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Math-459-A2 · revision r1 · on Math-459

OpenAI claim summary for Math-459

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.

The asserted transformation is invertible and ergodic on a standard nonatomic space, and has zero entropy, hence finite entropy. The claimed nonexistence of every compact finite-dimensional smooth positive-density model includes the smooth-volume-form models requested by this problem. Allowing smooth boundary or nonorientability on the forbidden model enlarges the excluded class. The obstruction is specifically to smooth positive volume; singular-measure models remain outside this target.

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

Manuscript

Contents
  1. Smooth Realization of Finite-Entropy Ergodic Transformations
  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

Smooth Realization of Finite-Entropy Ergodic Transformations

OpenAI's claim

A zero-entropy invertible ergodic nonatomic probability system is claimed to have no realization by any finite-dimensional smooth diffeomorphism preserving a strictly positive smooth density. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims an ergodic invertible probability-preserving transformation $(X,\mu,T)$ on a standard nonatomic probability space with \[ h_\mu(T)=0, \] which is not conjugate measurably after deleting invariant sets of measure zero to any smooth diffeomorphism of a compact manifold of finite dimension whose preserved probability measure has smooth everywhere-positive density. The exclusion covers nonorientable models as well as manifolds with a smooth boundary. The measure class is positive smooth volume; arbitrary singular invariant measures are not covered.

Outline of the argument

Manuscript 1 (family 152, 2026-09-25): For smooth positive volume, a dyadic switching estimate joins the future of one nearby point to the past of another with controlled output measure and summable cell complexity. Derivative-log bounds and finite boundary-value calculations establish that estimate; the constructed system violates the resulting measure-theoretic switching constraint in every finite dimension.

Formal verification

For family 152, the scope page documents the following selections. The linked obstruction only requires finite measure entropy. The manuscript's sharper zero-entropy property is beyond that declaration, whereas exclusion of smooth positive-volume models includes compact manifolds with boundary. This entry does not assert that the selected declarations formalize every part of manuscript 1's main conclusion. Its selected declaration also covers the clause used for this problem. Comparator files: SmoothObstruction.lean. Declarations: OAI.SmoothObstruction.main.

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