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Math-2699 · mathematics · Analysis

Metric Markov cotype two of ell1

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does the real Banach space ell1 have finite metric Markov cotype two, uniformly over all finite stationary reversible Markov chains and all positive integer times?

Short English statement

Does the real Banach space ell1 have finite metric Markov cotype two, uniformly over all finite stationary reversible Markov chains and all positive integer times?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2699-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2699

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.

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

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