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Math-2890 · mathematics · Probability

Partition and mean-length exponents for macroscopic triangular honeycomb chords

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the source critical boundary-to-boundary chord ensemble in an equilateral pure-cut honeycomb triangle of scale R, restricted to diameter at least R/100, is its partition mass comparable to $R^{3/4}$ and its expected visited length $R^(4/3+o(1))$? Ask also for the same mean exponent for ordinary center-to-center critical-weight walks with endpoints on the inner boundary layer.

Short English statement

For the source critical boundary-to-boundary chord ensemble in an equilateral pure-cut honeycomb triangle of scale R, restricted to diameter at least R/100, is its partition mass comparable to $R^{3/4}$ and its expected visited length $R^(4/3+o(1))$? Ask also for the same mean exponent for ordinary center-to-center critical-weight walks with endpoints on the inner boundary layer.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2890-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2890

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.

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.

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