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Math-2940 · mathematics · Probability

Height-averaged logarithmic bridge-length windows for critical honeycomb self-avoiding walks

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
On the honeycomb lattice with critical weight $X^length$, $X=(2+\operatorname{sqrt}(2))^{-1/2}$, use self-avoiding strict port-to-port bridges of integer height H and a fixed start per sum, with center-count length. Do there exist $c,C,h_0>0$ such that for all real $h>=h_0$ the summed mass, over $1<=H<=2h(\log $ $h)^{1/64}$, of bridges with length between $h^{4/3}(\log $ $h)^{-1/8}$ and $2h^{4/3}(\log $ $h)^{1/2}$ is at least c $h^{3/4}(\log $ $h)^{-C}$?

Short English statement

On the honeycomb lattice with critical weight $X^length$, $X=(2+\operatorname{sqrt}(2))^{-1/2}$, use self-avoiding strict port-to-port bridges of integer height H and a fixed start per sum, with center-count length. Do there exist $c,C,h_0>0$ such that for all real $h>=h_0$ the summed mass, over $1<=H<=2h(\log $ $h)^{1/64}$, of bridges with length between $h^{4/3}(\log $ $h)^{-1/8}$ and $2h^{4/3}(\log $ $h)^{1/2}$ is at least c $h^{3/4}(\log $ $h)^{-C}$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2940-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2940

The logarithmic bridge-length window theorem is the literal estimate in the question, with R_h = h(log h)^(1/64), lower length h^(4/3)(log h)^(-1/8), upper length 2h^(4/3)(log h)^(1/2), height range 1 <= H <= 2R_h, and lower mass c h^(3/4)(log h)^(-C). It retains the fixed source, free terminal lateral port, center-count length, and height averaging. Inventory does not list this manuscript in Lean Scope 237; the family doc lists other companion papers, so its main Lean flag is null. Within these assumptions, this is a claimed affirmative answer to the stated question.

Height-averaged logarithmic bridge-length windows for critical honeycomb self-avoiding walks

OpenAI's claim

OpenAI presents a proof in the selected manuscript. After summing heights through order $h(\log h)^{1/64}$, a logarithmic window around $h^{4/3}$ is claimed to carry at least $h^{3/4}(\log h)^{-C}$ bridge mass.

Mathematical statement of the claimed result

Result 1.
At honeycomb critical fugacity $X=(2+\sqrt2)^{-1/2}$, put $\mu(\omega)=X^{|\omega|}$, counting visited vertices. For large real $h$, define $R_h=h(\log h)^{1/64}$, $\ell_-=h^{4/3}(\log h)^{-1/8}$, $\ell_+=h^{4/3}(\log h)^{1/2}$. The claim gives \[ \sum_{1\le H\le2R_h}\sum_{\omega\in\mathcal B_H,\ \ell_-\le|\omega|\le2\ell_+}\mu(\omega) \ge c h^{3/4}(\log h)^{-C}. \] This averages over integer bridge heights. Separate finite estimates are $B_H\asymp H^{-1/4}$, arch mass $A_g\asymp g^{-5/4}$, with a lower bound retained at diameter $O(g)$. A fixed-width $TH$ bridge corridor retains mass $c_TH^{-1/4}$ and has length-weighted mass at most $C_TH^{13/12}$, even after fixing either weak endpoint-displacement sign. For strip arches/bridges, diameter tail mass is at most $CH^{-1/4}e^{-cr/H}$ for $r\ge CH$. Polygon shells satisfy $\sum_{R\le\operatorname{diam}P<2R}\mu(P)|P|^2\le CR^{2/3}(1+\log\log R)$. In convex finite triangle domains of diameter $C_0r$ and bounded side count, all-endpoint chord length-weighted mass is at most $Cr^{25/12}$. The polygon sum ranges over unrooted polygons modulo primitive lattice translations, so each translation class is counted once.

Outline of the argument

Finite-circumference loop expansions are recalibrated against positive cap amplitudes. Winding-pressure tails and divided-slice enclosing estimates transfer this normalization to strip crossings, concentrating a controlled mass in the specified length window after height averaging.

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

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