OpenAI · 151-01 · 2026-09-25 · Dynamical systems and ergodic theory
A C^1 Counterexample to the Entropy Conjecture
preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026
Family 151. A counterexample to Shub’s entropy conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Constructs a noninvertible self-map of a compact smooth manifold with zero topological entropy but eigenvalue on second homology. This disproves the homological entropy lower bound for general self-maps: homological growth need not force positive orbit complexity.
We disprove the general C1 self-map formulation of Shub's entropy conjecture. We construct a noninvertible C1 self-map of a compact smooth manifold without boundary whose topological entropy is zero, while its action on second real homology has eigenvalue 2. Thus the topological entropy is strictly smaller than the logarithm of the homological spectral radius.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 151: A <i>C</i><sup>1</sup> counterexample to the entropy conjecture. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.