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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Qualified van Hove and tempered-exterior thermodynamic limits

Physical scope

Classical positional Borel pair systems in dimension at least two; van Hove free-wall limits require superstability and regularity, while deterministic exterior limits additionally require a genuine positive divergent core in the supplier's infinity norm and a fixed uniformly tempered exterior class.

Statement

Assume AC, as required by the current mathematical definition and its shape suppliers. Let $d \geq 2$ , $E_{0} = 1 J$ , $\hat{φ} = φ / E_{0}$ , and assume $\hat{φ}$ is Borel measurable and superstable and regular as in Superstable regular pair interactions and configurational ensemble notation. Use $\hat{H} = H / E_{0}$ and $\hat{β} = E_{0} (k_{B} T)^{- 1} > 0$ in the mathematical rates. Let $(Λ_{k})$ be a van Hove sequence as in Van Hove regions and uniformly tempered exterior preparations. First, for free walls, the positional grand pressure along $Λ_{k}$ converges to the free-cube pressure of Stable-tempered free-cube grand pressure. For any positive interior attainable density $ρ$ (particle count per dimensionless reference volume, with physical density $ρ / ℓ_{0}^{d}$ in $m^{- d}$ ) and exact integers $N_{k}$ with $N_{k} / ∣ Λ_{k} ∣ \to ρ$ , the exact- $N_{k}$ free-wall canonical log-partition rate converges to the same cube canonical rate. These two shape statements require superstability and regularity; stability alone is not asserted to suffice for arbitrary van Hove shapes.

Second, strengthen the potential hypothesis: for some $r_{0} > 0$ , $\hat{φ} (x) \geq \hat{χ} (∥ x ∥_{\infty}) > 0$ whenever $0 < ∥ x ∥_{\infty} \leq r_{0}$ , where $\hat{χ} : (0, r_{0}] \to (0, \infty)$ is finite-valued and nonincreasing and $\int_{0}^{r_{0}} \hat{χ} (r) r^{d - 1} d r = + \infty$ . For each fixed $0 \leq t < \infty$ , the free-wall grand pressure in $Λ_{n}$ with exterior $ζ \in Ω (t)$ then converges uniformly in $ζ$ to the same free-cube pressure. The exterior interaction is the deterministic positional term of Van Hove regions and uniformly tempered exterior preparations, not a volume-filling external field. This genuine divergent-core condition is the exact stronger hypothesis of the current exterior supplier; it is not inferred from superstability or replaced by a weaker source condition.

Facts & Assumptions

[F1]

Stable regularly tempered classical positional preparation: $φ$ is even measurable, stable, and has an integrable decreasing tail. With $E_{0} = 1 J$ , $\hat{H} = H / E_{0}$ and $\hat{β} = E_{0} (k_{B} T)^{- 1}$ leave each Boltzmann factor unchanged.

[F2]

Van Hove regions and uniformly tempered exterior preparations: the van Hove condition uses the same two-sided Euclidean boundary-layer condition as the mathematical shape suppliers, and $Ω (t)$ uses the same centered unit cells and $t \geq 0$ square-count bound as the exterior mathematical supplier.

[A1]

Finite-volume equilibrium ensemble sampling: all grand and canonical laws are positional and use the adopted indistinguishable measure; for $β > 0$ , infinite-energy configurations receive zero weight. The mathematical comparison uses the same extended-real extension, $e^{- \hat{β} \hat{H}} = 0$ at $\hat{H} = + \infty$ .

[L0]

Superstable regular pair interactions and configurational ensemble notation: the current definition assumes AC, requires an even Borel potential, a stable-plus-nonnegative superstable splitting with a positive lower bound on an $ℓ^{\infty}$ -ball, and its lower/upper regularity tails; it separately defines the genuine divergent positive core. The definition carrier is present as a draft; its proof-contract entry passes, with current owner Step 1 proof-readiness recorded and independent review pending.

[L1]

Free-wall pressure on arbitrary van Hove shapes under superstability: the current statement gives free-wall grand-pressure convergence to the free-cube rate on every van Hove sequence under superstability and regularity; its text expressly does not assert stability alone. The item and draft proof text are present and the scoped batch-18 proof-contract check passes, but current owner Step 1 proof-readiness is recorded and independent review remains pending; this use remains conditional.

[L2]

Exact canonical limit on arbitrary van Hove shapes: the current statement gives the free-wall exact- $N_{k}$ positional log-partition rate on van Hove regions for $N_{k} / ∣ Λ_{k} ∣ \to ρ$ at a positive interior attainable density, under superstability and regularity. Its current draft statement/proof carrier and scoped batch-18 proof-contract entry pass; current owner Step 1 proof-readiness is recorded and independent review remains pending; this use remains conditional.

[L3]

Uniform exterior-pressure limit for tempered configurations: the current statement gives uniform free-wall grand-pressure convergence for each fixed exterior class $Ω (t)$ under stable regularity and the genuine divergent positive core $\hat{φ} (x) \geq \hat{χ} (∥ x ∥_{\infty})$ with divergent radial integral. Its item and draft proof text are present and the scoped batch-18 proof-contract check passes, but current owner Step 1 proof-readiness is recorded and independent review remains pending; this use remains conditional.

Proof

Given: The physical finite-volume law and numerical energy conversion, AC, definitions of geometry and exterior preparation in [F2], and the actual draft mathematical supplier statements [L1]–[L3]. Their unresolved proof status remains conditional input.

1.1givenF1F2A1L0L1

Under the Borel superstable regular hypotheses of [L0] and the physical pair assumptions [F1], the van Hove sequence in the statement satisfies the same two-sided Euclidean boundary-layer condition as the shape supplier in [L1]. The free-wall grand-pressure result then identifies its limit with the free-cube pressure. The physical grand normalizer and pressure are the ones fixed in [A1] and the earlier free-cube theorem.

2.1givenF1F2A1L2step 1.1

For an interior attainable $ρ$ and the stated $N_{k}$ , [L2] gives convergence of the exact- $N_{k}$ free-wall canonical log-partition rate to the same cube canonical rate. Its boundary-layer hypothesis is precisely the van Hove condition in [F2].

2.2givenF1F2A1L0L3step 1.1

Under the additional positive divergent-core condition defined in [L0], every $ζ \in Ω (t)$ meets the fixed exterior class in [F2]; [L3] gives uniform convergence of its grand pressure to the free-cube pressure. This conclusion is uniform only for the fixed $t$ class and does not cover a potential applied throughout a volume.

3.1step 1.1step 2.1step 2.2L1L2L3∎

Combining the three conditional limits proves exactly the two stated boundary regimes. The superstability and genuine-core assumptions remain distinct, so neither result is extended to stability alone or to arbitrary external fields.

Qualified van Hove and tempered-exterior thermodynamic limits

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources