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Shub's entropy conjecture for C1 self-maps

Theorem 1.1: for some $q\ge1$ there is a $C^1$ self-map $f$ of $M=(\mathbb R/100\mathbb Z)\times(S^2)^{q+1}$ with $h_{\mathrm{top}}(f)=0$ while $f_*$ on $H_2(M;\mathbb R)$ has eigenvalue 2, so $h_{\mathrm{top}}(f)<\log\rho(f_*)$ . The map is noninvertible, $\dim M=2q+3\ge5$ , and the eigenvalue sits in intermediate degree, consistent with the known first-homology, top-degree and $C^\infty$ positive results. It does not address $C^1$ diffeomorphisms, $C^r$ maps for $r\ge2$ , or the conjecture in low dimensions.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Shub's entropy conjecture for C1 self-maps · Shub's entropy conjecture (C1 maps)

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: Shub's entropy conjecture for C1 self-maps

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture in its general $C^1$ self-map form; it gives a noninvertible $C^1$ map of $(\mathbb R/100\mathbb Z)\times(S^2)^{q+1}$ with zero entropy and eigenvalue 2 on $H_2$ , which contradicts that form. The map is not a diffeomorphism, so the $C^1$ diffeomorphism version is untouched, as the paper indicates. The release's lean/docs/151.md points to ComparatorChallenges/C1EntropyCounterexample.json, which is not in the formalization catalogue; its solution module OAI/Dynamics/EntropyCounterexample/Counterexample.lean exists at the pinned commit. The statement was read here: for some $q>0$ a map of the unit circle times $(S^2)^{q+1}$ that is $C^1$ in the sense of local $C^1$ extensions in the ambient Euclidean space, with Mathlib cover entropy 0, a nonzero real singular $H_2$ class with eigenvalue 2, and not bijective. This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.