LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

Finite Gibbs response and covariance identities

Statement

Let $S$ be finite nonempty, let $H_{0}, M : S \to R$ , fix $β > 0$ , and set $H_{t} = H_{0} - t M$ and $Z (t) = \sum_{s \in S} e^{- β H_{t} (s)}$ for $t \in R$ . For the normalized finite Gibbs law $γ_{t} (s) = Z (t)^{- 1} e^{- β H_{t} (s)}$ , $\frac{d}{d t} \log Z (t) = β γ_{t} (M), \frac{d^{2}}{d t^{2}} \log Z (t) = β^{2} {Var}_{γ_{t}} (M) \geq 0,$ where ${Var}_{γ} (M) : = γ (M^{2}) - γ (M)^{2}$ . For fixed $H : S \to R$ , the corresponding inverse-temperature identities are $\frac{d}{d β} \log Z (β) = - γ_{β} (H), \frac{d^{2}}{d β^{2}} \log Z (β) = {Var}_{γ_{β}} (H) \geq 0.$ These are finite-state identities; they do not justify differentiating an infinite-volume limit.

Facts & Assumptions

Given: A finite nonempty state set, real-valued observables and energies, and a positive finite parameter $β$ .

[A1]

For finite energies and $β > 0$ , the finite Gibbs weights normalize to a strictly positive probability law (Finite Gibbs relative entropy and free-energy minimum).

[F1]

The real exponential is differentiable with derivative $\exp$ (The exponential function is smooth and $(\exp)^{'} = \exp$ ).

[F2]

The natural logarithm is differentiable on $(0, \infty)$ with derivative $1 / x$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t).

[F3]

Finite sums, products, quotients with nonzero denominator, and compositions of differentiable real functions obey the sum, product, quotient, and chain rules (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

Proof

1.1givenA1F1F2F3

Since $S$ is finite, repeated use of the sum rule and chain rule [F1, F3] differentiates each summand and gives $Z^{'} (t) = β \sum_{s} M (s) e^{- β H_{t} (s)}$ . Positivity of $Z (t)$ follows from [A1]; the logarithm chain rule and [F2, F3] give $(\log Z)^{'} = Z^{'} / Z = β γ_{t} (M)$ .

2.1A1F1F3step 1.1

Differentiate the finite-sum numerator $\sum_{s} M (s) e^{- β H_{t} (s)}$ and denominator $Z (t)$ by [F1, F3], then apply the quotient rule from [F3]. This gives $\frac{d}{d t} γ_{t} (M) = β (γ_{t} (M^{2}) - γ_{t} (M)^{2})$ . Multiplying by the factor $β$ in step 1.1 gives the stated second derivative, which is nonnegative because it equals $β^{2} γ_{t} ((M - γ_{t} (M))^{2})$ .

3.1givenA1F1F2F3∎

For the inverse-temperature family, finite-sum differentiation using [F1, F3] gives $Z^{'} (β) = - \sum_{s} H (s) e^{- β H (s)}$ and $Z^{''} (β) = \sum_{s} H (s)^{2} e^{- β H (s)}$ . Applying the logarithm chain rule and quotient rule from [F2, F3] yields the stated mean and variance identities.

Finite Gibbs response and covariance identities

Statement

Facts & Assumptions

Proof

Depends on

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