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Math-2389-A1 · revision r1 · on Math-2389

OpenAI claim summary for Math-2389

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 at unsolvedmath.com: 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 requested inequalities concern mean-square displacement of the endpoint in dimensions two and three. The selected result instead controls the diameter of a uniformly sampled fixed-length walk on the honeycomb lattice. Diameter estimates do not supply the requested endpoint lower bound, and this source gives no three-dimensional conclusion or corresponding estimate for the original lattice model. The manuscript is related to the question; it does not establish a stated subcase of these displacement inequalities.

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

Manuscript

Contents
  1. Displacement Exponents for Low-Dimensional Self-Avoiding Walks
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Displacement Exponents for Low-Dimensional Self-Avoiding Walks

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Uniform honeycomb walks of every sufficiently large length are claimed to have diameter scale $3/4$, while local counts and covers simultaneously have exponent $4/3$, with a polynomial failure bound.

Mathematical statement of the claimed result

Result 1.
Let $\gamma$ be uniform among length-$n$ self-avoiding honeycomb-lattice walks from a fixed vertex, with unrestricted endpoint. Write $D(\gamma)$ for Euclidean diameter, $M_\gamma(z,s)$ for the number of visited vertices in the closed radius-$s$ ball at $z$, and $\mathcal N_\gamma(s)$ for the minimum number of such balls covering the visited set. For every $\delta,k>0$, the source supplies $C_{\delta,k}<\infty$ and $n_0$, so every $n\ge n_0$ has an event of probability at least $1-C_{\delta,k}n^{-k}$ on which \[ n^{3/4-\delta}\le D(\gamma)\le n^{3/4+\delta}, \] \[ n^{-\delta}\min\{n,s^{4/3}\}\le M_\gamma(z,s)\le n^\delta\min\{n,s^{4/3}\}, \] \[ n^{-\delta}(1+ns^{-4/3})\le\mathcal N_\gamma(s)\le n^\delta(1+ns^{-4/3}). \] The latter bounds hold simultaneously for all visited $z$ and every real $1\le s\le n$.

Outline of the argument

Critical strip, nesting, polygon, and arc inputs bound path masses in boxes. Localized bridges with recoverable seams permit conditioning on exact lengths; multiscale covering and local-visit estimates are then made simultaneous without restricting to a density-one length set.

Formal verification

For family 237, the selected Lean statements give critical-strip mass and displacement exponents, bridge summability, and existence of pulled free energy. The $4/3$ bridge-length and $3/4$ spatial laws and the other polygon/chord exponents are outside these statements. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources