DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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.

Bernoulli relative free-energy functional

Definition

For $u \in (0, 1)$ define the Bernoulli free-energy density $f (u) : = u \log u + (1 - u) \log (1 - u)$ . For a reference density $\overset{ˉ}{ρ} \in (0, 1)$ define $h (u ∣ \overset{ˉ}{ρ}) : = f (u) - f (\overset{ˉ}{ρ}) - (\log \overset{ˉ}{ρ} - \log (1 - \overset{ˉ}{ρ})) (u - \overset{ˉ}{ρ})$ . For $d \geq 1$ , a smooth profile $ρ : T^{d} \to [ε, 1 - ε]$ with some $0 < ε < 1 / 2$ , and normalized product Haar measure $d x$ on $T^{d}$ , define $H_{\overset{ˉ}{ρ}} (ρ) : = \int_{T^{d}} h (ρ (x) ∣ \overset{ˉ}{ρ}) d x$ . If $\int_{T^{d}} ρ d x = \overset{ˉ}{ρ}$ , then $H_{\overset{ˉ}{ρ}} (ρ) = \int_{T^{d}} f (ρ (x)) d x - f (\overset{ˉ}{ρ})$ because the linear term integrates to zero. All logarithm arguments are positive on the stated domains.

Bernoulli relative free-energy functional

Definition

Definition references

Depends on

Dependency tree · 2 levels

Sources