Dynamical systems and ergodic theory·OpenAI Math·Result 151·Lean formalisation listed
A C1 counterexample to the entropy conjecture
One of the 372 results in OpenAI's public mathematics catalogue, filed under dynamical systems and ergodic theory: 1 preprint, the latest dated 25 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a proof of a C1 counterexample to the entropy conjecture (dynamical systems and ergodic theory) in a preprint dated 25 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More, is on YouTube.
Chapter 13 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 4:10 · watch from there on YouTube →
What OpenAI claims
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. (Lean)
Paper
A C1 Counterexample to the Entropy Conjecture
OpenAI's abstract 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.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 25 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- 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. (Lean)
- Where is the proof?
- In the paper, A C1 Counterexample to the Entropy Conjecture (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 13).
Also in
- Chapter 6 of 9 in Birkhoff Conjecture, Zero Entropy and 7 More in Dynamical Systems and Ergodic Theory · starts at 1:46 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · Compilation · 29:10 · chapter 13 from 4:10 · description, sources and chapters →
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