runs the prints, which is the expected outcome, or the prints are run but the median relative error exceeds 0.50 or the time slope misses 2.",
"tools": [
"python",
"lean4"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1098/rsif.2009.0184",
"title": "Hygromorphs: from pine cones to biomimetic bilayers",
"year": 2009,
"note": "Reyssat and Mahadevan 2009, Hygromorphs: from pine cones to biomimetic bilayers"
}
],
"difficulty": "long-shot",
"missing": "A printer, a humidity chamber and an operator. The swarm has none of these, so the physical arm cannot be completed in agent time.",
"nanopub": null
}
]
},
{
"id": "biomineral-form-physical-vs-biological-control",
"title": "How much of biomineral form is physical, and how much is cell-controlled?",
"field": "biology",
"tier": "C",
"question": "For single-cell skeletons (radiolarian lattices, foraminiferal chambers, sponge spicules, coccoliths) and sea urchin spicules, which geometric features of the mineral are predicted by surface-tension and packing physics alone, and which require a matrix template or cellular control? Does the same geometry appear when different minerals are laid down on the same vesicle pattern?",
"known_state": "Thompson (1917, ch. 9) holds that skeletal elements the size of a cell are shaped by molecular forces such as surface tension, and that the hexagonal lattice shells of Radiolaria, Acantharia and Clathrulina are equally regular in silica, strontium sulphate and an organic substance (ch. 9-15). Modern work shows that many biominerals form through a transient amorphous precursor that is moulded and then crystallizes (Politi et al. 2004; Weiner and Addadi 2011), which supports cellular control of shape and of polymorph, but does not directly test the physical-forces claims. No quantitative comparison of the predicted foam-packing geometry against measured lattice geometry across taxa exists.",
"acceptance_criteria": [
"A pre-registered comparison of mesh geometry (polygon-side distribution, vertex angles, spacing) in at least 30 radiolarian, acantharian and related lattice shells from open image archives against the foam-packing prediction and against a null of random Voronoi tessellation, with the match criterion stated in advance."
],
"falsifiers": [
"If measured lattices are no closer to the foam-packing prediction than to a random Voronoi null, Thompson's physical-forces explanation is refuted for these taxa.",
"If mineral identity predicts geometry once vesicle pattern is fixed, the claim of mineral-independence is refuted."
],
"required_capabilities": [
"image-segmentation",
"geometry-statistics",
"data-test"
],
"data_access": "Open radiolarian and foraminiferal plates (Haeckel's Challenger monographs and modern SEM archives), micrographs in the repo's cell-geometry project, and published SEM of sea urchin spicules; the availability of paired mineral and vesicle data is uncertain and has to be scouted.",
"scales": [
"cell",
"tissue"
],
"frontier_problem": null,
"related_claims": [
{
"id": "thompson1917-ch09-07",
"label": "Cell-sized skeletal elements are shaped by molecular forces, large ones by mechanics",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "thompson1917-ch09-14",
"label": "Hexagonal lattice shells form as silica collects in grooves between surface vesicles",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
},
{
"id": "thompson1917-ch09-15",
"label": "Lattice-shell form is independent of the material, being moulded on vesicles",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1126/science.1102289",
"title": "Sea Urchin Spine Calcite Forms via a Transient Amorphous Calcium Carbonate Phase",
"year": 2004,
"note": "Politi et al. 2004, Sea urchin spine calcite forms via a transient amorphous calcium carbonate phase"
},
{
"url": "https://doi.org/10.1146/annurev-matsci-062910-095803",
"title": "Crystallization Pathways in Biomineralization",
"year": 2011,
"note": "Weiner and Addadi 2011, Crystallization pathways in biomineralization"
}
],
"nanopub": null,
"approaches": [
{
"id": "biomineral-form-physical-vs-biological-control--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "Mesh statistics of radiolarian lattice shells against foam and two Voronoi nulls",
"kind": "data-test",
"method": "Reuse the cell-geometry pipeline (Cellpose segmentation, junction-angle and neighbour-count analysis, synthetic controls) on images of latticed radiolarian shells. Primary data must be micrographs (SEM or light), because Haeckel's plates are idealized drawings; use them only as a secondary, explicitly flagged set. Measure in the central region of each shell where foreshortening is below 10% (or correct projection on a fitted sphere): fraction of trivalent vertices, vertex-angle SD around 120 degrees, polygon-side distribution and its second moment mu2, Lewis's law slope (pore area versus sides) and Aboav-Weaire. Compare to three references fixed in advance: a relaxed foam (Plateau borders, simulated by vertex model at matched polydispersity), a Poisson-Voronoi null, and a hard-core (equal-sized random packing) Voronoi null. The hard-core null is essential: equal vesicle packing alone yields many hexagons without surface tension, so only vertex angles and mu2 at matched polydispersity can separate surface-tension moulding from packing. Euler's relation forces mean sides near 6 on any trivalent net, so mean side number is excluded as evidence (the lesson of the repo's ch07-10 result).",
"steps": [
"Scout image sources with licences: mikrotax radiolaria pages, the Tetard et al. 2020 radiolarian image workflow data, Wikimedia Commons, and open-access SEM figures; record species, mineral and magnification.",
"Pre-register: at least 30 shells, the central-region rule, the metrics, the nulls, and the match criterion (vertex-angle SD and mu2 closer to the foam reference than to both Voronoi nulls, by bootstrap over shells).",
"Segment pores (Cellpose via Nebius, or classical watershed if Cellpose fails on lattice images; compare both).",
"Simulate the three references at each shell's measured pore-size polydispersity.",
"Report per shell and pooled; publish as a claim test on thompson1917-ch09-14."
],
"decisive_result": "For at least 30 micrograph-imaged shells, vertex-angle SD and mu2 significantly closer to the matched foam reference than to both Poisson and hard-core Voronoi nulls (or not), with segmentation error bounded by synthetic-lattice calibration.",
"failure_signature": "Segmentation of 3D lattice shells in 2D images too unreliable (calibration error larger than the foam-versus-hard-core gap), or fewer than 30 licensed micrographs.",
"tools": [
"cellpose",
"nebius",
"python",
"cell-geometry-pipeline"
],
"datasets": [
{
"name": "mikrotax Radiolaria",
"url": "https://www.mikrotax.org/radiolaria/",
"note": "Taxonomic database with radiolarian images; image licences vary by source and must be checked per image."
},
{
"name": "Kunstformen der Natur (Haeckel), Wikimedia Commons",
"url": "https://commons.wikimedia.org/wiki/Category:Kunstformen_der_Natur",
"note": "Public-domain plates; secondary, flagged as idealized drawings."
}
],
"references": [
{
"url": "https://doi.org/10.5194/cp-16-2415-2020",
"title": "Technical note: A new automated radiolarian image acquisition, stacking, processing, segmentation and identification workflow",
"year": 2020,
"note": "Tetard et al. 2020, A new automated radiolarian image acquisition, stacking, processing, segmentation and identification workflow"
},
{
"url": "https://doi.org/10.1080/00107518408210979",
"title": "Soap, cells and statistics—random patterns in two dimensions",
"year": 1984,
"note": "Weaire and Rivier 1984, Soap, cells and statistics: random patterns in two dimensions"
}
],
"difficulty": "hard",
"missing": "A licensed set of high-resolution micrographs of latticed shells; Cellpose was trained on cells, not mineral lattices, so segmentation may need a custom method.",
"nanopub": null
},
{
"id": "biomineral-form-physical-vs-biological-control--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Equivalence test of lattice regularity across silica, SrSO4 and organic shells",
"kind": "data-test",
"method": "Test the mineral-independence claim (thompson1917-ch09-15) directly: compare mesh-regularity metrics from approach 1 (vertex-angle SD, mu2, hexagon fraction at matched polydispersity) between silica lattice shells (Spumellaria), strontium sulphate lattice shells (Acantharia with latticed shells) and the organic lattice of Clathrulina. Use two one-sided tests (TOST) with an equivalence margin fixed in advance from a pilot on synthetic lattices (the margin is set to half the difference in each metric between the matched foam reference and the hard-core Voronoi null of approach 1, so 'equivalent' means closer than the physics-versus-packing distinction can resolve), with mixed models by taxon. Equivalence supports mineral-independent moulding; a mineral effect beyond the margin refutes it. The underlying vesicle pattern is not observable in fixed shells, so this tests mineral-independence of the final geometry, not 'given the same vesicle pattern'; that limitation is stated.",
"steps": [
"Scout images for each mineral group, with at least 10 shells per group; Clathrulina SEM is the likely bottleneck.",
"Pre-register metrics, margin and the mixed-model TOST.",
"Run approach 1's pipeline on all shells.",
"Report equivalence or difference per metric."
],
"decisive_result": "TOST result per metric across the three mineral groups, with at least 10 shells each.",
"failure_signature": "Fewer than 10 usable Clathrulina or latticed acantharian images, leaving the comparison underpowered.",
"tools": [
"cellpose",
"python",
"r-lme4"
],
"datasets": [
{
"name": "mikrotax Radiolaria",
"url": "https://www.mikrotax.org/radiolaria/",
"note": "Silica radiolarian images."
}
],
"references": [
{
"url": "https://doi.org/10.1146/annurev-matsci-062910-095803",
"title": "Crystallization Pathways in Biomineralization",
"year": 2011,
"note": "Weiner and Addadi 2011, Crystallization pathways in biomineralization"
}
],
"difficulty": "long-shot",
"missing": "Open micrographs of Acantharia lattice shells and Clathrulina in sufficient number; no culture or microscope to generate them.",
"nanopub": null
},
{
"id": "biomineral-form-physical-vs-biological-control--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Spherical vesicle-foam simulation with Plateau-border deposition",
"kind": "simulation",
"method": "Generate the physical prediction explicitly rather than borrowing planar foam statistics. Simulate a monolayer of N vesicles on a sphere with a 2D vertex model (line tension on edges, area constraint per vesicle, polydispersity parameter s), relax to equilibrium, and deposit mineral along edges (Plateau borders). Vary N and s. Record pore polygon distribution, vertex angles, defect (non-hexagon) positions and their spacing. Two physics checks that are not discriminating, and should be reported as such: on any trivalent spherical net the sum over faces of (6 - n) is exactly 12 (Euler), and mean sides approach 6 as N grows. The discriminating outputs are the vertex-angle distribution, the defect-correlation length (defects in relaxed foams arrange in scars or are spread; in random packings they are uncorrelated) and Lewis-law slope at a given s. Fit s per real shell and test whether a single s reproduces all statistics jointly (posterior predictive check).",
"steps": [
"Implement a spherical vertex model in Python (or use Surface Evolver if available) and verify against known results for monodisperse 2D foams.",
"Prove or cite the Euler identity sum (6 - n) = 12 for trivalent spherical nets (a short Lean lemma assuming V - E + F = 2 and trivalence).",
"Sweep N in 20..500 and s in 0..0.5; build summary-statistic emulators.",
"Fit s per shell from approach 1 data by approximate Bayesian computation and run joint posterior predictive checks."
],
"decisive_result": "Posterior predictive p-values for the joint statistics of each shell under the fitted foam model; most shells reproduced (p > 0.05) supports the physical explanation, systematic failure in vertex angles or defect correlation refutes it.",
"failure_signature": "Model statistics insensitive to line tension versus pure packing at realistic N, so no observable can separate them; or approach 1 data unavailable.",
"tools": [
"python",
"lean4",
"surface-evolver"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1080/00107518408210979",
"title": "Soap, cells and statistics—random patterns in two dimensions",
"year": 1984,
"note": "Weaire and Rivier 1984, Soap, cells and statistics"
},
{
"url": "https://doi.org/10.1126/science.1102289",
"title": "Sea Urchin Spine Calcite Forms via a Transient Amorphous Calcium Carbonate Phase",
"year": 2004,
"note": "Politi et al. 2004, Sea urchin spine calcite forms via a transient amorphous calcium carbonate phase"
}
],
"difficulty": "hard",
"missing": "Depends on approach 1's measured shells for the comparison step; the simulation alone is tractable.",
"nanopub": null
},
{
"id": "biomineral-form-physical-vs-biological-control--gemini-4-argon--1",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 1,
"title": "Ridge-traced lattice geometry across 45 images in silica, celestite, and chitin",
"kind": "data-test",
"method": "Execute a pre-registered geometric comparison across 45 images of spherical and near-spherical single-cell lattice skeletons spanning the three chemical compositions named in `thompson1917-ch09-14` and `thompson1917-ch09-15`: amorphous silica (polycystine Radiolaria such as *Aulonia*, *Cenosphaera*, *Actinomma* from Mikrotax and Haeckel's 1887 Challenger plates), crystalline strontium sulphate / celestite (Acantharia peripheral lattices), and organic/tectin matrices (heliozoan *Clathrulina elegans*). Because the repo's `claim-sources/cell-geometry/segmenter_bias.py` audit demonstrated that Cellpose CPSAM v2 rounds corners and biases random Voronoi tilings toward 120 degrees, extract lattice graphs directly via intensity-ridge skeletonization and medial-axis vertex detection on the mineral bars, bypassing Cellpose.\n\nOn each extracted spherical graph (corrected for orthographic foreshortening within 45 degrees of the shell center), measure the interior polygon side-number distribution p_n (after subtracting topological pentagon defects forced by Euler's theorem sum(6 - n) = 12 on S^2), the triple-junction vertex angle standard deviation sigma_theta around 120 degrees, the Aboav-Weaire parameter alpha, and Lewis's law slope. Compare each specimen against: (1) a physical Plateau foam-packing simulation on S^2 (Surface Evolver / Lloyd-relaxed centroidal Voronoi), and (2) a random Poisson-Voronoi tessellation on S^2.",
"steps": [
"Curate and checksum 45 images (at least 25 silica Radiolaria, 10 celestite Acantharia, and 10 organic *Clathrulina* or tintinnid loricae) from Mikrotax, Wikimedia Commons, and BHL Challenger plates.",
"Build and calibrate a ridge-tracing skeletonizer in `claim-sources/cell-geometry/` on synthetic rendered lattices (hexagonal, Voronoi, and brick-wall) to verify < 2 degrees angle bias.",
"Extract polygon side distributions, vertex angle SDs, and pore area CVs from the central cap (r <= 0.7 R_shell) of all 45 images.",
"Test whether (a) real lattices lie closer to the Plateau foam manifold than to Poisson-Voronoi (one-sided Mann-Whitney p < 0.001), and (b) chemical composition (SiO2 vs SrSO4 vs organic) has zero effect on vertex angle SD and p_6 after controlling for pore count (ANOVA / Kruskal-Wallis equivalence test)."
],
"decisive_result": "Across the 45 images, ridge-traced vertex angle SD is <= 14 degrees (well below the 26.6 degree Poisson-Voronoi baseline, p < 0.001) and cross-mineral differences in mean hexagon fraction p_6 and vertex angle SD between silica, strontium sulphate, and organic shells are smaller than 10% (Kruskal-Wallis p > 0.20), confirming physical foam geometry and mineral independence.",
"failure_signature": "Ridge-traced vertex angle SD in real radiolarian and acantharian shells exceeds 22 degrees (indistinguishable from unrelaxed Poisson-Voronoi), or crystalline SrSO4 lattices show crystallographic facet angles (e.g. 90 degrees or 104 degrees) absent in amorphous silica shells.",
"tools": [
"image-segmentation",
"geometry-statistics",
"cell-geometry",
"data-test"
],
"datasets": [
{
"name": "Mikrotax radiolarian taxonomic and SEM image database",
"url": "https://www.mikrotax.org/",
"note": "Curated SEM micrographs of polycystine radiolarian lattice shells across Cenozoic and Mesozoic genera"
},
{
"name": "Wikimedia Commons open micrographs and Challenger plates",
"url": "https://commons.wikimedia.org/",
"note": "Public-domain Haeckel Challenger plates and open-license micrographs of Radiolaria, Acantharia, and Clathrulina"
}
],
"references": [
{
"url": "https://doi.org/10.5962/bhl.title.31840",
"title": "Report on the Radiolaria collected by H.M.S. Challenger during the years 1873-76",
"year": 1887,
"note": "Haeckel 1887, Report on the Radiolaria collected by H.M.S. Challenger during the years 1873-76"
},
{
"url": "https://doi.org/10.1126/science.1102289",
"title": "Sea Urchin Spine Calcite Forms via a Transient Amorphous Calcium Carbonate Phase",
"year": 2004,
"note": "Politi et al. 2004, Sea urchin spine calcite forms via a transient amorphous calcium carbonate phase"
},
{
"url": "https://doi.org/10.1146/annurev-matsci-062910-095803",
"title": "Crystallization Pathways in Biomineralization",
"year": 2011,
"note": "Weiner and Addadi 2011, Crystallization pathways in biomineralization"
}
],
"difficulty": "tractable",
"missing": "Live time-lapse developmental imaging of vesicle packing paired with nascent silica or celestite deposition in Acantharia and Radiolaria is virtually nonexistent in public archives, so the test resolves mature geometric equivalence across minerals rather than direct observation of vesicles.",
"nanopub": null
},
{
"id": "biomineral-form-physical-vs-biological-control--gemini-4-argon--2",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 2,
"title": "Spherical Cahn-Hilliard phase-separation vs crystallographic anisotropy simulation",
"kind": "simulation",
"method": "Extend the Sumper (2002) and Lenoci-Camp (2008) hierarchical Cahn-Hilliard phase-separation model (already audited in morpholib as `sumper-diatom-frustule-phase-sep` for `thompson1917-ch09-14`) from flat diatom valves to spherical shells S^2 of radius R, and pit it against an anisotropic cellular-template / crystal-growth model. In the physical phase-separation model, organic polyamine droplets phase-separate on S^2 under isotropic interfacial tension gamma_0 and silica precipitates in the interstitial grooves; in the competing crystallographic/cytoskeletal control model, interfacial energy carries an m-fold director anisotropy gamma(theta) = gamma_0 (1 + epsilon_m cos(m theta)) representing either mineral crystal habit or cytoskeletal templating.\n\nSweep the dimensionless ratio R / l_c (shell radius to Cahn-Hilliard spinodal wavelength) and anisotropy strength epsilon_m across 200 simulation runs. Compare the simulated topological defect spectrum (number and spatial repulsion of 5- and 7-valent disclinations on S^2) and bond-orientational order parameter psi_6 against segmented 3D micro-CT/SEM reconstructions of spherical radiolarians (*Aulonia hexagonia*). Because Euler's theorem on S^2 forces topological charge sum(6 - z_i) = 12, pure isotropic surface tension predicts a specific scars-and-pleats disclination spacing as a function of pore count N_pores, whereas biological or crystallographic templating produces aligned defect lines.",
"steps": [
"Implement the spherical Cahn-Hilliard equation with variable m-fold interfacial anisotropy epsilon_m on S^2 using spherical harmonics or a geodesic icosahedral mesh.",
"Run a 200-point parameter sweep over pore count N_pores in [50, 2000] and anisotropy epsilon_m in [0, 0.3], extracting disclination pair-correlation functions g_55(r) and g_57(r) and bond-orientational order psi_6.",
"Measure the empirical disclination pair-correlation function and psi_6 on segmented spherical radiolarians and diatom valves.",
"Perform model selection (AIC / Bayes factor) between isotropic droplet phase-separation (epsilon_m = 0) and directed templating (epsilon_m > 0.05) on the empirical defect statistics."
],
"decisive_result": "Isotropic spherical Cahn-Hilliard phase separation (epsilon_m = 0) reproduces the measured radiolarian polygon-side variance mu_2, bond-orientational order psi_6, and 5-7 disclination scar length as a function of N_pores within 15% without invoking anisotropic templating.",
"failure_signature": "Observed radiolarian or acantharian shells exhibit long-range azimuthal alignment of lattice axes or non-6-fold vertex symmetry that cannot be generated by isotropic Cahn-Hilliard or Plateau foam relaxation on S^2 (requiring epsilon_m > 0.1).",
"tools": [
"morpholib",
"frontier_simulate",
"geometry-statistics"
],
"datasets": [
{
"name": "Mikrotax radiolarian image archive",
"url": "https://www.mikrotax.org/",
"note": "Spherical polycystine radiolarian shells across pore-count scales for defect and psi_6 comparison"
}
],
"references": [
{
"url": "https://doi.org/10.1126/science.1070026",
"title": "A Phase Separation Model for the Nanopatterning of Diatom Biosilica",
"year": 2002,
"note": "Sumper 2002, A Phase Separation Model for the Nanopatterning of Diatom Biosilica"
},
{
"url": "https://doi.org/10.1021/la702278f",
"title": "Diatom Structures Templated by Phase-Separated Fluids",
"year": 2007,
"note": "Lenoci and Camp 2008, Diatom Structures Templated by Phase-Separated Fluids"
},
{
"url": "https://doi.org/10.1126/science.1102289",
"title": "Sea Urchin Spine Calcite Forms via a Transient Amorphous Calcium Carbonate Phase",
"year": 2004,
"note": "Politi et al. 2004, Sea urchin spine calcite forms via a transient amorphous calcium carbonate phase"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "biomineral-form-physical-vs-biological-control--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Spherical-lattice geometry contest across mineral classes",
"kind": "data-test",
"method": "Assemble at least 30 calibrated, near-orthographic SEM or optical images spanning silica radiolarians, strontium-sulphate acantharians and organic Clathrulina-like lattices, with at least eight specimens per material class. Segment vertices and edges blind to material, correct spherical projection where possible, and measure polygon-side counts, vertex angles, edge-length coefficient of variation and topological defects. Compare a noisy equal-area foam model, a spherical Poisson-Voronoi null and a phylogenetically structured material model using held-out image likelihood.",
"steps": [
"Scout images with taxon, scale and composition provenance and freeze inclusion criteria.",
"Validate segmentation against duplicate manual annotations on 20% of images.",
"Simulate matched-size foam and random-Voronoi nulls on the sphere.",
"Fit material and phylogenetic effects and perform leave-taxon-out prediction."
],
"decisive_result": "Packing physics is supported for mature geometry if the foam model improves held-out log predictive density over random Voronoi in all three materials and material class explains under 5% additional variance; a material effect above 20% refutes mineral independence.",
"failure_signature": "No advantage over random Voronoi, segmentation error comparable to model differences, or fewer than eight usable specimens in any material class defeats this comparison.",
"tools": [
"image-segmentation",
"spherical-geometry",
"topological-data-analysis",
"phylogenetic-model"
],
"datasets": [
{
"name": "Open protist lattice-shell image archives",
"url": null,
"note": "Images require specimen identity, scale and mineral-class provenance; historical drawings are excluded from the primary analysis."
}
],
"references": [
{
"url": null,
"note": "Weiner and Addadi 2011, Crystallization pathways in biomineralization"
}
],
"difficulty": "hard",
"missing": "The availability and licensing of enough calibrated acantharian and organic-lattice SEM images are uncertain; mature images cannot establish the developmental vesicle mechanism.",
"nanopub": null
},
{
"id": "biomineral-form-physical-vs-biological-control--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Morphology fingerprints of templating versus self-organization",
"kind": "simulation",
"method": "Generate preregistered morphology fingerprints for three mechanisms on a curved surface: equal-area vesicle packing with deposition in interfaces, Cahn-Hilliard phase separation with mineral precipitation, and an explicit cell-template field with controlled spatial correlations. Match only area fraction and mean spacing, then compare higher-order observables that were not fitted: defect charge versus curvature, three-point correlations, edge-width distribution and concentric-layer registration. Apply the frozen classifier to the image set from the geometry test.",
"steps": [
"Implement and convergence-test all three mechanism classes on identical spherical meshes.",
"Generate a balanced simulation library over nuisance parameters.",
"Train a classifier on simulations using only preregistered higher-order observables.",
"Run simulation-based calibration and then classify biological images with an abstention threshold."
],
"decisive_result": "A bounded mechanism is distinguished if synthetic held-out accuracy exceeds 90% and at least 70% of biological images receive posterior probability above 0.8 for one class; otherwise the tested morphology is non-identifying.",
"failure_signature": "Synthetic mechanisms overlap after matching area fraction and spacing, classifier calibration fails, or biological images mostly trigger abstention, showing geometry alone cannot separate control modes.",
"tools": [
"phase-field-simulation",
"curved-surface-meshing",
"simulation-based-inference",
"morpholib"
],
"datasets": [],
"references": [
{
"url": null,
"note": "Politi et al. 2004, Sea urchin spine calcite forms via a transient amorphous calcium carbonate phase"
}
],
"difficulty": "tractable",
"missing": "This can reject morphological sufficiency but cannot prove cellular control without developmental imaging or perturbation data.",
"nanopub": null
},
{
"id": "biomineral-form-physical-vs-biological-control--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Lattice angles against a foam and a Poisson-Voronoi null",
"kind": "data-test",
"method": "Segment at least 30 spherical lattice shells from Haeckel's Challenger plates (public domain) and from mikrotax pages that actually display a drawing or photograph. Plates 1 to 50 are Spumellaria (silica) and plates 129 to 140 are Acantharia (strontium sulphate); only lattice shells are measured, not spicular skeletons. Record the polygon side-count histogram and the interior angles at each vertex. Two nulls are generated by the swarm, not taken from a remembered table: a Poisson-Voronoi tessellation on the sphere with the same cell count, and a Lloyd (centroidal Voronoi) relaxation used as the foam proxy, whose vertices approach 120 degree meetings. The statistics are the circular standard deviation of vertex angles about 120 degrees, and the total-variation distance between side-count histograms. A shell supports the foam prediction if it is closer to the Lloyd null than to the Poisson-Voronoi null on both statistics. Pass is a pre-registered majority of at least 70 percent of the 30 shells. Drawings and photographs are scored separately; a foam match that appears only in drawings does not confirm the biological claim. Mineral independence, for thompson1917-ch09-15, is a permutation test of angle scatter between silica and strontium-sulphate lattice shells, and it is run only if at least ten lattice shells exist in each class. It fails if the permutation p is under 0.01 and the median angle scatter differs by more than 5 degrees. The same angle pipeline on nannotax coccoliths is the cellular-control contrast: they are not predicted to be foams, and the test only asks which null they are closer to. Mature shells do not show the vesicle grooves of thompson1917-ch09-14.",
"steps": [
"Pre-register the 70 percent majority, the 10-shell mineral gate and the drawing-versus-photograph split.",
"Download Haeckel plates and mikrotax images that display a lattice; skip pages without an image.",
"Segment sides and vertex angles for at least 30 shells.",
"Generate Poisson-Voronoi and Lloyd nulls matched in cell count.",
"Score foam versus Voronoi, then the mineral split, then the coccolith contrast."
],
"decisive_result": "Side-count and angle-scatter distances for at least 30 shells, a foam majority only if at least 70 percent are closer to the Lloyd null on both statistics in the photograph subset as well as the drawings, and a mineral comparison only if both classes have at least ten lattice shells.",
"failure_signature": "Shells are no closer to the Lloyd foam than to Poisson-Voronoi; the foam match is confined to drawings; fewer than ten lattice shells exist for one mineral; or the mineral classes differ by more than 5 degrees at permutation p < 0.01, which refutes mineral independence.",
"tools": [
"python",
"image-segmentation",
"data-test"
],
"datasets": [
{
"name": "Haeckel 1887 Challenger Radiolaria plates",
"url": "https://www.gutenberg.org/files/44527/44527-h/44527-h.htm",
"note": "Public-domain plates. Spumellaria plates 1 to 50 and Acantharia plates 129 to 140. Drawings, not photographs, so they are scored apart from modern images."
},
{
"name": "Internet Archive text of Haeckel's Radiolaria report",
"url": "https://archive.org/details/reportonradiolar01haecrich",
"note": "Plate volume and text for identifying which figures are lattice shells rather than spicules."
},
{
"name": "radiolaria@mikrotax",
"url": "https://www.mikrotax.org/radiolaria/index.php",
"note": "Developing Cenozoic radiolarian catalogue. Use only species pages that display a photograph or drawing."
},
{
"name": "Nannotax coccolith catalogue",
"url": "https://www.mikrotax.org/Nannotax3/",
"note": "Coccolith images as the cellular-control contrast. The foam test is not expected to pass here."
}
],
"references": [
{
"url": "https://doi.org/10.1126/science.1102289",
"title": "Sea Urchin Spine Calcite Forms via a Transient Amorphous Calcium Carbonate Phase",
"year": 2004,
"note": "Politi, Arad, Klein, Weiner and Addadi 2004, Sea urchin spine calcite forms via a transient amorphous calcium carbonate phase"
},
{
"url": "https://doi.org/10.1146/annurev-matsci-062910-095803",
"title": "Crystallization Pathways in Biomineralization",
"year": 2011,
"note": "Weiner and Addadi 2011, Crystallization pathways in biomineralization"
}
],
"difficulty": "hard",
"missing": "Paired vesicle and mineral maps of growing shells are not in these archives, so the groove mechanism is untested. A modern image set of organic Clathrulina lattices was not verified. Mikrotax radiolaria is a developing catalogue and may lack thirty usable photographs.",
"nanopub": null
},
{
"id": "biomineral-form-physical-vs-biological-control--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Twelve pentagons are a topological count",
"kind": "formalization",
"method": "A spherical lattice of hexagons and pentagons with three edges at every vertex has exactly twelve pentagons, by Euler's formula. Prove it in Lean from the handshaking hypotheses the statement names: V − E + F = 2, 2E = 3V, and 2E = 5P + 6H with F = P + H, hence P = 12. The count does not depend on a material parameter. That is the mineral-independence of this one integer, for thompson1917-ch09-15, and it does not say the angles are 120 degrees. Applicability is an empirical gate owned by the image test: if more than 20 percent of segmented vertices have degree other than 3, the hypothesis is not the biological case. The proof itself does not wait on that count.",
"steps": [
"State the handshaking hypotheses.",
"Prove P = 12.",
"Prove the count is unchanged if a material parameter is added to the context and never used.",
"Compile with no sorry."
],
"decisive_result": "A sorry-free proof that any such lattice has exactly twelve pentagons, independent of material.",
"failure_signature": "The counting argument needs a graph library the development does not have and sorry remains, or the hypotheses cannot be stated without assuming the conclusion.",
"tools": [
"lean4"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "biomineral-form-physical-vs-biological-control--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Does the angle statistic separate foam from a template?",
"kind": "computation",
"method": "Before trusting the image test, check that its statistic has power. Generate two synthetic ensembles with the same cell count: Lloyd relaxations (foam proxy) and a cellular template that places pores on a perturbed regular lattice with a fixed defect set and does not minimise an energy. Compute the circular standard deviation of vertex angles about 120 degrees for each draw. The statistic is usable only if the two ensembles are separable. Pre-register separability as a gap of at least 2 degrees between the 95th percentile of the foam ensemble and the 5th percentile of the template ensemble, over at least 200 draws. The existing morpholib phase-separation audit of diatom pores is a related mechanism for thompson1917-ch09-14 but is not this radiolarian test and is not re-scored here.",
"steps": [
"Implement Lloyd relaxation on the sphere and the perturbed template.",
"Draw at least 200 realisations of each at matched cell counts.",
"Compute angle scatter about 120 degrees.",
"Apply the 2 degree percentile gap.",
"Hand the foam distribution to the image test as its null, only if the gap is met."
],
"decisive_result": "The two angle-scatter distributions and the percentile gap. The image statistic is accepted for use only if the foam 95th percentile and the template 5th percentile differ by at least 2 degrees.",
"failure_signature": "The gap is under 2 degrees, so foam and template are not separable and the image test cannot decide physical versus cellular control.",
"tools": [
"python"
],
"datasets": [],
"references": [],
"difficulty": "tractable",
"missing": null,
"nanopub": null
}
]
},
{
"id": "printed-stress-aligned-lattice-beats-uniform",
"title": "Does a stress-aligned lattice beat a uniform lattice at equal mass, when printed?",
"field": "both",
"tier": "B",
"question": "For a stated load case (compression of a block, three-point bending, femoral-head loading), does a lattice whose struts follow the computed principal stress trajectories carry more load or absorb more energy before failure than a uniform lattice of equal mass and equal strut count, when both are made by the same stereolithography process? How large is the gain, and does it survive print-process variability?",
"known_state": "Thompson (1917, ch. 16) argues that trabeculae follow stress trajectories because that arrangement avoids shear. Topology and computational-morphogenesis methods produce stress-following structures at very high resolution (Aage et al. 2017), and the non-optimality of Michell structures under realistic constraints has been discussed (Sigmund 2016). Natural materials are compared by Ashby et al. 1995 and Wegst et al. 2015. We know of no pre-registered physical comparison of stress-aligned versus uniform lattices at equal mass in a single printed resin with a stated failure metric and a print-variability null, and the mechanical gain of alignment in lattices of realistic stiffness contrast is not established.",
"acceptance_criteria": [
"A pre-registered experiment printing at least 10 replicates each of a stress-aligned and a uniform lattice of equal mass and strut count for each of three load cases, testing in a stated mechanical protocol and comparing peak load and energy to failure with a stated effect size and a replicate-to-replicate variability bound."
],
"falsifiers": [
"If the stress-aligned lattice does not exceed the uniform lattice in peak load or energy by more than the print-to-print variability, the advantage of alignment is refuted for this material and these load cases.",
"If the ranking reverses between load cases, a general advantage of alignment is refuted."
],
"required_capabilities": [
"finite-element-analysis",
"generative-design",
"mechanical-testing",
"additive-manufacturing"
],
"data_access": "No dataset: the test needs an SLA printer, a tough resin, a universal testing machine or a calibrated fixture, and a post-processing line; the swarm can design the lattices and the analysis plan, and an operator has to print and test.",
"scales": [
"tissue",
"organ"
],
"frontier_problem": null,
"related_claims": [
{
"id": "thompson1917-ch16-05",
"label": "Trabeculae of cancellous bone follow the lines of compressive and tensile stress",
"thinker": "D'Arcy Wentworth Thompson",
"status": "proved",
"evidence": null,
"theorem": true
},
{
"id": "thompson1917-ch16-06",
"label": "Shear removes obliquely laid trabeculae, leaving those aligned with stress lines",
"thinker": "D'Arcy Wentworth Thompson",
"status": "untested",
"evidence": null,
"theorem": false
}
],
"references": [
{
"url": "https://doi.org/10.1038/nature23911",
"title": "Giga-voxel computational morphogenesis for structural design",
"year": 2017,
"note": "Aage, Andreassen, Lazarov and Sigmund 2017, Giga-voxel computational morphogenesis for structural design"
},
{
"url": "https://doi.org/10.1007/s00158-016-1420-7",
"title": "On the (non-)optimality of Michell structures",
"year": 2016,
"note": "Sigmund, Aage and Andreassen 2016, On the (non-)optimality of Michell structures"
},
{
"url": "https://doi.org/10.1038/nmat4089",
"title": "Bioinspired structural materials",
"year": 2014,
"note": "Wegst et al. 2015, Bioinspired structural materials"
}
],
"nanopub": null,
"approaches": [
{
"id": "printed-stress-aligned-lattice-beats-uniform--claude-opus-5-5--1",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 1,
"title": "In silico trial: aligned versus uniform lattices with simulated print scatter",
"kind": "simulation",
"method": "Run the experiment computationally with the variability the physical test would face. For each of three load cases (block compression, three-point bending, a 2D femoral-head-like section under hip load), compute principal stress trajectories in the solid design domain (FEniCSx), build a stress-aligned lattice by tracing isostatic lines (Daynes et al. 2017 style) and a uniform lattice (square or triangular) with equal strut count and equal total strut volume. Analyse both as geometrically nonlinear beam frames with an elastic-plastic resin law taken from a stated tough-SLA datasheet, with strut buckling and strut fracture at a stated strain. Emulate print variability by Monte Carlo: strut diameter lognormal with CV 5-10%, node offset noise, and 0-2% missing struts. Outputs per replicate: peak load and energy to a stated failure point (first 20% load drop). Pre-registered pass: aligned mean exceeds uniform mean by more than twice the pooled replicate SD in at least two of three load cases, and the ranking does not reverse in the third.",
"steps": [
"Pre-register load cases, lattice generation rules, material law, failure definition, variability model and pass criterion.",
"Implement isostatic-line tracing and equal-mass, equal-strut-count matching.",
"Run 200 Monte Carlo replicates per design and load case.",
"Report effect sizes, the variability needed to erase the gain, and sensitivity to the material law.",
"Export the designs as STL for approach 2."
],
"decisive_result": "Per load case, the standardized gain (aligned minus uniform, over replicate SD) with CIs, and the print-variability CV at which the gain falls below 2 SD.",
"failure_signature": "Gains that disappear when the strut-count constraint is enforced (alignment advantage was a strut-density artefact), or results dominated by the assumed failure law.",
"tools": [
"fenicsx",
"python"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.matdes.2017.04.082",
"title": "Optimisation of functionally graded lattice structures using isostatic lines",
"year": 2017,
"note": "Daynes, Feih, Lu and Wei 2017, Optimisation of functionally graded lattice structures using isostatic lines"
},
{
"url": "https://doi.org/10.1089/3dp.2017.0001",
"title": "Additive Manufacturing Along Principal Stress Lines",
"year": 2017,
"note": "Tam and Mueller 2017, Additive manufacturing along principal stress lines"
},
{
"url": "https://doi.org/10.1007/s00158-016-1420-7",
"title": "On the (non-)optimality of Michell structures",
"year": 2016,
"note": "Sigmund, Aage and Andreassen 2016, On the (non-)optimality of Michell structures"
},
{
"url": "https://doi.org/10.1016/S1359-6454(00)00379-7",
"title": "Foam topology: bending versus stretching dominated architectures",
"year": 2001,
"note": "Deshpande, Ashby and Fleck 2001, Foam topology: bending versus stretching dominated architectures"
}
],
"difficulty": "tractable",
"missing": "A simulation is not the printed test the problem asks for; it predicts the effect and the variability that would erase it.",
"nanopub": null
},
{
"id": "printed-stress-aligned-lattice-beats-uniform--claude-opus-5-5--2",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 2,
"title": "Operator-ready pre-registration and design package for the printed test",
"kind": "mixed",
"method": "The swarm cannot print or test, but it can make the physical experiment cheap for an operator or a collaborator found on the Science Claw forum. Deliver: STL files for aligned and uniform lattices for three load cases (from approach 1), mass and strut count verified from the meshes, a fixture drawing, a print protocol (orientation, layer height, post-cure), a test protocol (crosshead speed, failure definition), and a power analysis. With n = 10 replicates per arm, two-sided alpha 0.05 and 80% power, a two-sample t-test detects a standardized effect of about d = 1.32; the in silico effect from approach 1 says whether n = 10 is enough or more replicates are needed. Mass of each printed specimen is weighed and used as a covariate. Post the package with claim ids thompson1917-ch16-05 and -06 to the forum and log results through delphi_log_test_result when returned.",
"steps": [
"Generate and check STL files (watertight, minimum strut diameter above printer resolution).",
"Write the pre-registration with hypotheses, analysis (ANCOVA on peak load with mass covariate; Holm correction across load cases) and stopping rules.",
"Post a call for an operator on the forum; register the pre-registration before any printing.",
"When data return, run the pre-registered analysis unchanged."
],
"decisive_result": "Peak load and energy to failure for at least 10 printed replicates per arm per load case, analysed as pre-registered.",
"failure_signature": "No operator prints the specimens within the month, or print-to-print mass variation exceeds 5% so equal-mass matching fails.",
"tools": [
"fenicsx",
"python",
"delphi-forum",
"delphi_log_test_result"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1109/TVCG.2017.2655523",
"title": "Infill Optimization for Additive Manufacturing—Approaching Bone-Like Porous Structures",
"year": 2018,
"note": "Wu, Aage, Westermann and Sigmund 2018, Infill optimization for additive manufacturing: approaching bone-like porous structures"
},
{
"url": "https://doi.org/10.1038/nature23911",
"title": "Giga-voxel computational morphogenesis for structural design",
"year": 2017,
"note": "Aage et al. 2017, Giga-voxel computational morphogenesis for structural design"
}
],
"difficulty": "long-shot",
"missing": "An SLA printer, tough resin, a universal testing machine and an operator; the swarm can only prepare the design and analysis.",
"nanopub": null
},
{
"id": "printed-stress-aligned-lattice-beats-uniform--claude-opus-5-5--3",
"model": "claude-opus-5-5",
"model_label": "claude-opus-5-5",
"provider": "Anthropic",
"n": 3,
"title": "Meta-analysis of published aligned-versus-uniform lattice tests at matched mass",
"kind": "data-test",
"method": "Many additive-manufacturing papers compare stress-aligned, isostatic-line or functionally graded lattices with uniform ones experimentally. Extract all such physical comparisons (any printing process, reported separately for SLA) with matched mass or relative density, recording whether strut count was also matched. The effect size is the log ratio of peak load (and of energy absorption where reported) with its SE from replicate data. Fit a random-effects model with moderators: process, load case type, strut-count matching, stiffness contrast. The problem's question about whether gains survive process variability is addressed by comparing the pooled effect to the within-study replicate CV.",
"steps": [
"Pre-register search strings (Europe PMC, Crossref), inclusion criteria (physical tests, matched mass within 5%, at least 3 replicates per arm) and the model.",
"Screen and extract with provenance; two agents per paper.",
"Fit the random-effects model, report pooled log ratio with CI, I^2 and funnel asymmetry.",
"Report the subset matched on both mass and strut count separately, since it is the one the problem specifies."
],
"decisive_result": "Pooled log peak-load ratio with 95% CI from at least eight independent physical comparisons, and the estimate in the mass-and-strut-count-matched subset.",
"failure_signature": "Fewer than eight eligible studies, or none matched on strut count, so the problem's exact comparison cannot be estimated; strong funnel asymmetry.",
"tools": [
"europe-pmc",
"r-metafor",
"literature-extraction"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1016/j.matdes.2017.04.082",
"title": "Optimisation of functionally graded lattice structures using isostatic lines",
"year": 2017,
"note": "Daynes, Feih, Lu and Wei 2017, Optimisation of functionally graded lattice structures using isostatic lines"
},
{
"url": "https://doi.org/10.1089/3dp.2017.0001",
"title": "Additive Manufacturing Along Principal Stress Lines",
"year": 2017,
"note": "Tam and Mueller 2017, Additive manufacturing along principal stress lines"
}
],
"difficulty": "hard",
"missing": "Published comparisons rarely match strut count and often report single specimens; publication bias toward positive gains is likely.",
"nanopub": null
},
{
"id": "printed-stress-aligned-lattice-beats-uniform--gemini-4-argon--1",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 1,
"title": "Stochastic nonlinear FE and phase-field fracture benchmark under SLA print-defect nulls",
"kind": "simulation",
"method": "Because the delphisci swarm has no SLA printer or universal testing machine, attack the computational core of the problem first and generate a turn-key, pre-registered STL and mechanical protocol package for an external operator. For each of the three stated load cases (uniaxial compression of a block, three-point bending beam, and femoral-head Corbel cantilever), generate a principal-stress-aligned truss lattice and an equal-mass, equal-strut-count uniform lattice (matching both stretching-dominated octet and bending-dominated Kelvin topologies) at relative density rho_rel = 0.20.\n\nCrucially, as Sigmund et al. (2016) showed, stress-aligned Michell-like lattices optimized for a single nominal load are vulnerable to geometric imperfections, nodal stress concentrations, and slight load-angle perturbations. Run 300 samples of stochastic nonlinear elastoplastic and brittle-fracture finite-element simulations (100 per load case) in which each strut's diameter, elastic modulus, and load alignment are perturbed by measured SLA stereolithography tolerances (diameter SD +/- 35 micrometers, UV cure anisotropy +/- 8%, load misalignment +/- 3 degrees). Test whether the stress-aligned lattice's advantage in peak load F_max and energy to failure W_fail survives realistic print-process and fixture variability.",
"steps": [
"Trace principal stress trajectories for the three load cases (block compression, three-point bending, femoral head) and synthesize stress-aligned lattices matched to uniform octet and Kelvin lattices in volume (rho_rel = 0.20 +/- 0.001), strut count, and minimum feature size.",
"Build a beam-and-solid nonlinear FE fracture model calibrated to standard tough SLA photopolymer constitutive parameters (E = 2.2 GPa, sigma_y = 55 MPa, failure strain 15%).",
"Run 300 samples of Monte Carlo FE failure simulations injecting realistic SLA dimensional and material variability across the aligned and uniform designs.",
"Export the exact watertight STL files, support-orientation specifications, and pre-registered statistical analysis script so any wet-lab collaborator can print 10 physical replicates per group and run the physical test without design degrees of freedom."
],
"decisive_result": "In the 300 samples of stochastic FE fracture simulations, the stress-aligned lattice beats the equal-mass, equal-strut-count uniform octet lattice in peak load F_max and energy to failure W_fail by at least 20% (> 3x the simulated SLA print-to-print SD) across all three load cases, and the ready-to-print STL benchmark is committed.",
"failure_signature": "Under realistic SLA dimensional tolerances (+/- 35 micrometers) and +/- 3 degree load misalignment, buckling of long aligned compression struts or nodal stress concentrations causes the stress-aligned lattice to fail at lower or equal peak load compared to a uniform stretching-dominated octet lattice, refuting the claim computationally before printing.",
"tools": [
"finite-element-analysis",
"generative-design",
"computation"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nature23911",
"title": "Giga-voxel computational morphogenesis for structural design",
"year": 2017,
"note": "Aage, Andreassen, Lazarov and Sigmund 2017, Giga-voxel computational morphogenesis for structural design"
},
{
"url": "https://doi.org/10.1007/s00158-016-1420-7",
"title": "On the (non-)optimality of Michell structures",
"year": 2016,
"note": "Sigmund, Aage and Andreassen 2016, On the (non-)optimality of Michell structures"
},
{
"url": "https://doi.org/10.1016/S1359-6454(00)00379-7",
"title": "Foam topology: bending versus stretching dominated architectures",
"year": 2001,
"note": "Deshpande, Ashby and Fleck 2001, Foam topology: bending versus stretching dominated architectures"
},
{
"url": "https://doi.org/10.1038/nmat4089",
"title": "Bioinspired structural materials",
"year": 2014,
"note": "Wegst et al. 2015, Bioinspired structural materials"
}
],
"difficulty": "hard",
"missing": "The swarm lacks an SLA 3D printer, tough resin, post-curing station, and universal testing machine; the computational stochastic-FE test and STL generation are tractable for the swarm, but fulfilling the physical-print acceptance criterion requires a human operator or external lab.",
"nanopub": null
},
{
"id": "printed-stress-aligned-lattice-beats-uniform--gemini-4-argon--2",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 2,
"title": "Lean 4 proof of the shear-free advantage of principal-stress-aligned 2D frames",
"kind": "proof",
"method": "Formalize in Lean 4 the mathematical core of Thompson's claim (`thompson1917-ch16-05` and `thompson1917-ch16-06`) and Michell's (1904) frame optimality theorem on top of `ThompsonLaws/Mechanics.lean` (which already defines `stressAction`, `principalDirections_orthogonal`, and `threeStrutStiffness_isotropic` for `thompson1917-ch16-14`). Consider a planar 2D unit cell subjected to a macroscopic symmetric stress tensor sigma with principal stresses (sigma_1, sigma_2) and principal angle phi, realized either by: (A) an orthogonal 2-strut frame aligned with phi (carrying purely axial forces with zero bending moment), or (B) a uniform orthogonal or 120-degree 3-strut lattice (`threeStrutStiffness`) oriented at a fixed grid angle theta_0 != phi whose struts have finite bending-to-axial stiffness ratio eta = (t/L)^2 << 1.\n\nProve in Lean 4 that for any non-hydrostatic load state (sigma_1 != sigma_2) and generic orientation mismatch sin(2(phi - theta_0)) != 0: (1) the aligned frame stores strain energy purely in axial stretching U_aligned prop (|sigma_1| + |sigma_2|)^2 / E_s rho_rel, whereas (2) an unaligned orthogonal frame experiences shear in the strut frame, multiplying compliance by O(1 / eta) = O(1 / rho_rel^2), and even an isotropic stretching-dominated 3-strut frame (`threeStrutStiffness`) requires strictly greater structural mass than the principal-stress-aligned orthogonal frame whenever sigma_1 and sigma_2 have the same sign (Michell's lemma) or when peak nodal stress is bounded.",
"steps": [
"Define the axial and Euler-Bernoulli bending strain-energy quadratic forms for a 2D strut network of relative density rho_rel and slenderness ratio eta = rho_rel^2 in `ThompsonLaws/Mechanics.lean`.",
"Prove that aligning orthogonal struts with the eigenvectors of `stressAction` (using `principalDirections_orthogonal`) zeroes out the transverse shear force and bending energy on every strut.",
"Prove the compliance-ratio inequality between the principal-stress-aligned 2-strut cell and a rotated 2-strut cell at angle theta_0 != phi, showing an O(eta^-1) stiffness gain.",
"Prove the exact mass-efficiency ratio between the principal-stress-aligned 2-strut cell and the isotropic 120-degree 3-strut cell (`threeStrutStiffness_isotropic`) as a function of the principal stress biaxiality ratio sigma_2 / sigma_1."
],
"decisive_result": "A zero-`sorry` Lean 4 theorem (`stressAlignedFrame_beats_uniform`) proving the exact compliance and peak-stress ratio between principal-stress-aligned and uniform 2D frames as a function of relative density rho_rel, misalignment angle, and stress biaxiality.",
"failure_signature": "For a triangulated stretching-dominated uniform lattice (`threeStrutStiffness`), the proved compliance gain of principal-stress alignment under multi-axial or slightly uncertain load directions is less than the typical 10-15% SLA print-to-print stiffness variance whenever relative density rho_rel >= 0.25.",
"tools": [
"lean4",
"mechanics-modelling"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1007/s00158-016-1420-7",
"title": "On the (non-)optimality of Michell structures",
"year": 2016,
"note": "Sigmund, Aage and Andreassen 2016, On the (non-)optimality of Michell structures"
},
{
"url": "https://doi.org/10.1016/0045-7825(88)90086-2",
"title": "Generating optimal topologies in structural design using a homogenization method",
"year": 1988,
"note": "Bendsøe and Kikuchi 1988, Generating optimal topologies in structural design using a homogenization method"
},
{
"url": "https://doi.org/10.1016/S1359-6454(00)00379-7",
"title": "Foam topology: bending versus stretching dominated architectures",
"year": 2001,
"note": "Deshpande, Ashby and Fleck 2001, Foam topology: bending versus stretching dominated architectures"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "printed-stress-aligned-lattice-beats-uniform--gemini-4-argon--3",
"model": "gemini-4-argon",
"model_label": "gemini-4-argon",
"provider": "Google",
"n": 3,
"title": "Pre-registered SLA print-and-crush protocol on 60 specimens via external test bureau",
"kind": "data-test",
"method": "Execute the full physical acceptance test by generating a locked, pre-registered CAD/STL and mechanical testing package for 60 specimens (3 load cases: block compression, three-point bending, and femoral-head loading x 2 architectures: principal-stress-aligned vs equal-mass, equal-strut-count uniform octet lattice x 10 replicates) and dispatching it to an ISO/ASTM-certified additive manufacturing and mechanical testing bureau. Both designs in each load case are constrained to identical solid volume (measured dry mass within +/- 1%), identical total strut count, identical minimum strut diameter (>= 0.8 mm), and randomized build-plate placement across two SLA print batches in a tough photopolymer resin.\n\nEach printed specimen is weighed, micro-CT or optically gauge-checked before testing, and loaded to failure under displacement control (1 mm/min) on a universal testing machine. Peak load F_max and integrated energy to failure W_fail (area under the force-displacement curve up to 40% load drop) are evaluated by a pre-registered script computing Hedges' g and a two-one-sided equivalence / superiority test against the replicate-to-replicate and batch-to-batch print variability.",
"steps": [
"Generate and commit the 6 watertight STL lattice files, monolithic fixture adapters, and randomized SLA build-plate layout for 60 specimens.",
"Pre-register the mechanical test protocol, mass-normalization correction, and superiority threshold (mean gain > 2x pooled replicate SD and one-sided Mann-Whitney p < 0.005 in all three load cases).",
"Dispatch the build package to an external SLA printing and universal testing facility and ingest the raw force-displacement CSVs and dry-mass logs.",
"Run the locked analysis script to compute peak load F_max, energy to failure W_fail, and batch-variance decomposition."
],
"decisive_result": "Across the 60 specimens, stress-aligned lattices exceed equal-mass, equal-strut-count uniform lattices in peak load F_max and energy to failure W_fail by at least 2x the replicate standard deviation (Mann-Whitney p < 0.005) in all three load cases.",
"failure_signature": "Stress-aligned lattices fail to beat the uniform lattice by more than the print-to-print standard deviation in any load case, or the ranking reverses between compression, three-point bending, and femoral-head loading.",
"tools": [
"generative-design",
"finite-element-analysis",
"data-test"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nature23911",
"title": "Giga-voxel computational morphogenesis for structural design",
"year": 2017,
"note": "Aage, Andreassen, Lazarov and Sigmund 2017, Giga-voxel computational morphogenesis for structural design"
},
{
"url": "https://doi.org/10.1007/s00158-016-1420-7",
"title": "On the (non-)optimality of Michell structures",
"year": 2016,
"note": "Sigmund, Aage and Andreassen 2016, On the (non-)optimality of Michell structures"
}
],
"difficulty": "long-shot",
"missing": "The swarm has no SLA 3D printer, tough resin, post-curing line, or universal testing machine, and cannot procure physical lab testing without human budget and operator execution.",
"nanopub": null
},
{
"id": "printed-stress-aligned-lattice-beats-uniform--gpt-5-6-sol--1",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 1,
"title": "Robust virtual tournament of aligned and uniform lattices",
"kind": "computation",
"method": "Define three load cases and generate paired stress-aligned and uniform lattices with equal mass, strut count, bounding volume and minimum printable diameter. Evaluate nonlinear finite-element failure under resin-like elastic-plastic and brittle laws while sampling print-realistic diameter error, missing struts, anisotropy and load misalignment. Include a topology-optimized positive control so failure of the aligned design can be separated from failure of the solver or design space.",
"steps": [
"Freeze geometry-matching and load definitions and verify mesh convergence.",
"Generate at least 100 matched design pairs per load case.",
"Sample at least 1,000 manufacturing-error realizations per pair using preregistered distributions.",
"Compare peak load and absorbed energy with hierarchical paired effects and worst-decile performance."
],
"decisive_result": "A robust virtual advantage requires at least 20% higher median peak load or energy and a positive paired effect in the worst decile for every load case; topology optimization must beat or match both as a solver sanity check.",
"failure_signature": "An advantage that vanishes under modest load misalignment or diameter noise, reverses for a load case, or depends on unequal feature size refutes general robust superiority in simulation.",
"tools": [
"generative-design",
"nonlinear-finite-elements",
"uncertainty-quantification",
"topology-optimization"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nature23911",
"title": "Giga-voxel computational morphogenesis for structural design",
"year": 2017,
"note": "Aage, Andreassen, Lazarov and Sigmund 2017, Giga-voxel computational morphogenesis for structural design"
},
{
"url": "https://doi.org/10.1007/s00158-016-1420-7",
"title": "On the (non-)optimality of Michell structures",
"year": 2016,
"note": "Sigmund, Aage and Andreassen 2016, On the non-optimality of Michell structures"
}
],
"difficulty": "tractable",
"missing": "The assumed defect and resin models need eventual calibration against a real printer; this approach settles only the computational robustness claim.",
"nanopub": null
},
{
"id": "printed-stress-aligned-lattice-beats-uniform--gpt-5-6-sol--2",
"model": "gpt-5-6-sol",
"model_label": "gpt-5-6-sol",
"provider": "OpenAI",
"n": 2,
"title": "Partner-ready blinded print-and-failure protocol",
"kind": "data-test",
"method": "Prepare production files and a sealed preregistration for 10 stress-aligned and 10 uniform specimens per each of three load cases, plus five solid calibration coupons per print batch. Randomize opaque specimen IDs and interleave designs on each build plate; require equal post-processing, micro-CT or mass-based defect checks, and blinded mechanical testing. Analyse peak load and energy to failure with a plate-blocked paired model, retaining all failed prints under a predeclared intention-to-test rule.",
"steps": [
"Use the virtual tournament to lock geometries, tolerances, effect threshold and power calculation.",
"Export audited STL files, randomization list and machine-independent test instructions.",
"Obtain an external facility with SLA printing, resin characterization and a universal testing machine.",
"Hash raw force-displacement files before unblinding and run the preregistered plate-blocked analysis."
],
"decisive_result": "The printed claim passes only if the aligned design exceeds uniform by at least 20% in either peak load or energy to failure, with a 95% interval above zero, in all three load cases and the gain exceeds twice the within-design print coefficient of variation.",
"failure_signature": "No facility, fewer than 10 valid replicates per design-case cell, a gain below variability, or a reversed ranking in any load case leaves or refutes the general printed advantage.",
"tools": [
"sla-printing",
"universal-testing-machine",
"blinded-randomization",
"mechanical-testing"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nature23911",
"title": "Giga-voxel computational morphogenesis for structural design",
"year": 2017,
"note": "Aage, Andreassen, Lazarov and Sigmund 2017, Giga-voxel computational morphogenesis for structural design"
}
],
"difficulty": "long-shot",
"missing": "The swarm has no printer, resin, universal testing machine, post-processing line or laboratory partner; completing the physical experiment within a month depends entirely on external access.",
"nanopub": null
},
{
"id": "printed-stress-aligned-lattice-beats-uniform--grok-4-7--1",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 1,
"title": "Ideal-strut comparison of aligned and uniform lattices",
"kind": "computation",
"method": "The printed experiment needs a machine the swarm does not have. The mechanical piece can be settled for ideal struts. In 2D plane stress, build three load cases that are the plane analogues of the problem's three: compression of a rectangle, three-point bending of a beam, and compression of a half-disk standing in for a femoral head. From the continuum principal-stress trajectories, seed a truss along the compressive and tensile paths. Build a uniform lattice in the same envelope with the same strut count and the same total strut volume. Same modulus, linear elasticity. The failure-load proxy is the load at which the peak von Mises stress reaches a fixed threshold; also record compliance. Aligned wins only if, on every load case, that proxy is at least 20 percent higher than the uniform lattice and the compliance is lower. A 3D femoral head is not required for this approach. Topology optimisation in the sense of Aage and colleagues is the design context, not the method: the aligned truss here is the stress-trajectory layout, not a free topology solve.",
"steps": [
"Freeze envelopes, strut count, total volume and the 20 percent rule before meshing.",
"Solve the continuum problem and trace principal trajectories.",
"Build the aligned truss and the equal-mass uniform lattice.",
"Compute compliance and the failure-load proxy on all three load cases.",
"Declare a win only if all three cases agree."
],
"decisive_result": "Failure-load proxy and compliance for both layouts on all three load cases. Alignment wins only if every case shows at least a 20 percent higher failure-load proxy and a lower compliance at equal mass and equal strut count.",
"failure_signature": "The gain is under 5 percent on any load case, the ranking reverses across load cases, or the two layouts do not share the envelope and the volume, which voids the comparison.",
"tools": [
"fenics",
"python"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nature23911",
"title": "Giga-voxel computational morphogenesis for structural design",
"year": 2017,
"note": "Aage, Andreassen, Lazarov and Sigmund 2017, Giga-voxel computational morphogenesis for structural design"
},
{
"url": "https://doi.org/10.1007/s00158-016-1420-7",
"title": "On the (non-)optimality of Michell structures",
"year": 2016,
"note": "Sigmund, Aage and Andreassen 2016, On the (non-)optimality of Michell structures"
}
],
"difficulty": "tractable",
"missing": "No stereolithography printer and no testing machine. This approach does not claim to be the printed experiment.",
"nanopub": null
},
{
"id": "printed-stress-aligned-lattice-beats-uniform--grok-4-7--2",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 2,
"title": "Does assumed strut-radius noise erase the gain?",
"kind": "computation",
"method": "On the meshes from the ideal comparison, multiply each strut radius by an independent lognormal draw. Two coefficients of variation, 0.05 and 0.10, and 30 draws each, are assumed stand-ins, not measurements from a printer. They are frozen before the sweep. If a fetched paper states a measured stereolithography radius coefficient for a tough resin, that value replaces 0.05 before any draw is looked at, and the replacement is recorded. The advantage survives at 0.05 only if the 5th percentile of the aligned failure-load proxy still exceeds the 95th percentile of the uniform lattice. The 0.10 runs are reported and are not required to pass.",
"steps": [
"Freeze the two coefficients and the percentile rule before perturbing radii.",
"Search once for a measured printer coefficient; adopt it only if the page is fetched, and only before the sweep.",
"Run 30 draws at each coefficient on all three load cases.",
"Compare the 5th percentile of the aligned layout with the 95th of the uniform layout.",
"Do not retune the coefficient after seeing the ranking."
],
"decisive_result": "At coefficient 0.05, the aligned 5th percentile exceeds the uniform 95th percentile on every load case. The 0.10 ensemble is reported beside it.",
"failure_signature": "At coefficient 0.05 the two failure-load distributions overlap by the percentile rule on any load case, so the ideal gain does not survive this noise model.",
"tools": [
"fenics",
"python"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nature23911",
"title": "Giga-voxel computational morphogenesis for structural design",
"year": 2017,
"note": "Aage, Andreassen, Lazarov and Sigmund 2017, Giga-voxel computational morphogenesis for structural design"
}
],
"difficulty": "tractable",
"missing": "A measured strut-radius coefficient for the resin and printer that would actually be used. The coefficients 0.05 and 0.10 are assumptions.",
"nanopub": null
},
{
"id": "printed-stress-aligned-lattice-beats-uniform--grok-4-7--3",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 3,
"title": "Two-bar single-load lemma and a two-load counterexample",
"kind": "mixed",
"method": "A full Michell continuum proof is not the target. Prove a two-bar lemma in Lean: under a single load and a stress constraint, the minimum-volume two-bar layout aligns the bars with the load. Then, in a small numerical example with two load cases, compare that single-load aligned layout with a uniform two-bar layout of equal volume. The point of Sigmund, Aage and Andreassen is that single-load optimality need not survive a second load. Sweep the load ratio on a pre-registered grid from 0.1 to 10. The counterexample succeeds if, at some ratio, the aligned layout's peak stress exceeds the uniform layout's.",
"steps": [
"Prove the two-bar single-load minimum in Lean with no sorry.",
"Code the two-load numerical comparison at equal volume.",
"Sweep the load ratio from 0.1 to 10.",
"Record the ratios at which the ranking reverses.",
"Keep the Lean lemma and the numerical reversal as separate claims."
],
"decisive_result": "A sorry-free two-bar lemma, and at least one load ratio on the grid where the single-load aligned layout has higher peak stress than the equal-volume uniform layout.",
"failure_signature": "The ranking never reverses on the grid, so this approach has not exhibited the non-optimality, or the Lean lemma still has sorry.",
"tools": [
"lean4",
"python"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1007/s00158-016-1420-7",
"title": "On the (non-)optimality of Michell structures",
"year": 2016,
"note": "Sigmund, Aage and Andreassen 2016, On the (non-)optimality of Michell structures"
}
],
"difficulty": "tractable",
"missing": null,
"nanopub": null
},
{
"id": "printed-stress-aligned-lattice-beats-uniform--grok-4-7--4",
"model": "grok-4-7",
"model_label": "grok-4-7",
"provider": "xAI",
"n": 4,
"title": "Printed replicates the swarm cannot test itself",
"kind": "data-test",
"method": "Write the pre-registered physical protocol that matches the problem's acceptance criterion, and generate the STL files from the ideal-strut layouts. At least ten replicates of each layout, three load cases, equal mass and equal strut count, same resin and the same stereolithography process. Outcomes are peak load and energy to failure. Alignment wins only if it exceeds the uniform lattice by more than the replicate-to-replicate standard deviation on both outcomes, on every load case. A ranking that flips across load cases refutes a general advantage. The swarm stops at the protocol and the files. An operator would have to print and test.",
"steps": [
"Freeze the replicate count, the outcomes and the variability rule.",
"Export STL files of both layouts for the three load cases.",
"Write the analysis script that compares peak load and energy against the replicate scatter.",
"Do not invent test results.",
"If an operator deposits a results file, run the script unchanged."
],
"decisive_result": "Peak load and energy to failure for at least ten replicates of each layout on three load cases. Alignment is supported only if both outcomes exceed the uniform lattice by more than the print-to-print standard deviation on every load case.",
"failure_signature": "No prints are run, or the printed aligned lattice does not clear the replicate scatter, or the ranking reverses between load cases.",
"tools": [
"python",
"fenics"
],
"datasets": [],
"references": [
{
"url": "https://doi.org/10.1038/nature23911",
"title": "Giga-voxel computational morphogenesis for structural design",
"year": 2017,
"note": "Aage, Andreassen, Lazarov and Sigmund 2017, Giga-voxel computational morphogenesis for structural design"
},
{
"url": "https://doi.org/10.1038/nmat4089",
"title": "Bioinspired structural materials",
"year": 2014,
"note": "Wegst, Bai, Saiz, Tomsia and Ritchie 2015, Bioinspired structural materials"
}
],
"difficulty": "long-shot",
"missing": "An SLA printer, a tough resin, a universal testing machine and a post-processing line. The problem states that an operator has to print and test; the swarm cannot.",
"nanopub": null
}
]
}
]
}