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Phys-12-A2 · revision r1 · on Phys-12

OpenAI claim summary for Phys-12

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 all-$n$ finite-volume theorem gives exponential decay for each $n\ge3$ and every finite inverse temperature, uniformly over the specified bounded ferromagnetic interactions. Taking its free-boundary plane limit yields the displayed infinite-volume target. The O(3) continuum and O(4) mass papers add specialized results; their narrower spin dimensions do not limit the coverage supplied by the all-$n$ theorem.

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

Manuscript

Contents
  1. Polyakov Conjecture: Absence of a Phase Transition for Planar O(n) Models with n >= 3
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Result 2.
  6. Result 3.
  7. Result 4.
  8. Outline of the argument
  9. Formal verification
  10. Status caveat
  11. Sources

Polyakov Conjecture: Absence of a Phase Transition for Planar O(n) Models with n >= 3

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; Square-lattice classical $O(n)$ spin correlations for $n\geq3$ are claimed to decay exponentially at every finite positive inverse temperature; 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 $\beta$; The two-dimensional $O(3)$ lattice model is claimed to have a unique interacting massive continuum limit under susceptibility and correlation-length normalization. Together they provide the following support within the scopes described below.

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.
On any finite nearest-neighbor square-lattice subgraph $G=(V,E)$, put spins $\sigma_x\in S^{n-1}$ and give their free-boundary law density proportional to \[ \exp\!\left(\sum_{\{x,y\}\in E}b_{xy}\sigma_x\cdot\sigma_y\right) \prod_{x\in V}d\omega_{n-1}(\sigma_x),\qquad 0\le b_e\le\beta. \] For every integer $n\ge3$ and finite $\beta>0$, the asserted constants $A(n,\beta)<\infty$, $m(n,\beta)>0$ work for every such graph, coupling array, and pair of vertices: \[ 0\le\langle\sigma_x\cdot\sigma_y\rangle\le A e^{-m\|x-y\|_2}. \] There is no fixed boundary spin, applied field, or added interaction.
Result 3.
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.
Result 4.
For square-lattice $O(3)$ spins, the source specifies dyadic $L>1$, terminal coupling $H>0$, field normalizations $B_N>0$, and \[ a_N=L^{-N},\qquad\beta_N=H+\frac{\log L}{2\pi}N+O(\log(N+1)). \] The smeared fields $\phi_N(f)=B_Na_N^2\sum_x f(a_Nx)\sigma_x$ are claimed to converge jointly in law and all moments, with Schwinger distributions converging as tempered distributions at every order. The limiting distributions are Euclidean and internally $O(3)$ invariant, reflection positive, symmetric, clustering, and satisfy $|S_n(F)|\le C^nn!\sup_x(1+|x|^2)^{2n}|F(x)|$. The reconstructed $1+1$-dimensional local unitary theory has a unique vacuum, nonzero orthogonal complement, and \[ H_{\mathrm{phys}}|_{\Omega^\perp}\ge mI\quad(m>0). \] The connected distribution of order four does not vanish on smooth tests with separated times, excluding Wick factorization. For some $b_0>0$ and every $b\ge b_0$, the preliminary lattice mixing bound is exponential on scale $X(b)=Ce^{(4\pi-\eta)b}$, $\eta>0$, uniformly for $0\le\beta_e\le b$, including free subgraphs and sufficiently large unpinned tori; the homogeneous periodic local limit is unique. The spin law uses normalized spherical measure on $S^2$ and density proportional to $\exp(\sum_e\beta_e\sigma_x\cdot\sigma_y)$. For supports $A,A'$ at maximum-norm separation $d$, taking periodic distance on tori, define \[ L_F=\|F\|_\infty+\max_{x\in A}\operatorname{Lip}_x(F),\qquad |\operatorname{Cov}(F,G)|\le C(1+b+|A|+|A'|+d)^p L_FL_G e^{-cd/X(b)}. \] Each coordinate Lipschitz constant uses round distance on $S^2$, keeping other spins fixed. Constants $C,c,p,\eta$ are positive. The estimate applies to finite free subgraphs and tori with each side at least $CX(b)$. Write $Z_b(n,w)$ for the homogeneous torus partition function. Given an aspect bound $\kappa\ge1$, for even side lengths obeying $\kappa^{-1}\le n/w\le\kappa$ and $\min(n,w)\ge CX(b)$, the constants can be chosen to give \[ 0\le4\log Z_b(n,w)-\log Z_b(2n,2w)\le C(1+b+n+w)^p e^{-c\min(n,w)/X(b)}. \]

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. Rotation identities and multiscale estimates force a quantitative loss of effective stiffness. Once a suitable coarse scale is reached, block estimates give exponential transmission bounds uniformly for free finite graphs and bounded nonnegative couplings.

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. A prescribed physical block scale first produces limiting Schwinger fields with uniform finite-volume and moment estimates. Renormalization comparisons remove subsequence dependence and convert the block prescription to canonical normalization; reflection positivity and clustering reconstruct a gapped theory with a nonzero connected four-point function.

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. The listed Comparator files are ClassicalON.lean. Named declarations include OAI.ClassicalON.main. 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