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The smooth realization problem: does finite entropy suffice for a smooth volume-preserving model?

Claims Theorem 1.1: there is an ergodic invertible transformation of a standard nonatomic probability space with $h_\mu(T)=0$ that is not measurably conjugate on invariant conull sets to any $C^\infty$ diffeomorphism of a compact finite-dimensional manifold (boundary allowed, orientation not required) preserving a strictly positive smooth probability density. A single symbolic example excludes every dimension. So finite entropy, even zero entropy, does not suffice. It does not exclude models whose invariant measure is only equivalent to volume, and it does not characterize which systems are smoothly realizable.

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: The smooth realization problem: does finite entropy suffice for a smooth volume-preserving model? · Smooth realization problem

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: The smooth realization problem: does finite entropy suffice for a smooth volume-preserving model?

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No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the problem as the manuscript frames it. The proof was not refereed. The paper's own scope limits: the model must preserve a strictly positive smooth density; it does not claim an obstruction for invariant measures merely equivalent to volume (the Foreman-Weiss formulation), nor for Z^2 actions. Lean-checked on the release's own Comparator challenge SmoothObstruction together with its solution module OAI.Dynamics.SmoothObstruction.Main, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statement (OAI.SmoothObstruction.main) gives a standard nonatomic ergodic system with FINITE entropy and no smooth positive-density model on any compact manifold of any dimension, with or without boundary. That is a full negative answer to the posed question, but it is weaker than the paper's zero-entropy headline; the zero-entropy conclusion is not formalized (the release's own scope note for family 152 says so).