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Math-2898 · mathematics · Probability

The exact low-temperature transfer gap of the planar O(4) model

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the nearest-neighbor two-dimensional O(4) lattice model and its full transfer-gap mass $m_lat(\beta )$, is $lim_{\beta \to \infty }$ exp(pi $\beta )m_lat(\beta )/\operatorname{sqrt}(\beta )=32$ $\exp (\pi /4-1/2)$?

Short English statement

For the nearest-neighbor two-dimensional O(4) lattice model and its full transfer-gap mass $m_lat(\beta )$, is $lim_{\beta \to \infty }$ exp(pi $\beta )m_lat(\beta )/\operatorname{sqrt}(\beta )=32$ $\exp (\pi /4-1/2)$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2898-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2898

The exact-mass manuscript claims the displayed coefficient $32e^{\pi/4-1/2}$ in the physical lattice-time normalization, so it supplies the precise asymptotic asked for here. The earlier sharp-mass manuscript gives positivity and bounds up to constant factors; those bounds alone do not identify the coefficient. The exact theorem supplies the full match.

The exact low-temperature transfer gap of the planar O$4$ model

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: The proposed large-$\beta$ $O(4)$ mass is $32e^{\pi/4-1/2}\sqrt\beta e^{-\pi\beta}(1+o(1))$ in original lattice time units; The full square-lattice $O(4)$ transfer gap is claimed to be positive at every finite coupling and comparable to $\sqrt\beta e^{-\pi\beta}$ at large $\beta$. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
For the nearest-neighbor square-lattice $O(4)$ model, spins lie in $S^3$ and the periodic Gibbs density is proportional to $\exp(\beta\sum_{\langle xy\rangle}\sigma_x\cdot\sigma_y)$, each bond counted once. In the periodic local limit, let $T_\beta$ be the full Osterwalder--Schrader one-time-step transfer operator and $\Omega_\beta$ its constant-function vacuum. Define \[ m_{\mathrm{lat}}(\beta)=-\log\|T_\beta|_{\Omega_\beta^\perp}\|. \] The exact claimed asymptotic is \[ \lim_{\beta\to\infty}\frac{e^{\pi\beta}}{\sqrt\beta}m_{\mathrm{lat}}(\beta) =32e^{\pi/4-1/2}. \] The gap includes every sector generated by local observables, not only spin correlations.
Result 2.
For the homogeneous nearest-neighbor square-lattice $O(4)$ model, the source claims a unique periodic local limit for sufficiently large $\beta$, and full transfer gap \[ c\sqrt\beta\,e^{-\pi\beta}\le\operatorname{gap}(\beta) \le C\sqrt\beta\,e^{-\pi\beta},\qquad\beta\ge\beta_0, \] with $0<c<C<\infty$. Fixed dyadic $L$, terminal coupling $H$, and integers $N_0,M_0$ give couplings $\beta_N$ whose exact $N$-step transform ends at $H$, satisfying \[ \frac{\log2}{M_0L^N}\le\operatorname{gap}(\beta_N)\le\frac{\log8}{L^N}\quad(N\ge N_0). \] The preliminary result uses $\Xi(\beta)=\exp[\pi\beta+C\sqrt{(1+\beta)\log(2+\beta)}]$. For unpinned free subgraphs, or rectangular tori with sides at least $C\Xi$, couplings $0\le\beta_e\le\beta$, and local Lipschitz observables at distance $d$, it bounds covariance by $C(1+\beta+|A|+|A'|+d)^{20}L_FL_Ge^{-cd/\Xi}$. It also gives \[ 0\le4\log Z_\beta(n,n)-\log Z_\beta(2n,2n) \le C(1+\beta)^pn^p e^{-cn/\Xi(\beta)} \] for even $n\ge C\Xi(\beta)$. Spins lie in $S^3$ with normalized spherical measure, and the Gibbs density is proportional to $\exp(\sum_e\beta_e\sigma_x\cdot\sigma_y)$. Reflection positivity of the periodic limiting state supplies a vacuum $\Omega$ and positive one-step translation contraction $T_\beta$, with $T_\beta\Omega=\Omega$. The gap convention is \[ \operatorname{gap}(\beta)=-\log\|T_\beta|_{\Omega^\perp}\|. \] For a local observable $F$ with support $A$, define $L_F=\|F\|_\infty+\max_{x\in A}\operatorname{Lip}_x(F)$, using round-sphere distance in that coordinate while all others are held fixed. The separation $d$ is maximum-norm distance between supports, with periodic distance on tori. The preliminary estimates hold for every $\beta\ge0$; $Z_\beta(n,w)$ is the homogeneous torus partition function. The constants $L,H,N_0,M_0$ above are selected once, with $H$ sufficiently large.

Outline of the argument

Odd and even observable sectors express the full gap through ratios of twisted partition functions. A Brownian regulator permits an exact auxiliary spectral calculation, and regulator comparison transports its leading constant back to the square-lattice model.

Quaternionic rotation and current identities drive the scale-by-scale stiffness decrease. A massive block expansion provides two-sided gap bounds, and comparison of polynomial local-observable sectors ensures the spin estimate controls the entire transfer gap.

Formal verification

For family 215, the selected theorem gives exponential finite-free-graph $O(n)$ spin decay for $n\geq3$ and bounded ferromagnetic edge strengths. Plane limits, full $O(4)$ spectral gaps, and continuum $O(3)$ results are beyond that entry. 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.

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