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.

Grand-response derivatives are covariances

Statement

Let $(X, Σ, μ)$ be a measure space, let $w : X \to [0, \infty)$ and $X_{a} : X \to R$ ( $1 \leq a \leq r$ ) be measurable, and let $U \subseteq R^{r}$ be open. For $θ = (θ_{1}, \dots, θ_{r})$ , define $θ \cdot X (x) : = \sum_{a = 1}^{r} θ_{a} X_{a} (x)$ . Set $Z (θ) : = \int w \exp (θ \cdot X) d μ$ and assume $0 < Z (θ) < \infty$ on $U$ . Suppose for every compact $K \subset U$ there is an integrable $G_{K} \geq 0$ such that $w e^{θ \cdot X} (1 + \sum_{a} ∣ X_{a} ∣^{2}) \leq G_{K}$ pointwise for all $θ \in K$ . For the normalized density $p_{θ} = w e^{θ \cdot X} / Z (θ)$ , define $E_{θ} Y : = \int Y p_{θ} d μ$ when integrable and ${Cov}_{θ} (X_{a}, X_{b}) : = E_{θ} [(X_{a} - E_{θ} X_{a}) (X_{b} - E_{θ} X_{b})]$ . Then $\partial_{a} \log Z = E_{θ} X_{a}$ and $\partial_{b} E_{θ} X_{a} = {Cov}_{θ} (X_{a}, X_{b})$ . Thus the Hessian of $\log Z$ is positive semidefinite. This is a mathematical model identity; it assumes no independence or absence of correlations.

Facts & Assumptions

Given: As stated in the item.

[A1]

$(X, Σ, μ)$ is a measure space; $w$ is finite-valued nonnegative measurable and each $X_{a}$ is real-valued measurable; $U$ is open, $Z$ is positive finite, and the displayed compact-parameter envelope is integrable.

[F1]

Measure space and integral: measurable integrals and densities use the fixed reference measure $μ$ .

[F2]

Dominated convergence for the measure integral: a measurable pointwise-limit representative of measurable functions dominated by an integrable envelope is integrable, and the limit passes through the integral. On a noncomplete measure space the named representative must be measurable; the canonical measurable representative always exists. The limits used below are measurable by [F7].

[F3]

The real exponential function and the number $e$ by a power series and The exponential function is smooth and $(\exp)^{'} = \exp$ : exponential weights are differentiable, with derivative equal to the exponential.

[F4]

The natural logarithm as the inverse of the exponential function and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log Z$ differentiates as $Z^{'} / Z$ when $Z > 0$ .

[F5]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : scalar difference quotients of the exponential are bounded by intermediate derivatives.

[F6]

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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ and 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 parameterized exponential and quotient have the usual derivative rules.

[F7]

Arithmetic and lattice operations preserve measurability whenever they are defined and Composition with a Borel measurable outer map preserves measurability: finite sums/products and composition with continuous scalar functions are measurable.

Proof

Given: The parameterized measure model and its compact-parameter integrable envelopes.

1.1A1F1F2F3F5F6F7

For each fixed parameter, measurable energy/observable coordinates and [F7] make every finite linear combination, exponential weight, weighted first/second moment and parameter difference quotient measurable; their displayed pointwise derivative limits are the same measurable compositions. For a coordinate direction $a$ , the difference quotient of $w e^{θ \cdot X}$ tends pointwise to $w X_{a} e^{θ \cdot X}$ . On a compact neighborhood $K^{'}$ of $θ$ , the mean-value theorem [F5] bounds its absolute value by $w ∣ X_{a} ∣ e^{η \cdot X}$ for an intermediate parameter $η \in K^{'}$ , at most $G_{K^{'}}$ . Dominated convergence [F2] therefore gives $\partial_{a} Z = \int w X_{a} e^{θ \cdot X} d μ$ . Applying the same argument to the first-moment numerator gives $\partial_{b} \int w X_{a} e^{θ \cdot X} d μ = \int w X_{a} X_{b} e^{θ \cdot X} d μ$ ; the same envelope controls this quotient because $∣ X_{a} X_{b} ∣ \leq 1 + \sum_{c} ∣ X_{c} ∣^{2}$ .

2.1A1F1F4step 1.1

Divide the first derivative by $Z > 0$ and use [F4] to obtain $\partial_{a} \log Z = E_{θ} X_{a}$ . The quotient rule for the normalized first moment gives $\partial_{b} E_{θ} X_{a} = E_{θ} (X_{a} X_{b}) - E_{θ} X_{a} E_{θ} X_{b}$ , the covariance.

3.1A1step 2.1∎

For any $v \in R^{r}$ , $v^{T} (\nabla^{2} \log Z) v = E_{θ} [(v \cdot X - E_{θ} (v \cdot X))^{2}] \geq 0$ . This proves positive semidefiniteness without any independence assumption.

Grand-response derivatives are covariances

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources