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Math-2901 · mathematics · Probability

A first-order temperature singularity for a stable radial continuum gas

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does there exist a bounded continuous radial pair potential phi on $R^3$, stable in the sense $sum_{i<j}$ $\phi (|x_i-x_j|)\geq -BN$, with $|\phi (r)|\leq Cr^{-3-1/32}$, and a nonempty density interval I, such that one $\beta _c\in [7/8,9/8]$ is a strict temperature derivative jump of the finite canonical free-cube thermodynamic free energy for every $\rho \in I$?

Short English statement

Does there exist a bounded continuous radial pair potential phi on $R^3$, stable in the sense $sum_{i<j}$ $\phi (|x_i-x_j|)\geq -BN$, with $|\phi (r)|\leq Cr^{-3-1/32}$, and a nonempty density interval I, such that one $\beta _c\in [7/8,9/8]$ is a strict temperature derivative jump of the finite canonical free-cube thermodynamic free energy for every $\rho \in I$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2901-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2901

The radial phase-transition manuscript gives the bounded continuous stable potential, decay $O(r^{-3-1/32})$, common critical inverse temperature in $[7/8,9/8]$ and an open density interval requested here. The earlier temperature-singularity construction permits an unbounded core and a different parameter interval, so it would not cover these extra conditions alone. The bounded-potential construction is the basis for the full match.

A first-order temperature singularity for a stable radial continuum gas

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: A stable three-dimensional radial pair potential with a divergent core and integrable tail is claimed to exhibit one finite-temperature canonical derivative jump throughout a positive density interval; A bounded continuous stable radial potential with decay $O(r^{-3-1/32})$ is claimed to give the same common-temperature first-order singularity over an open density interval. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
There exists a radial Borel pair potential $\Phi(x)=\phi(|x|)$ on $\mathbb R^3$ with $\phi(0)=+\infty$, finite locally bounded values on $(0,\infty)$, and \[ \phi(r)\to+\infty\ (r\downarrow0),\quad \phi(r)\le-a\ (r\in[r_1,r_2]),\quad \int_R^\infty r^2|\phi(r)|\,dr<\infty,\quad \phi(r)=o(r^{-3}), \] for suitable $a>0$, $0<r_1<r_2$ and finite $R$. Its interaction satisfies $U_N=\sum_{i<j}\Phi(x_i-x_j)\ge-BN$ on every finite configuration. For free cubes $\Lambda_L=[-L/2,L/2]^3$, let \[ Z_{L,N}(\beta)=\frac1{N!}\int_{\Lambda_L^N}e^{-\beta U_N}\,dx_1\cdots dx_N,\qquad f_\Phi(\beta,\rho)=-\beta^{-1}\lim_{L\to\infty}L^{-3}\log Z_{L,\lfloor\rho L^3\rfloor}(\beta). \] This limit exists and is finite for every $\beta,\rho>0$. For some nonempty open density interval $I$ and a single $\beta_c\in(1/2,3/2)$, \[ D^-_\beta f_\Phi(\beta_c,\rho)>D^+_\beta f_\Phi(\beta_c,\rho)\quad(\rho\in I), \] with finite one-sided derivatives. Position integration uses temperature-independent Lebesgue measure.
Result 2.
A second construction gives a bounded continuous radial pair potential $\phi$ on $\mathbb R^3$ and finite $B,C>0$ satisfying \[ \sum_{i<j}\phi(|x_i-x_j|)\ge-BN,\qquad |\phi(r)|\le Cr^{-3-1/32}\quad(r\ge1). \] Let $p=\lim_{L\to\infty}L^{-3}M_L$, where $M_L$ is the maximum number of points in $[0,L]^3$ with pairwise separation at least one. For the free-cube partition function \[ Z_{L,N}^\phi(\beta)=\frac1{N!}\int_{([0,L]^3)^N} e^{-\beta\sum_{i<j}\phi(|x_i-x_j|)}\,dx_1\cdots dx_N, \] the limit $f(\beta,\rho)=-\beta^{-1}\lim_LL^{-3}\log Z_{L,\lfloor\rho L^3\rfloor}^\phi(\beta)$ is finite for every $\beta>0$ and every $\rho$ in an open interval $I$ centred at $5p/3$. At a common $\beta_c\in[7/8,9/8]$, \[ f'_-(\beta_c,\rho)>f'_+(\beta_c,\rho)\quad(\rho\in I), \] with both derivatives finite.

Outline of the argument

Four-particle cluster bounds give stability and canonical thermodynamic limits. Successive Kac-type attractions preserve a triple of supporting slopes; perturbation estimates keep the slope separation while the range grows, yielding one potential and one common singular temperature.

Packing bounds control cell occupations and ensure stability of the bounded core. Long-range attraction is compared quantitatively with an ideal density transform, and pressure supports are preserved through the range iteration; the chosen scales produce the explicit algebraic tail.

Formal verification

For family 228, the documented constructions include both the divergent-core singularity and the bounded continuous algebraically decaying potential, with one common inverse temperature over their respective density intervals. The listed Comparator files are ContinuumTransition.lean, RadialDensityInterval.lean, RadialTransition.lean. Named declarations include OAI.ContinuumTemperature.main_theorem, OAI.RadialTransition.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.

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