Constructs a noninvertible C1 self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C1 self-maps: homological growth need not force positive orbit complexity. (OpenAI’s Lean note)
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The paper states
In A C^1 Counterexample to the Entropy Conjecture
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
There are a positive integer q and a C^1 self-map f of the compact smooth manifold
such that \htop(f)=0 and f_*:H_2(M;\R)\to H_2(M;\R) has eigenvalue 2. In particular,
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C1EntropyCounterexampleThe paper states
There are a positive integer q and a C^1 self-map f of the compact smooth manifold
such that \htop(f)=0 and f_*:H_2(M;\R)\to H_2(M;\R) has eigenvalue 2. In particular,
The words they share: log, manifold, integer, self, map, entropy, counterexample, positive, compact, smooth, eigenvalue.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
C1EntropyCounterexample.lean, the release lists it as a supporting statement.
theorem exists_C1_entropy_counterexample : ∃ q : ℕ, 0 < q ∧ ∃ f : CounterexampleManifold q → CounterexampleManifold q, ∃ hf : IsC1SelfMap q f, Dynamics.coverEntropy f (Set.univ : Set (CounterexampleManifold q)) = (0 : EReal) ∧ (∃ v : RealSingularHomology 2 (CounterexampleManifold q), v ≠ 0 ∧ realHomologyMap 2 f hf.1 v = (2 : ℝ) • v) ∧ 0 < Real.log (2 : ℝ) ∧ Real.log (2 : ℝ) ≤ Real.log (totalRealHomologySpectralRadius f hf.1) ∧ ¬ Function.Bijective f ∧ 5 ≤ 2 * q + 3 := by sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note: Statement: ∃ q > 0 and a non-bijective C¹ self-map f of S¹ × (S²)^(q+1) (C¹ = locally the restriction of an ambient C¹ map) with Dynamics.coverEntropy f univ = 0, a nonzero v ∈ H₂(M;ℝ) with f_* v = 2v, and log 2 ≤ log(total real-homology spectral radius): exactly the summary’s counterexample. Mathlib’s coverEntropy is the topological entropy on compact metric spaces.
Definitions the group asks readers to compare with the paper: totalRealHomologySpectralRadius f := sSup {r | ∃ n a b u v, (u ≠ 0 ∨ v ≠ 0) ∧ f_* u = a•u − b•v ∧ f_* v = b•u + a•v ∧ r = √(a²+b²)} (custom; moduli of complex eigenvalues over all degrees; if the set were unbounded sSup = 0 and the clause log 2 ≤ log radius would fail, so it cannot trivialise)
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
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cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
Shub’s entropy conjecture predicts that a smooth self-map’s topological entropy is at least the logarithm of the spectral radius of its action on real homology. The formalization gives a counterexample to the general $C^1$ self-map version: a noninvertible $C^1$ map on a compact smooth manifold without boundary has topological entropy zero, while its action on second real homology has a nonzero eigenvector with eigenvalue $2$. Thus the homological lower bound is strictly positive and fails for this map.
A $C^1$ counterexample to the entropy conjecture C1EntropyCounterexample
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a noninvertible C¹ self-map of the paper’s compact manifold with zero topological entropy whose action on second real homology has eigenvalue 2.
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C1EntropyCounterexampleTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
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Why this passage is shown
Theorem 1.1
In A C¹ Counterexample to the Entropy Conjecture
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, C1EntropyCounterexample
Proof-verification report
MathVet, C1EntropyCounterexample
Lean statement
C1EntropyCounterexample