LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 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.

Finite Gibbs response of an observable is $β$ times its covariance with the field-coupled observable

Statement

Let $E$ be a finite nonempty set, let $H_{0}, M, F : E \to R$ , fix $β > 0$ , and for $t \in R$ define $H_{t} (x) : = H_{0} (x) - t M (x)$ , $Z (t) : = \sum_{x \in E} e^{- β H_{t} (x)}$ , and $γ_{t} (x) : = Z (t)^{- 1} e^{- β H_{t} (x)}$ . For functions $F, M : E \to R$ , define ${Cov}_{γ_{t}} (F, M) : = \sum_{x} γ_{t} (x) F (x) M (x) - (\sum_{x} γ_{t} (x) F (x)) (\sum_{x} γ_{t} (x) M (x))$ . Then $t \mapsto \sum_{x} γ_{t} (x) F (x)$ is differentiable on $R$ and

$\frac{d}{d t} \sum_{x \in E} γ_{t} (x) F (x) = β {Cov}_{γ_{t}} (F, M) .$

Statement references

Finite Gibbs relative entropy and free-energy minimum

The exponential function is smooth and $(\exp)^{'} = \exp$

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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$

Facts & Assumptions

Given: A finite nonempty set $E$ , real-valued functions $H_{0}, M, F : E \to R$ , and $β > 0$ .

[F1]

For every finite nonempty state set and finite real energy, the Gibbs partition sum is finite and strictly positive, and its normalized weights form a probability vector (Finite Gibbs relative entropy and free-energy minimum).

[F2]

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

[F3]

Finite sums, products, and quotients with nonzero denominator obey the ordinary derivative 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 $g (c) \neq 0$ ).

[F4]

A differentiable real function composed with a differentiable real function obeys the chain rule (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

1.1givenF1algebra

For every $t \in R$ , each $e^{- β H_{t} (x)}$ is positive and finite, and $E$ is finite nonempty, so $0 < Z (t) < \infty$ by [F1]. Thus $γ_{t}$ is a probability vector, and both $Z (t)$ and $N_{F} (t) : = \sum_{x} F (x) e^{- β H_{t} (x)}$ are finite sums with nonzero denominator $Z (t)$ .

2.1F2F3F4step 1.1

Since $H_{t} (x) = H_{0} (x) - t M (x)$ , [F2] and [F4] give $\frac{d}{d t} e^{- β H_{t} (x)} = β M (x) e^{- β H_{t} (x)}$ . Differentiating the finite sums term by term with [F3] therefore gives $Z^{'} (t) = β \sum_{x} M (x) e^{- β H_{t} (x)}$ and $N_{F}^{'} (t) = β \sum_{x} F (x) M (x) e^{- β H_{t} (x)}$ .

3.1F3step 1.1step 2.1∎

The denominator $Z (t)$ is nonzero by step 1.1. The quotient rule [F3] applied to $N_{F} (t) / Z (t) = \sum_{x} γ_{t} (x) F (x)$ yields $β {γ_{t} (F M) - γ_{t} (F) γ_{t} (M)}$ after dividing numerator and denominator by $Z (t)$ ; by the definition of covariance in the Statement this is $β {Cov}_{γ_{t}} (F, M)$ .

Finite Gibbs response of an observable is $β$ times its covariance with the field-coupled observable

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Finite Gibbs response of an observable is ββ times its covariance with the field-coupled observable

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Finite Gibbs response of an observable is $β$ times its covariance with the field-coupled observable