TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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.

DLR states attain the bounded lattice pressure variational principle

Statement

Assume Countable Choice, a bounded translation-invariant finite-range lattice interaction on a finite alphabet, and $β > 0$ . Let $p$ be its dimensionless pressure per site, and let $μ$ be a translation-stationary probability with entropy density $h (μ)$ and energy density $u (μ)$ as in Stationary lattice entropy and energy densities. Then $h (μ) - β u (μ) \leq p .$ If $μ$ is a DLR state for the specification, equality holds. Since stationary DLR states exist under these hypotheses, the supremum of $h (μ) - β u (μ)$ over stationary probabilities is attained and equals $p$ . The converse “every maximizing stationary probability is DLR” is not asserted.

Facts & Assumptions

Given: A bounded finite-range translation-invariant interaction, $β > 0$ , a stationary lattice probability, and Countable Choice for the DLR existence input.

[A1]

Countable Choice is retained in the finite-alphabet DLR measure construction (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Stationary finite-alphabet laws have entropy and energy densities with the stated limits (Stationary lattice entropy and energy densities).

[F2]

The free-boundary pressure along cubes and van Hove sets has a finite limit $p$ (Bounded lattice pressure on van Hove sets).

[F3]

For each finite state set, the Gibbs relative-entropy identity is $D (ρ ∥ γ) = - H_{S h} (ρ) + β ρ (H) + \log Z \geq 0$ (Finite Gibbs relative entropy and free-energy minimum).

[F4]

For the stated finite-range interaction and $β > 0$ , its candidate Gibbs kernels are proper, consistent and cylinder-local (Conditional Gibbs kernel consistency). Under Countable Choice, such a specification has a nonempty DLR state set (Finite-range lattice DLR existence and closed convex state set). For a translation-invariant interaction with a nonempty DLR set, translation averaging then gives a stationary DLR state (Translation-averaged stationary Gibbs state).

Proof

1.1F1F2F3givenalgebra

Under the Countable Choice assumption [A1], apply [F3] to the $B_{n}$ marginal of $μ$ and the free Gibbs law for $H_{B_{n}}^{f r e e}$ . Rearranging gives $H_{n} - β E_{μ} H_{B_{n}}^{f r e e} \leq \log Z_{B_{n}}$ . Divide by $∣ B_{n} ∣$ and use [F1] and [F2] to obtain $h (μ) - β u (μ) \leq p$ .

1.2givenF1F3algebra

Suppose $μ$ is DLR. Its $B_{n}$ marginal is a mixture of the conditional finite-volume Gibbs laws $γ_{B_{n}} (\cdot ∣ η)$ . Let $R$ be the interaction range, $\partial_{R} B_{n} = {x \in B_{n} : {dist}_{\infty} (x, B_{n}^{c}) \leq R}$ , and $C = \sum_{r \in R} ∣ S_{r} ∣ ∥ Φ_{r} ∥_{\infty}$ . A support meeting both $B_{n}$ and its complement contains an interior site in $\partial_{R} B_{n}$ ; for each such site and type $r$ there are at most $∣ S_{r} ∣$ possible anchors. Thus the conditional energy differs from $H_{B_{n}}^{f r e e}$ by at most $D_{n} = C ∣ \partial_{R} B_{n} ∣$ . Termwise comparison shows the likelihood ratio of each conditional law to the free Gibbs law lies in $[e^{- 2 β D_{n}}, e^{2 β D_{n}}]$ ; the same holds for their mixture. Hence $0 \leq D (μ_{B_{n}} ∥ γ_{B_{n}}^{f r e e}) \leq 2 β D_{n}$ .

2.1F1F2F3step 1.2algebra

Since $D_{n} / ∣ B_{n} ∣ \to 0$ , the finite relative-entropy identity [F3], divided by volume and passed to the limits in [F1]–[F2], gives $h (μ) - β u (μ) = p$ .

3.1A1F4step 1.1step 2.1∎

The kernel consistency and existence results in [F4] first give a nonempty DLR set under [A1]; translation averaging in [F4] then gives a stationary DLR state and hence attains equality from step 2.1. Combining with the upper bound of step 1.1 proves the supremum statement, without any converse implication.

DLR states attain the bounded lattice pressure variational principle

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources