TheoremStatement: Literature-sourcedProof: 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.

Specific Relative Entropy Density

Statement

Assume AC and let $P \in P_{s t a t}^{1} (Ω)$ , with this class defined in Local-Tame Topology, be stationary for the translations of Poisson Configuration Reference. For a bounded Borel set $G$ , write $P_{G}$ and $Q_{G}$ for the restrictions of $P$ and the unit-intensity Poisson reference $Q$ to configurations in $G$ . For $L > 0$ , put $Λ_{L} = [0, L)^{d}$ and $D_{L} : = D (P_{Λ_{L}} ∥ Q_{Λ_{L}}) .$ Then the specific relative entropy density $I (P) : = \lim_{L \to \infty} \frac{D_{L}}{L^{d}}$ exists in $[0, + \infty]$ and satisfies $I (P) = \sup_{m \in N, m \geq 1} \frac{D (P_{Λ_{m}} ∥ Q_{Λ_{m}})}{m^{d}} .$ If $I (P) < \infty$ , every bounded Borel restriction has finite relative entropy. This is an information functional relative to the specified unit-intensity Poisson law; changing the reference changes its value.

Facts & Assumptions

Given: AC, dimension $d \geq 1$ , a stationary finite-intensity configuration law $P$ , and the unit-intensity reference $Q$ .

[A1]

The Axiom of Choice is inherited from the standard-Borel Poisson reference and its countable product construction (The Axiom of Choice, Poisson Configuration Reference).

[F1]

For independent Poisson reference restrictions on disjoint bounded sets, the finite-region relative entropy is nonnegative, obeys the conditional chain rule, and does not increase under restriction (Poisson Reference Counts and Translation Invariance, Relative Entropy of Probability Measures, Relative Entropy Variational, Chain, and Conditioning Rules).

[F2]

For the stated dimension $d \geq 1$ , the domain $P_{s t a t}^{1} (Ω)$ consists of translation-invariant laws with finite unit-cell count, and stationarity of $P$ and $Q$ gives $D (P_{G + x} ∥ Q_{G + x}) = D (P_{G} ∥ Q_{G})$ for every bounded Borel $G$ and translation $x$ (Local-Tame Topology, Poisson Configuration Reference, Poisson Reference Counts and Translation Invariance).

[F3]

If a cube restriction has finite relative entropy, then every bounded subregion has no larger relative entropy (Relative Entropy Variational, Chain, and Conditioning Rules, claim 2).

Proof

Given: AC, $P$ , and $Q$ as in the Statement.

1.1A1F1algebra

If $G_{1}, G_{2}$ are disjoint bounded Borel sets, [F1] decomposes $D (P_{G_{1} \cup G_{2}} ∥ Q_{G_{1}} \otimes Q_{G_{2}})$ into $D (P_{G_{1}} ∥ Q_{G_{1}})$ plus the mean conditional relative entropy of $G_{2}$ given $G_{1}$ . That conditional increment is at least $D (P_{G_{2}} ∥ Q_{G_{2}})$ by the conditioning monotonicity in [F1], comparing observation by $G_{1}$ with no observation. Thus relative entropy is superadditive over disjoint regions; if the left side is infinite the inequality is immediate, and otherwise all terms in the chain rule are finite.

2.1F1F2step 1.1construct

Fix an integer $m \geq 1$ and let $q_{N} = ⌊ N / m ⌋$ . Tile $[0, q_{N} m)^{d}$ by $q_{N}^{d}$ disjoint translates of $Λ_{m}$ , leaving the Borel remainder $Λ_{N} ∖ [0, q_{N} m)^{d}$ . Iterating step 1.1 and using nonnegativity on the remainder gives $D (P_{Λ_{N}} ∥ Q_{Λ_{N}}) \geq q_{N}^{d} D (P_{Λ_{m}} ∥ Q_{Λ_{m}}),$ because stationarity gives the same entropy for each translated $m$ -cube. If $D (P_{Λ_{m}} ∥ Q_{Λ_{m}}) = + \infty$ , this forces $D (P_{Λ_{N}} ∥ Q_{Λ_{N}}) = + \infty$ for every $N \geq m$ .

3.1step 2.1algebra

Put $s = \sup_{m \geq 1} m^{- d} D (P_{Λ_{m}} ∥ Q_{Λ_{m}})$ . If $s = + \infty$ , step 2.1 makes the liminf of $N^{- d} D_{N}$ exceed every finite bound, so the integer-cube limit is $+ \infty = s$ . If $s < + \infty$ , step 2.1 gives ${lim inf}_{N \to \infty} N^{- d} D_{N} \geq m^{- d} D (P_{Λ_{m}} ∥ Q_{Λ_{m}})$ for every $m$ , since $q_{N}^{d} / N^{d} \to m^{- d}$ . Taking the supremum gives liminf at least $s$ , while the definition of $s$ gives $N^{- d} D_{N} \leq s$ for every integer $N$ . Hence the integer-cube limit exists and equals $s$ .

4.1F2F3step 3.1algebra

For real $L \to \infty$ , let $N = ⌊ L ⌋$ . The inclusions $Λ_{N} \subseteq Λ_{L} \subseteq Λ_{N + 1}$ and [F3] give $D_{N} \leq D_{L} \leq D_{N + 1}$ . Since $N / L \to 1$ and $(N + 1) / L \to 1$ , step 3.1 squeezes $L^{- d} D_{L}$ to the same extended limit $s$ ; if some integer-cube entropy is infinite, monotonicity gives the same conclusion directly.

5.1F2F3step 3.1construct∎

If $I (P) < \infty$ and $A$ is bounded, choose an integer cube $Λ_{N} + x$ containing $A$ . Stationarity [F2] and restriction monotonicity [F3] give $D (P_{A} ∥ Q_{A}) \leq D (P_{Λ_{N} + x} ∥ Q_{Λ_{N} + x}) = D (P_{Λ_{N}} ∥ Q_{Λ_{N}}) \leq N^{d} I (P) < \infty .$ Thus every bounded restriction has finite relative entropy.

Specific Relative Entropy Density

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources