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Math-2889-A1 · revision r1 · on Math-2889

OpenAI claim summary for Math-2889

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.

Chords with prescribed separated ports are claimed to have length $R^{4/3+o(1)}$ in regular hexagons, with analogous half-plane and cut laws. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

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

Manuscript

Contents
  1. The length exponent of critical honeycomb chords with fixed endpoints
  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

The length exponent of critical honeycomb chords with fixed endpoints

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Chords with prescribed separated ports are claimed to have length $R^{4/3+o(1)}$ in regular hexagons, with analogous half-plane and cut laws.

Mathematical statement of the claimed result

Result 1.
A honeycomb chord joins two boundary-edge midpoints, enters through triangle centres, visits each centre at most once, and makes no additional boundary exit. Its length $L$ counts visited centres. Give each chord probability proportional to $x_*^L$, where $x_*=(2+\sqrt2)^{-1/2}$. Let $D_R$ be convex hexagons built from complete lattice rows with $R^{-1}D_R$ tending to a regular hexagon. Suppose the prescribed endpoints $a_R,b_R$, after division by $R$, tend to different interior points of sides. For every $\varepsilon>0$, \[ \mathbb P_{D_R;a_R,b_R}\{R^{4/3-\varepsilon}\le L\le R^{4/3+\varepsilon}\}\longrightarrow1. \] The conclusion also applies in a lattice half-plane with endpoint distance $R$; in a strip whose height and $R$ have comparable ratio, with endpoints on opposite cuts and separation $O(R)$; and in a wedge of angle $\pi/3$ or $2\pi/3$ with one endpoint on each ray, distance $O(R)$ from the apex, and distance comparable to $R$ from the opposite cut. Bounds are uniform over compact families of these nondegenerate rescaled configurations. Each partition sum is at least $cR^{-5/4}$.

Outline of the argument

Finite estimates under the irreducible bridge law first control a fixed source and terminal intervals. Strip partition regularity and exact height/displacement atoms sharpen the result to specified endpoints, while parallel and nonparallel cut comparisons give the different geometries and averaged-height mean statements.

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