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Math-2257-A1 · revision r1 · on Math-2257

OpenAI claim summary for Math-2257

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: partially solved. 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 theorem excludes every two-sided geodesic in the planar nearest-neighbor model under a continuous edge law and the minimum-of-four second-moment assumption. The problem statement asks natural iid models on Z^d without limiting d to two or imposing those law hypotheses. This entry records partial support; complete coverage of the question is not established by this catalogue.

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

Manuscript

Contents
  1. Absence of bigeodesics in first-passage percolation
  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

Absence of bigeodesics in first-passage percolation

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Planar iid nonatomic first-passage weights are claimed to admit no doubly infinite minimizing path when the minimum of four weights has a finite second moment.

Mathematical statement of the claimed result

Result 1.
Give the nearest-neighbor edges of $\mathbb Z^2$ independent nonnegative passage times with an atomless common law $G$, and let $T(x,y)$ be the minimal passage cost. Assume \[ \mathbb E[\min\{X_1,X_2,X_3,X_4\}^2]<\infty, \qquad X_i\text{ independent with law }G. \] The assertion is that almost surely no doubly infinite self-avoiding adjacent sequence $(v_i)_{i\in\mathbb Z}$ satisfies \[ \sum_{k=i}^{j-1}t_{\{v_k,v_{k+1}\}}=T(v_i,v_j) \quad\text{for all integers }i<j. \] Every finite segment would otherwise be geodesic.

Outline of the argument

Busemann labels reduce potential bigeodesics to compact sectors. Finite increases of passage times create marked paths with a definite extra cost; deterministic label bins and telescoping Busemann comparisons impose a finite budget that cannot support all the marks, including environment-selected exceptional directions.

Formal verification

For family 212, the documented entries give differentiability and continuously differentiable boundaries for planar Gamma shapes, including the exponential law. They do not supply strict convexity or the separate no-bigeodesic theorem. 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