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Math-449-A2 · revision r1 · on Math-449

OpenAI claim summary for Math-449

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 151, 2026-09-25): The selected C1 self-map conjecture does not require invertibility. The paper’s compact-manifold example has entropy zero and a real homology eigenvalue two, directly contradicting its logarithmic spectral-radius lower bound.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Shub entropy conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Shub entropy conjecture

OpenAI's claim

A noninvertible $C^1$ self-map of a closed smooth manifold is claimed to have zero topological entropy although its second-homology action has eigenvalue two, disproving the general $C^1$ entropy inequality. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For some integer $q>0$, OpenAI claims a continuously differentiable self-map $f$ of \[ M=(\mathbb R/100\mathbb Z)\times(S^2)^{q+1} \] with zero topological entropy and an eigenvalue $2$ on $f_*:H_2(M;\mathbb R)\to H_2(M;\mathbb R)$. Thus \[ h_{\mathrm{top}}(f)=0<\log2\le\log\rho(f_*). \] The map need not be invertible. The spectral radius is for the induced real-homology action; the claim is a $C^1$ self-map counterexample rather than a smooth diffeomorphism counterexample.

Outline of the argument

Manuscript 1 (family 151, 2026-09-25): A slowing circle clock controls a main sphere map and a collection of signal registers. A uniformly differentiable infinite prediction series forces every orbit toward two zero-entropy invariant sets; a direct separated-set argument localizes entropy there, while projection retains degree two on homology without providing a dynamical factor map.

Formal verification

For family 151, the scope page documents the following selections. The selection gives a noninvertible map with $C^1$ regularity, zero topological entropy, and an eigenvalue-two vector in second real homology. It does not supply a smooth or invertible counterexample. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: C1EntropyCounterexample.lean. Declarations: OAI.Problem340.exists_C1_entropy_counterexample.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources