OpenAI claim summary for Math-2918

Massive interface limits under thermal FK–Ising perturbation

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. At fixed nonzero thermal mass, FK--Ising interfaces on angle-controlled isoradial lattices are claimed to converge to a lattice-independent massive $\mathrm{SLE}_{16/3}$ law.

Mathematical statement of the claimed result

Result 1.

Fix a bounded simply connected $\Omega\subset\mathbb C$, interior basepoint $z_*$, different boundary prime ends $a,b$, and $m\ne0$. Approximate it by finite simply connected isoradial domains of circumradius $\delta$, with rhombus half-angles $\eta\le\theta_e\le\pi/2-\eta$ for fixed $\eta>0$. Assume kernel convergence from $z_*$ and convergence of the two marks under normalized maps $\chi_\delta:\Omega^\delta\to\mathbb D$ and $\chi:\Omega\to\mathbb D$. Choose nomes $r_\delta$ with $r_\delta/\delta\to|m|/2$ and elliptic parameters $k_\delta$ satisfying \[ r_\delta=\exp[-\pi K(\sqrt{1-k_\delta^2})/K(k_\delta)],\quad U_k(t)=\int_0^t(1-k^2\sin^2s)^{-1/2}\,ds,\quad K(k)=U_k(\pi/2). \] Let $A_\delta(\theta)$ solve $U_{k_\delta}(A_\delta(\theta))=2K(k_\delta)\theta/\pi$, and set $\widehat\theta_e=A_\delta(\theta_e)$ for $m>0$, or $\pi/2-A_\delta(\pi/2-\theta_e)$ for $m<0$. Along the counterclockwise boundary, wire $(b^\delta a^\delta)$ primally and $(a^\delta b^\delta)$ dually. The interface is oriented from $a^\delta$ to $b^\delta$. An open-edge configuration $E$, with $\ell(E)$ closed medial loops, has probability proportional to \[ (\sqrt2)^{\ell(E)}\prod_{e\in E}\sin(\widehat\theta_e/2) \prod_{e\notin E}\sin(\pi/4-\widehat\theta_e/2). \] For its oriented interface $\gamma_\delta$, the full-sequence limit is \[ \operatorname{Law}(\chi_\delta\circ\gamma_\delta)\Longrightarrow\chi_*\mu_{\Omega,a,b,m} \] in uniform curve distance modulo increasing reparametrization. The physical prime-end law is independent of the admissible lattices and nome sequence. It is the uniquely characterized thermal massive $\mathrm{SLE}_{16/3}$ law for $m>0$; for $m<0$, reverse the positive-$|m|$ law with exchanged endpoints and primal/dual labels. No regularity of the physical boundary beyond the stated prime-end hypotheses is imposed. The conformal maps satisfy $\chi_\delta(z_*)=\chi(z_*)=0$, with both derivatives at $z_*$ positive and real.

Outline of the argument

A massive boundary observable is carried through the interface exploration. Green-potential estimates bound the moving-point drift in local energy, and uniqueness of the resulting driving equation identifies the limit; localization treats general domain approximations, while duality and reversal handle negative mass.

Formal verification

Family 223 has no Lean scope page in this release. Its selected result is not presented as formalized. 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

• OpenAI, Thermal FK–Ising interfaces and massive SLE, 2026-10-05; citation key OAI:Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI catalogue entry.