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Controlled continuum interval-energy thermodynamic limit

Physical scope

Superstable regular classical positional pair systems in periodic cubes at positive temperature, together with the expressly defined formal retained-hard-admissibility zero-inverse-temperature branch, with a nondegenerate density interval and strict upper-energy threshold; periodically continued local convergence requires the stated unique minimizer.

Statement

Assume AC, the ultrafilter extension principle, DC, and HB, as required by the current variational-pressure supplier. Let $E_{0} = 1 J$ , $\hat{φ} = φ / E_{0}$ , and assume $\hat{φ}$ is Borel measurable. Require the exact superstable regular hypotheses of Superstable regular pair interactions and configurational ensemble notation for this dimensionless potential: a stable part plus a nonnegative core part bounded below by a positive constant on an $ℓ^{\infty}$ -ball, with the supplier's lower and upper regular tail conditions. Use $\hat{H} = H / E_{0}$ ; then $\hat{β} \hat{H} = β H$ when $\hat{β} = E_{0} β$ .

Use the stationary finite-intensity law space $P_{s t a t}^{1}$ and the supplier's $ρ (P)$ (expected count in a dimensionless unit cube), $I (P)$ (specific relative entropy against the unit-intensity Poisson law), and $\hat{Φ} (P)$ (limiting numerical positional energy density) from Superstable regular continuum grand pressure and variational formula. Define $ν = \sup {ρ (P) : P \in P_{s t a t}^{1}, I (P) < \infty, \hat{Φ} (P) < \infty}, e_{0} (ρ) = \inf {\hat{Φ} (P) : P \in P_{s t a t}^{1}, ρ (P) = ρ, I (P) < \infty},$ with the supremum of an empty set equal to $0$ and the infimum of an empty set equal to $+ \infty$ . Here $ρ$ is particle count per dimensionless reference volume, corresponding to physical number density $ρ / ℓ_{0}^{d}$ in $m^{- d}$ . Let $D \subset [0, \infty)$ be an interval of positive length and let $\hat{e} \in R$ be a numerical energy per dimensionless reference volume. Require $\inf D < ν$ and $\hat{e} > \inf_{ρ \in D} e_{0} (ρ)$ , and set $e = E_{0} \hat{e}$ in joules per dimensionless reference volume. The interval-event supplier Interacting density-interval and upper-energy event limit supplies a multiplier pair $(\hat{β}, z)$ with $\hat{β} \geq 0$ and $z > 0$ for this same $(D, \hat{e})$ . If $\hat{β} > 0$ , take $β = \hat{β} / E_{0}$ in $J^{- 1}$ and $T = E_{0} / (k_{B} \hat{β}) > 0$ in kelvins. If $\hat{β} = 0$ , use only the formal retained-hard-admissibility branch below; it is not a finite-temperature preparation.

For integers $n \geq 0$ , set $L_{n} = 2 n + 1$ , $Λ_{n} = [- L_{n} / 2, L_{n} / 2)^{d}$ , and $V_{n} = ∣ Λ_{n} ∣ = L_{n}^{d}$ . Define $A_{n} = {ω : N_{Λ_{n}} (ω) / V_{n} \in D, H_{L_{n}, p e r} (ω) / (E_{0} V_{n}) \leq \hat{e}},$ where $H_{L_{n}, p e r}$ is the physical positional energy in joules with the image convention of Stable regularly tempered classical positional preparation. Equivalently, the upper-energy event is $H_{L_{n}, p e r} / V_{n} \leq e$ . For each $n$ with positive event probability, set $R_{n} (B) = Q_{Λ_{n}} (B \cap A_{n}) / Q_{Λ_{n}} (A_{n})$ on the finite count sigma-algebra. Equivalently, with $M_{n} = \sum_{N \geq 0} μ_{Λ_{n}, N}$ , $R_{n} (B) = M_{n} (B \cap A_{n}) / M_{n} (A_{n})$ , since $Q_{Λ_{n}} = e^{- V_{n}} M_{n}$ ; use the count-sigma restriction of these positional measures. This defines the density-interval/upper-energy conditional preparation.

For the positive- $\hat{β}$ branch, the finite-volume grand and conditional laws use the adopted positional preparation of Finite-volume equilibrium ensemble sampling at the matching $T$ and $z$ . For the $\hat{β} = 0$ branch, require a nonempty hard-core displacement set $B = {x : φ (x) = + \infty}$ and use the formal retained-admissibility continuation of Retained hard-core continuation at zero inverse temperature; do not interpret $0 \cdot (+ \infty)$ . Assume the singleton-minimizer postulate Selected grand-variational phase uniqueness holds on its stated finite-intensity stationary-law domain at this exact supplier-selected multiplier pair and branch.

For either branch, Interacting density-interval and upper-energy event limit gives eventual event positivity, the constrained entropy-rate limit, and common-minimizer accumulation points; at $\hat{β} = 0$ , Zero-inverse-temperature-parameter limit with forbidden pairs retained identifies the matching retained grand law and minimizer set. Under the stated singleton assumption, both the conditioned laws and the corresponding periodic grand laws converge locally to $P^{*}$ for every measurable local observable $f$ supported in a bounded region $A$ and satisfying $∣ f ∣ \leq c (1 + N_{A})$ . The energy-density center $e$ and widths for this model are in joules per dimensionless reference volume; the corresponding physical $d$ -volume density has units $J m^{- d}$ . This is an interval-density and upper-energy result, not a fixed-particle-number or shrinking-shell theorem.

Here local observables are evaluated on the periodic continuation of the prepared torus configuration. This is the same periodic boundary preparation, with no new physical sampling assumption; a configuration is not extended by the empty configuration outside its fundamental cube.

Facts & Assumptions

[F1]

Stable regularly tempered classical positional preparation: $H_{N}$ is positional pair energy in joules and the local tail/stability conditions are stated in dimensionless coordinates. On a periodic cube, applying the mathematical supplier's image-energy formula to $\hat{φ} = φ / E_{0}$ divides every term of $H_{L, p e r}$ by $E_{0}$ ; thus the numerical energy used by that supplier is $H_{L, p e r} / E_{0}$ .

[F2]

Retained hard-core continuation at zero inverse temperature: at $β = 0$ the free-wall weight is zero exactly when its pair energy is forbidden/infinite; in periodic boxes the retained weight also excludes every forbidden periodic-image pair, exactly as in the mathematical supplier. No value is assigned to $0 \cdot (+ \infty)$ .

[A1]

Finite-volume equilibrium ensemble sampling: for $β > 0$ finite-energy configurations have weight $z^{N} e^{- β H_{N}}$ relative to the indistinguishable positional measure and infinite-energy configurations have zero weight; the zero-beta case is separately defined by [F2]. In the mathematical comparison, set $e^{- \hat{β} \hat{H}} = 0$ at $\hat{H} = + \infty$ ; for finite energies $\hat{β} \hat{H} = β H$ .

[A2]

Selected grand-variational phase uniqueness: for the multiplier pair chosen by the applicable mathematical result, the positive-beta functional or retained-hard-core zero-beta functional has one stationary finite-intensity minimizer $P^{*}$ ; at positive beta its physical and numerical forms agree because $\hat{β} = E_{0} β$ and $\hat{Φ} = Φ / E_{0}$ .

[L0]

Superstable regular pair interactions and configurational ensemble notation: the current definition requires AC, an even Borel potential, a decomposition $\hat{φ} = φ_{s} + φ_{r}$ with $φ_{s}$ stable, $φ_{r} \geq 0$ , and $φ_{r} (x) \geq δ$ whenever $∥ x ∥_{\infty} \leq r_{0}$ for some $δ, r_{0} > 0$ , plus its specified lower and upper regularity tails. The definition carrier is present as a draft; its proof-contract entry passes with no proof obligation, and current owner Step 1 proof-readiness is recorded; independent review remains pending.

[L1]

Superstable regular continuum grand pressure and variational formula: the current statement assumes AC, the ultrafilter extension principle, DC, and HB; for superstable regular Borel pair potentials it gives the free/periodic Poisson-reference pressure limit as the negative minimum of $I + \hat{β} Φ - ρ \log z$ over stationary finite-intensity laws, with a nonempty compact minimizer set in $τ_{L}$ . Its draft proof text is present, and the scoped batch-18 proof-contract check passes, and current owner Step 1 proof-readiness is recorded; it remains a draft used conditionally pending independent review.

[L2]

Interacting density-interval and upper-energy event limit: the current statement assumes a positive-length density interval $D$ , $\inf D < ν$ , and an upper energy threshold above $\inf_{ρ \in D} e_{0} (ρ)$ ; for its multipliers $\hat{β} \geq 0, z > 0$ it gives eventual event positivity, the constrained entropy-rate limit, and common conditional/grand minimizer accumulation points, with convergence under uniqueness for periodically continued laws; its empirical-field concentration is established separately in its proof. The draft statement/proof carrier and scoped batch-18 proof-contract entry pass; current owner Step 1 proof-readiness is recorded, and the draft result remains conditional under [L1]’s axioms pending independent review.

[L3]

Zero-inverse-temperature-parameter limit with forbidden pairs retained: the current statement defines the $β = 0$ weight as $z^{N} 1_{a d m i s s i b l e}$ and gives free/periodic Poisson-reference pressure rates as a minimum over hard-admissible stationary laws, with compact minimizers and local convergence under uniqueness. The draft statement/proof carrier and scoped batch-18 proof-contract entry pass; current owner Step 1 proof-readiness is recorded, with independent review pending. The consumer keeps the declared nonempty-core branch and never interprets $0 \cdot (+ \infty)$ .

[L4]

Periodic preparation and local-tame transfer from entropy bounds: periodic translation invariance identifies the continued law with the empirical-field barycenter, and bounded finite-volume entropy density supplies count-tail uniformity and local-tame transfer. This draft mathematical result is used conditionally under its declared choice hypotheses.

Proof

Given: The exact superstable regular Borel branch, the stated set-theoretic assumptions and dimensionless energy conversion, the strict density/energy constraints, the positive- $β$ ensemble postulate or the explicitly defined formal retained-hard-core zero- $β$ law, and the actual draft supplier statements [L1]–[L3]. The multiplier pair and uniqueness condition must match the same branch.

1.1givenF1A1L0L1L2

Translate the centered cube $Λ_{n}$ by $L_{n} / 2$ in each coordinate to $[0, L_{n})^{d}$ . This preserves particle number, Lebesgue sector measures, every periodic pair displacement and the image energy; hence the physical event and conditional law match the supplier's cube convention. If $β > 0$ , the Borel superstable regular hypotheses are exactly those of [L0], and the physical conditional preparation agrees with the periodic Poisson-reference conditional law in [L2]: the factor $e^{- V_{n}}$ between $Q_{Λ_{n}}$ and the unnormalized positional measure is constant on the conditioning event and cancels in normalization. Under the stated strict inequalities, [L2] gives eventual event positivity, its rate limit, and the common-minimizer accumulation statement.

1.2givenF2A1L2L3

If $β = 0$ , the same centered-to- $[0, L_{n})^{d}$ translation preserves the periodic energy and the positional reference measure. The periodic indicator in [F2] excludes every forbidden pair, including periodic-image pairs, from the finite upper-energy event. Thus the physical event is exactly in the retained-admissibility model rather than an undefined $0 \cdot (+ \infty)$ expression. [L2] gives its event rate and conditional accumulation points; [L3] identifies the corresponding retained grand variational minimizers.

2.1givenA2L2L3L4step 1.1step 1.2∎

In either applicable branch, when the common variational minimizer set furnished by that branch is the singleton $P^{*}$ in [A2], every local-tame accumulation point of the periodically continued laws is $P^{*}$ . Their local-tame relative compactness, including uniformly integrable local counts through [L4], then gives convergence of the conditioned and corresponding grand laws for each stated observable. The conclusion remains conditional on the draft mathematical results and the selected uniqueness premise.

Controlled continuum interval-energy thermodynamic limit

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources