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Math-2699-A1 · revision r1 · on Math-2699

OpenAI claim summary for Math-2699

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 main theorem claims metric Markov cotype two for real $\ell_1$, with a constant at most $12\sqrt{21}$. Its definition ranges over every finite stationary reversible chain and every positive integer time, as the question requires; zero stationary coordinates are allowed. The extension theorem is an additional consequence. No part of the recorded cotype question lies outside this scope, so this is a full affirmative claim.

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

Manuscript

Contents
  1. Metric Markov cotype two of ell1
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Metric Markov cotype two of ell1

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 332; 2026-10-05), the asserted result is that real $\ell_1$ has metric Markov cotype two, giving uniformly controlled Lipschitz extension from Hilbert-space subsets to this target.

Mathematical statement of the claimed result

Manuscript 1.
For every finite reversible stochastic matrix $A=(a_{ij})$ with stationary distribution $\pi$, every integer $t\ge1$ and every $x_1,\ldots,x_n\in\ell_1(\mathbb R)$, the claim supplies $y_1,\ldots,y_n$ satisfying \[ \sum_i\pi_i\|x_i-y_i\|_1^2+t\sum_{i,j}\pi_i a_{ij}\|y_i-y_j\|_1^2 \le3024\sum_{i,j}\pi_i\left(\frac1t\sum_{s=1}^tA^s\right)_{ij}\|x_i-x_j\|_1^2. \] Thus the metric Markov cotype-two constant is at most $12\sqrt{21}$. The asserted extension consequence is one universal $K<\infty$ such that every Lipschitz $f:S\to\ell_1$ from a subset of any real Hilbert space $\mathcal H$ extends to $F:\mathcal H\to\ell_1$ with $\operatorname{Lip}(F)\le K\operatorname{Lip}(f)$.

Outline of the argument

For manuscript 1 (family 332; 2026-10-05), finite cut representations reconstruct an expected coordinatewise median of three stopped-walk endpoints. A flat-map martingale and quartic potential control the geometric endpoint costs, which a metric comparison converts to the Cesàro energy needed for cotype.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 332; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources