Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Energy-density concentration at differentiable lattice pressure

Physical scope

Conditional model probabilities for bounded finite-range lattice systems at a differentiability point of the limiting log-partition density; no concentration claim is made at a nondifferentiable point or for arbitrary observables.

Statement

For the lattice model and canonical law of Bulk log-partition pressure for a bounded finite-range lattice model, let $p_{L} (γ) = L^{- d} \log Z_{L} (γ)$ converge pointwise on an open interval $J_{β} \subset (0, \infty)$ to $p (γ)$ . If $p$ is differentiable at $β \in J_{β}$ , set $e = - p^{'} (β)$ in joules per site. Then the canonical mean energy density satisfies $E_{β, L} [H_{L}] / L^{d} \to e$ , and for every $ε > 0$ there exist $c > 0$ and $L_{0}$ such that $P_{β, L} ⁣ (∣ H_{L} / L^{d} - e ∣ \geq ε) \leq 2 e^{- c L^{d}} (L \geq L_{0}) .$ The probability is the model's finite-volume canonical probability, not an observed frequency. No conclusion is asserted when $p$ is nondifferentiable.

Facts & Assumptions

Given: The bounded finite-range lattice models, their canonical probabilities, pointwise convergence of $p_{L}$ on an open positive interval, and differentiability of the limit at $β$ .

[F1]

Bulk log-partition pressure for a bounded finite-range lattice model: $p_{L} = L^{- d} \log Z_{L}$ is the reduced lattice log-partition density and its canonical law is the finite-volume model law.

[L1]

Limiting derivative and concentration (current batch 11 draft): if the finite-volume lattice pressures converge pointwise on an open positive interval and the limiting pressure is differentiable at $β$ , then the finite-volume mean energy density converges to $- p^{'} (β)$ and both tails have an exponential bound. Its full current proof was inspected; its draft status remains conditional.

[F2]

Canonical sampling assumption for finite and normalized statistical models: the finite-volume probabilities are $e^{- β H_{L}} / Z_{L}$ .

[F3]

SI quantities and unit conventions: $H_{L}$ is in joules, $H_{L} / L^{d}$ and $e$ are in joules per site, $β$ in $J^{- 1}$ , and the probability bound is dimensionless.

[A1]

$β$ lies in an open interval $J_{β} \subset (0, \infty)$ , $p_{L} \to p$ on that interval, and $p$ is differentiable at the specified $β$ .

Proof

Given: The lattice model, its finite canonical law, the pointwise limiting pressure, and differentiability at $β$ .

1.1F1F2F3L1A1

The finite-volume energy law and $p_{L}$ in [F1]–[F2] match the model and probability hypotheses of [L1]. The condition $β > 0$ and differentiability at $β$ are exactly those in [A1].

2.1F1L1step 1.1∎

Applying [L1] yields $E_{β, L} [H_{L}] / L^{d} \to - p^{'} (β) = e$ and, for each fixed $ε > 0$ , a constant $c > 0$ and threshold $L_{0}$ with the stated two-tail exponential estimate. The result is a model probability statement and is silent at pressure nondifferentiability.

Energy-density concentration at differentiable lattice pressure

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources