LemmaStatement: 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.

Periodic preparation and local-tame transfer from entropy bounds

Statement

Assume AC, the ultrafilter extension principle, DC, and HB as in Compactness of Finite-Specific-Entropy Levels in the Local-Tame Topology. Let $L \to \infty$ and let $C_{L}$ be probability laws on finite configurations in $Λ_{L} = [0, L)^{d}$ , invariant under every translation modulo $L Z^{d}$ . Write $Q_{L}$ for unit-intensity Poisson measure and suppose $c_{L} = L^{- d} D (C_{L} ∥ Q_{L}), \sup_{L} c_{L} < \infty .$ Let ${\tilde{C}}_{L}$ be the law of the periodic continuation. Then ${\tilde{C}}_{L} = E_{C_{L}} R_{L, ω}$ ; these laws are stationary and are relatively sequentially compact in $τ_{L}$ as $L \to \infty$ . Their local counts are uniformly integrable. Along any convergent subsequence with limit $P$ , $I (P) \leq {lim inf}_{L} c_{L} .$ If the empirical fields $R_{L, ω}$ converge in probability in $τ_{L}$ to a fixed $P_{*}$ , then ${\tilde{C}}_{L} \to P_{*}$ in $τ_{L}$ , including expectations of every measurable local $f$ with $∣ f ∣ \leq b (1 + N_{A})$ for bounded $A$ . This concerns periodic continuation, not the zero extension of a configuration outside its fundamental cube.

Facts & Assumptions

Given: The stated choice hypotheses, torus-invariant laws, and bounded entropy densities.

[A1]

The Axiom of Choice, The ultrafilter extension principle (UL/BPI), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, and The real dominated-extension principle as an additional hypothesis over ZF: the explicit hypotheses for the weak-density compactness, marginal extension and subsequence extraction supplied by [F4].

[F1]

Periodized empirical field of a finite configuration and Stationarity and intensity of a periodized empirical field: the empirical field averages translations of the periodic continuation; its barycenter is stationary with intensity $E_{C_{L}} N / L^{d}$ when this expectation is finite.

[F2]

Relative Entropy Variational, Chain, and Conditioning Rules: relative entropy decreases under restriction, is superadditive over disjoint regions against a product reference, and satisfies the bounded-test variational inequality.

[F3]

Poisson Reference Counts and Translation Invariance: Poisson counts have all positive exponential moments.

[F4]

Dunford--Pettis for real $L^{1}$ on a finite measure space and Eberlein–Šmulian theorem: under the named choice hypotheses, uniformly integrable real $L^{1}$ densities on a probability space have weakly compact closures and weakly convergent subsequences. Weak topology on a normed space and Holder's inequality for integrals, including the endpoint cases make bounded measurable tests weakly continuous. The cell spaces in Poisson Configuration Reference are standard Borel; Assuming the Axiom of Choice, Kolmogorov extension for arbitrary families of standard Borel coordinate spaces extends their consistent finite-dimensional laws under AC. The local extraction required here is proved in steps 3.2 and 4.1, rather than attributed to compactness of specific-entropy levels.

[F5]

Specific Relative Entropy Density: for a stationary law, $I (P) = \sup_{a \in N, a \geq 1} a^{- d} D (P_{[0, a)^{d}} ∥ Q_{[0, a)^{d}})$ .

[F6]

Monotone convergence for the integral and Dominated convergence: bounded count truncations may increase to counts, and Poisson exponential domination justifies the tail limits below.

Proof

Given: The hypotheses and suppliers in Facts & Assumptions.

1.1givenF1F2F3F6algebra

For bounded local $f$ , torus invariance gives $E_{C_{L}} f (T_{x} ω^{(L)}) = E_{C_{L}} f (ω^{(L)})$ for every $x$ . Integrating over $Λ_{L}$ proves ${\tilde{C}}_{L} (f) = E_{C_{L}} R_{L, ω} (f)$ . In particular the continued law is stationary. The entropy inequality with bounded truncations of $a N_{Λ_{L}}$ and [F3] gives finite $E_{C_{L}} N$ , so the intensity assertion of [F1] applies.

2.1F2step 1.1algebra

Fix an integer $a \geq 1$ and take $L > a$ . The fundamental cube contains $m_{L} = ⌊ L / a ⌋^{d}$ disjoint translates of $W = [0, a)^{d}$ . Their Poisson restrictions are independent and their $C_{L}$ marginals are identical by torus invariance. Superadditivity gives $D (({\tilde{C}}_{L})_{W} ∥ Q_{W}) \leq D (C_{L} ∥ Q_{L}) / m_{L} = c_{L} L^{d} / m_{L} .$ For fixed $a$ , $L^{d} / m_{L} \to a^{d}$ . Translation invariance gives the same bound for every translate of $W$ , hence a uniform entropy bound on every bounded region.

3.1F2F3F6step 2.1algebra

On a bounded $A \subset W$ , apply the entropy inequality to $t \min (N_{A} 1_{{N_{A} > K}}, M)$ and let $M \to \infty$ . If $C_{W}$ is the uniform bound from step 2.1, then $t E_{{\tilde{C}}_{L}} [N_{A} 1_{{N_{A} > K}}] \leq C_{W} + \log E_{Q_{W}} e^{t N_{A} 1_{{N_{A} > K}}} .$ For fixed $t > 0$ , the logarithm tends to zero as $K \to \infty$ by [F3] and [F6]. First take $t$ large and then $K$ large. This proves uniform count-tail integrability.

3.2A1F2F4step 2.1algebra

Consider any sequence of periods tending to infinity and enumerate the integer cubes $W = a + [0, m)^{d}$ . For each fixed $W$ , step 2.1 gives a finite entropy bound $C_{W}$ on an eventual tail; it does not assert a bound for all periods on arbitrarily large cubes. For a probability density $h = d P_{W} / d Q_{W}$ with entropy at most $C_{W}$ , the inequalities $- u \log u \leq 1 / e$ for $0 \leq u \leq 1$ and $h \log h \geq h \log K$ on ${h > K}$ give $\int_{h > K} h d Q_{W} \leq (C_{W} + 1 / e) / \log K$ . These densities have norm one and are uniformly integrable. By [F4], discard the finite exceptional prefix for the first cube and extract a weakly convergent subsequence, then do so for the second cube and continue; diagonal extraction, using AC, gives weak convergence on every cube. Each limiting density is nonnegative with integral one, by tests against indicators and $1$ . The entropy variational formula [F2] is a supremum of weakly continuous affine functions, so it gives entropy at most $C_{W}$ for each limit.

4.1A1F2F4F5step 2.1step 3.1step 3.2construct

Consistency of restrictions and stationarity pass to these limits by testing bounded measurable local functions: every equality involves only finitely many cube coordinates. A finite set of unit cells lies in an integer cube; its induced marginal is independent of the containing cube by consistency through a larger cube. The standard-Borel extension [F4] gives a law on the countable cell product, hence on simple locally finite configurations. It is stationary. The bound of step 3.1 holds for its cube marginals as well, so its counts are integrable and their tails are controlled. Truncating $f$ on ${N_{A} \leq K}$ therefore upgrades bounded-local convergence of the extracted sequence to convergence for every tame $f$ , by step 3.1. This proves relative sequential compactness as the periods tend to infinity. Along any such convergent sequence, the same variational formula gives $D (P_{W} ∥ Q_{W}) \leq lim inf D (({\tilde{C}}_{L})_{W} ∥ Q_{W})$ . Taking a further subsequence realizing $lim inf c_{L}$ , step 2.1 gives $a^{- d} D (P_{[0, a)^{d}} ∥ Q_{[0, a)^{d}}) \leq lim inf c_{L}$ for every integer $a \geq 1$ ; [F5] yields $I (P) \leq lim inf c_{L}$ .

5.1step 1.1step 3.1step 4.1algebra∎

If the empirical fields converge in probability to $P_{*}$ , then for bounded local $f$ the bounded random variables $R_{L, ω} (f)$ converge in probability to $P_{*} (f)$ , hence their expectations converge. By step 1.1 this is convergence of ${\tilde{C}}_{L} (f)$ . For tame $f$ , truncate on ${N_{A} \leq K}$ ; step 3.1 makes the truncation error uniformly small, and the same is true for the limit law. Taking $L \to \infty$ and then $K \to \infty$ proves convergence for every stated tame observable.

Depends on

Dependency tree · 2 levels

Sources

H.-O. Georgii, Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction