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Math-2890-A1 · revision r1 · on Math-2890

OpenAI claim summary for Math-2890

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.

Critical macroscopic triangle chords with both ports summed are claimed to have partition order $R^{3/4}$ and mean-length exponent $4/3$; two vacuum caps select a cylinder amplitude of exponent $1/6$. 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. Partition and mean-length exponents for macroscopic triangular honeycomb chords
  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

Partition and mean-length exponents for macroscopic triangular honeycomb chords

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Critical macroscopic triangle chords with both ports summed are claimed to have partition order $R^{3/4}$ and mean-length exponent $4/3$; two vacuum caps select a cylinder amplitude of exponent $1/6$.

Mathematical statement of the claimed result

Result 1.
Let $\Delta_R$ be an equilateral triangle of side length $R$ in the triangular tiling dual to the honeycomb lattice. Let $\mathcal C_R$ consist of self-avoiding port-to-port chords lying in $\Delta_R$, with different boundary endpoints and diameter at least $R/100$. A port is a boundary-edge midpoint. If $m(\gamma)$ counts visited triangle centres, set \[ x_*=(2+\sqrt2)^{-1/2},\quad Z_R=\sum_{\gamma\in\mathcal C_R}x_*^{m(\gamma)},\quad \mathbb P_R(\gamma)=Z_R^{-1}x_*^{m(\gamma)}. \] Counting both endpoint orders, the claim is \[ Z_R\asymp R^{3/4},\qquad \mathbb E_Rm=R^{4/3+o(1)}. \] The mean-length exponent also holds for ordinary self-avoiding walks with centres as both endpoints inside $\Delta_R$ with endpoints in its inner boundary layer, diameter at least $R/100$, and weight $x_*$ per edge.

Outline of the argument

Transfer fusion and positive boundary mass build a normalized disk vacuum. A residue formula with initially independent twists identifies distinct-site cap moments before common-twist specialization; this amplitude comparison is then transferred to triangle chord sums.

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