we-solve (beta)

Math-2886-A1 · revision r1 · on Math-2886

OpenAI claim summary for Math-2886

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.

The cylinder-loop-weights manuscript asserts the displayed piecewise exponent for every positive loop fugacity, including the transition at two. It therefore covers the complete question. The signed-cylinder companion treats fugacity two and additional propagation bounds; it would supply only that special case if considered alone. The all-fugacity theorem is the basis for the full match.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. The fugacity-dependent cylinder exponent for critical honeycomb loops
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Result 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

The fugacity-dependent cylinder exponent for critical honeycomb loops

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Critical honeycomb loop partition exponents are claimed at every positive fugacity; fugacity two gives cylinder exponent $1/6$ and planar nesting exponent $1/12$; Signed propagation is claimed to recover the fugacity-two nesting exponents and macroscopic first-exit chord mass and length laws, together with a squared-polygon-length bound. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
On the balanced honeycomb cylinder of even circumference parameter $N$, sum over finite vertex-disjoint polygon families separating its two prescribed markers. With $\rho=(2+\sqrt2)^{-1/2}$, set $Z_N(\chi)=\sum_{\mathcal P}\prod_{P\in\mathcal P}\chi\rho^{|P|}$, including the empty family. For each fixed $\chi>0$, the source claims \[ Z_N(\chi)=N^{\tau(\chi)+o(1)},\qquad \tau(\chi)=\begin{cases} \frac16-\frac{2}{3\pi^2}\arccos(\chi/2)^2,&0<\chi\le2,\\ \frac16+\frac{2}{3\pi^2}\operatorname{arcosh}(\chi/2)^2,&\chi\ge2. \end{cases} \] Separation includes winding loops and contractible loops enclosing exactly one marker. Use the quotient of the embedded honeycomb plane by the translation $iN\sqrt3/2$, with $N=2m$. The two markers are $0$ and $m e^{2\pi i/3}$.
Result 2.
For the honeycomb cylinder polygon partition $G_m(g)$, each separating loop has weight $g x^{|\lambda|}$, $x=(2+\sqrt2)^{-1/2}$, with vertex-disjoint loop families. Along even $m$, the source asserts \[ G_m(2)=m^{1/6+o(1)},\qquad -\left.\frac{d^2}{dg^2}\log G_m(g)\right|_{g=0}=O(\log m), \] where the latter counts weighted ordered incompatible cycle pairs, including self-pairs. Locally uniformly for complex $\Theta$ with $|\operatorname{Re}\Theta|<\pi$, \[ |G_m(2\cos\Theta)|\le m^{\operatorname{Re}(1/6-2\Theta^2/(3\pi^2))+o(1)}. \] For disjoint polygons surrounding the central face of a regular side-$R$ hexagon, fugacity two gives $\mathcal Z(R)=R^{1/12+o(1)}$. The cylinder identifies horizontal coordinates modulo $2m$, and its markers are $(0,0)$ and $(m,0)$, with $m$ even.

Outline of the argument

Contour sectors are centered and compared with a coercive fluctuation spectrum. Compensated quadratic integration and an electric contour integral continue the imbalance sum, determining cylinder growth; winding pressure and separating-loop bounds transfer it to planar nesting.

Boundary-balance identities define a two-puncture spin observable. Angular transfer and bosonization control the horizontal gas, including possible Jordan blocks, while a Pfaffian normalization fixes the physical amplitude; a separate tilted two-bond estimate gives polygon moment control.

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