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Math-400-A2 · revision r1 · on Math-400

OpenAI claim summary for Math-400

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.

Manuscript 1 (family 145, 2026-09-23): The theorem exactly matches the invertible probability-preserving, no-rate, all-finite-orders and diverging-positive-gaps clauses of the problem definition.

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

Manuscript

Contents
  1. Rokhlin problem on multiple mixing
  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

Rokhlin problem on multiple mixing

OpenAI's claim

Every invertible probability-preserving mixing transformation is claimed to be mixing at every finite order. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
On any probability space $(X,\mathcal F,\mu)$, suppose $T$ is invertible, bimeasurable, measure preserving and mixing in the two-set sense. OpenAI claims that for every $k\ge3$ and fixed measurable $A_1,\ldots,A_k$, \[ \mu\left(A_1\cap T^{-n_1}A_2\cap\cdots\cap T^{-(n_1+\cdots+n_{k-1})}A_k\right) \longrightarrow\prod_{j=1}^k\mu(A_j) \] as the minimum of the positive integer gaps $n_1,\ldots,n_{k-1}$ tends to infinity. No countable-generation or standardness assumption is made.

Outline of the argument

Manuscript 1 (family 145, 2026-09-23): A hypothetical nonproduct joining yields moving labels coupled to a zero-entropy process. Dissociated label orbits are canceled by exponential estimates against subexponential word counts, while polynomial label orbits are reduced in degree by Hilbert-space averaging; ordinary mixing anchors the reduction and contradicts the preserved positive correlation.

Formal verification

For family 145, the scope page documents the following selections. The selected implication gives mixing of every finite order for one invertible probability-preserving mixing map, with arbitrary measurable-set inputs and divergent successive time gaps. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: Rokhlin.lean. Declarations: OAI.Rokhlin.mixing_all_finite_orders.

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