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Math-2888 · mathematics · Probability

Critical honeycomb polygon diameter and length exponents

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For simple unoriented honeycomb polygon translation classes with critical weight $w(P)=\rho _v^{|P|}$, define $A(r)=sum_(diam$ $P\geq r)w(P)$, $P(r)=sum_(diam$ $P\geq r)|P|w(P)$ and $M_2(r)=sum_(diam$ $P\leq r)|P|^2w(P)$. Are $A(r)=r^(-2+o(1))$, $P(r)=r^(-2/3+o(1))$ and $M_2(r)\leq r^(2/3+o(1))$, with every indicated sum finite?

Short English statement

For simple unoriented honeycomb polygon translation classes with critical weight $w(P)=\rho _v^{|P|}$, define $A(r)=sum_(diam$ $P\geq r)w(P)$, $P(r)=sum_(diam$ $P\geq r)|P|w(P)$ and $M_2(r)=sum_(diam$ $P\leq r)|P|^2w(P)$. Are $A(r)=r^(-2+o(1))$, $P(r)=r^(-2/3+o(1))$ and $M_2(r)\leq r^(2/3+o(1))$, with every indicated sum finite?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2888-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2888

The radial-transfer manuscript claims both weighted diameter-tail exponents, finiteness of the sums and the second length-moment bound requested here. The marked-polygon companion supplies the second-moment upper bound and related arc estimates, but not both sharp tails on its own. The full label uses the radial-transfer theorem, with the companion providing additional estimates.

Critical honeycomb polygon diameter and length exponents

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Critical polygons of diameter at most $H$ are claimed to have squared-length mass at most $H^{2/3+o(1)}$, with quantitative two-mark and unequal-arc bounds; Critical unrooted polygon diameter tails are claimed to have exponents $-2$ and $-2/3$ without and with length weighting, leading to diameter exponent $3/4$ under large-length conditioning. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
For simple unoriented honeycomb polygons, modulo primitive triangular-lattice translations and with no distinguished vertex, let $|P|$ count vertices and diameter be Euclidean. The source claims \[ \forall\eta>0\ \exists C_\eta<\infty:\quad \sum_{[P]:\operatorname{diam}P\le H}(2+\sqrt2)^{-|P|/2}|P|^2 \le C_\eta H^{2/3+\eta}\qquad(H\ge1). \] The translation convention retains the two honeycomb vertex types.
Result 2.
On the honeycomb lattice with edge length one, count finite simple unoriented polygons once modulo lattice translations. For a polygon $\gamma$, let $L(\gamma)$ count edges, let $D(\gamma)$ be its Euclidean diameter, and put $w(\gamma)=(2+\sqrt2)^{-L(\gamma)/2}$. The three sums \[ A(r)=\sum_{D(\gamma)\ge r}w(\gamma),\quad P(r)=\sum_{D(\gamma)\ge r}L(\gamma)w(\gamma),\quad M_2(r)=\sum_{D(\gamma)\le r}L(\gamma)^2w(\gamma) \] are finite and satisfy \[ A(r)=r^{-2+o(1)},\qquad P(r)=r^{-2/3+o(1)},\qquad M_2(r)\le r^{2/3+o(1)}\quad(r\to\infty). \] The exponent notation means bounds with every fixed positive error in the exponent, for sufficiently large $r$.

Outline of the argument

Finite cylinder vacua yield an exact marked-pair identity. A denominator estimate tolerates unbalanced multiplicities, and coercive magnetic trial actions handle unequal walls; the rare central scale then gives the uniform polynomial one-arc bound.

Two cylinder pressures and angular color-sum asymptotics are passed to a radial series. Determinant normalization and a one-mark Hilbert--Schmidt plateau bound control tails and moments; unequal-scale two-mark estimates complete the length-conditioned consequences.

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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