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Math-2885 · mathematics · Probability

The mass and mean length of critical bridges in a fixed-width corridor

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For critical honeycomb bridge weights spanning pure cuts at separation R, does there exist $C_0$ such that for every fixed corridor aspect $C\geq C_0$, the confined and unconfined total masses are comparable to $R^{-1/4}$, their first length masses are $R^(13/12+o(1))$, and both normalized mean lengths are $R^(4/3+o(1))$?

Short English statement

For critical honeycomb bridge weights spanning pure cuts at separation R, does there exist $C_0$ such that for every fixed corridor aspect $C\geq C_0$, the confined and unconfined total masses are comparable to $R^{-1/4}$, their first length masses are $R^(13/12+o(1))$, and both normalized mean lengths are $R^(4/3+o(1))$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2885-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2885

The disk-transfer manuscript claims the fixed-aspect corridor result appearing in the question: a sufficiently large fixed aspect preserves the $R^{-1/4}$ mass law, both first length masses have exponent $13/12$, and both normalized mean lengths have exponent $4/3$. The other manuscripts supply strip, logarithmic-window, renewal or marked-polygon estimates in their own bridge laws. Their individual restrictions do not weaken the complete fixed-corridor assertion of the disk-transfer theorem. The full classification rests on that theorem.

The mass and mean length of critical bridges in a fixed-width corridor

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: The critical honeycomb strip-crossing mass is claimed comparable to $N^{-1/4}$, and the displacement-weighted return mass to $N^{3/4}$; 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; Critically weighted all-length bridges with a fixed starting port are claimed to have mean length $R^{4/3+o(1)}$ even under confinement, with mass of order $R^{-1/4}$; Polynomial-vacuum calculations are claimed to give mean bridge length $h^{4/3+o(1)}$, central chord-visit exponent $-2/3$, and chord mean exponent $4/3$; The proposed spatial exponent is $3/4$ for infinite irreducible bridges, all compatible large even fixed-length bridges, and thermal laws; unrestricted endpoint lower bounds hold on one density-one set of lengths; Uniform two-mark polygon control is claimed to imply bridge mass $h^{-1/4}$ and finite length-mass bounds, with a definite portion on lengths at least order $h^{4/3}$. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
In a honeycomb strip of $N$ triangular bands, weight self-avoiding port paths by $\rho^{|\gamma|}$, $\rho=(2+\sqrt2)^{-1/2}$. From a fixed bottom source, let $\mathcal A_N$ be total return-arch mass, $\mathcal B_N$ total top-bridge mass, $K_N(k)$ the mass of arches ending at bottom port $k$, and $m_N=\sum_{k\ge1}kK_N(k)$. The claim gives finite sums and \[ \cos(3\pi/8)\mathcal A_N+\mathcal B_N=1,\quad m_{N+1}-m_N\asymp\mathcal B_N,\quad m_N\asymp N^{3/4},\quad\mathcal B_N\asymp N^{-1/4}, \] with $\mathcal B_N$ nonincreasing. Here $m_N$ is a horizontal-displacement moment, not a bridge mean length. Length counts visited honeycomb vertices, equivalently triangle centres; terminal half-edges contribute no weight. An arch or bridge has no boundary contact beyond its two endpoints.
Result 2.
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.
Result 3.
Between honeycomb walls separated by admissible distance $R\in(3/2)\mathbb Z_{>0}$, fix a source port and weight each self-avoiding bridge by $x^{\ell(\gamma)}$, $x=(2+\sqrt2)^{-1/2}$, counting vertices. Let $B(R)$ and $M(R)$ be bridge mass and length-weighted mass, and $B_C,M_C$ their restrictions to lateral displacement at most $CR$ throughout the path. The source claims some $C_0$ such that every fixed $C\ge C_0$ has \[ B_C(R)\asymp B(R)\asymp R^{-1/4},\qquad M_C(R)=R^{13/12+o(1)},\quad M(R)=R^{13/12+o(1)}. \] Both normalized mean vertex counts are therefore $R^{4/3+o(1)}$.
Result 4.
For critical honeycomb self-avoiding port bridges across $h$ triangular layers, let $b_h$ sum weights $x_*^{|\gamma|}$ over all top endpoints and $\ell_h$ sum $|\gamma|x_*^{|\gamma|}$, with $x_*=(2+\sqrt2)^{-1/2}$. The source asserts finite sums and \[ b_h=h^{-1/4+o(1)},\qquad\ell_h=h^{13/12+o(1)},\qquad \mathbb E_h|\gamma|=h^{4/3+o(1)}. \] Restricting every path to horizontal distance $h\log^2h$ from its source preserves all three exponents. Length is visited-vertex count.
Result 5.
Concatenate independent critical irreducible honeycomb bridges from the fixed source, with length $L_k$, height $H_k$, and cumulative height $S_H(k)$. The source's weaker-finite-input corollary claims, for the $n$-th vertex and its height $Y_n$, almost surely \[ Y_n\ge n^{3/4-o(1)},\qquad|\gamma_n-\gamma_0|\le n^{17/18+o(1)}. \] Sum the finite bridge measure across all heights using weights $\rho^Le^{-L/n}$. The resulting law has height at least $n^{3/4-\varepsilon}$ with probability tending to one. Long half-plane arch mass is at most $n^{-3/16+o(1)}$. For $V_h=\sum_kL_k\mathbf1_{S_H(k)\le h}$ and its version counting only completed prefixes confined laterally to $Ch\log^2h$, both expected rewards are $h^{4/3+o(1)}$. Consequently the expected vertex count before first disk exit is at least $R^{4/3-o(1)}$. Here $\rho=(2+\sqrt2)^{-1/2}$, and the independent irreducible pieces have probability $p(\eta)=\rho^{L(\eta)}$. A bridge is cut into these pieces at horizontal lines which the path crosses exactly once.
Result 6.
Let $\mathcal B_h$ be honeycomb port bridges of height $h$, critical weights $w(\gamma)=(2+\sqrt2)^{-\ell(\gamma)/2}$, and total mass $B_h$. The claimed constants $c,C>0$ satisfy, for sufficiently large $h$, \[ ch^{-1/4}\le B_h\le Ch^{-1/4},\qquad \sum_{\gamma\in\mathcal B_h}\ell(\gamma)w(\gamma)\le Ch^{13/12}, \] \[ \sum_{\gamma\in\mathcal B_h:\ell(\gamma)\ge ch^{4/3}}w(\gamma)\ge ch^{-1/4}. \] For all $h\ge1,x\ge0$, $\sum_{\gamma\in\mathcal B_h:\operatorname{diam}\gamma>x}w(\gamma)\le C(1+x)^{-1/4}$. The integer $h$ counts triangular layers, with physical strip width $h\sqrt3/2$. The source port is fixed and the opposite endpoint is summed freely; $\ell(\gamma)$ counts visited centres.

Outline of the argument

Positive strip flux is represented by finite transfer rows and a polynomial vacuum. An exact arch-moment formula becomes a positive homogeneous integral; isolating its smallest coordinate determines the strip exponent. 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.

Local exploration yields a disk transfer representation and an inverse-matrix identity along a rapidity orbit. A boundary deficit and two-gap cylinder contour calculation fix the normalization; two-bridge and winding-polygon bounds then control confinement and mean length. Row matrices construct the balanced puncture polynomial and its boundary asymptotics. Surrounding-polygon localization transports the cylinder observable into finite chord visits, while a marked-tripod comparison converts those visits into the strip bridge mean.

All laws are represented on a common renewal space. Exact-length and logarithmic-window conditioning are combined with recoverable geometric cuts, and controlled changes of measure extend the bounds to thermal and unrestricted ensembles; localized endpoint estimates also yield pulling and correlation-scale exponents. Positive interpolation gives the strip exponent, and a periodic vacuum pairing fixes physical normalization. Controlled-current contour expansions and cylinder barriers produce nesting and unequal-mark bounds; these retain the marked-separation power needed for finite bridge moments.

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. The listed Comparator files are CriticalStripMass.lean, HoneycombBridgeFiniteness.lean, HoneycombFreeEnergy.lean. The inventory does not supply a named declaration for these Comparator entries. 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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