ExampleConstruction: AI-generatedVerification: 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.

Equal-variance Gaussian location KL

Example

Let $P = N (a, τ^{2})$ and $Q = N (b, τ^{2})$ be real Gaussian laws with $τ > 0$ . Their directional divergence is $D (P ∥ Q) = \frac{(a - b)^{2}}{2 τ^{2}} \geq 0.$ Equality holds when $a = b$ . This calculation concerns equality of the two laws, not identifiability of arbitrary parameter labels.

Facts & Assumptions

Given: AC, real $a, b$ , and $τ > 0$ .

[A1]

The Axiom of Choice: the Gaussian KL lemma and KL nonnegativity theorem assume AC; the lemma derives its countable-choice use from AC.

[F1]

KL divergence between equal-variance Gaussian location laws: for equal-variance Gaussian location laws, $D (N (a, τ^{2}) ∥ N (b, τ^{2})) = (a - b)^{2} / (2 τ^{2})$ .

[F2]

Kullback–Leibler divergence is nonnegative: for probability laws with the published KL direction, divergence is nonnegative and equality means the laws are equal.

Verification

1.1A1F1given

Apply [F1] to the stated laws; its hypotheses are exactly $a, b \in R$ and $τ > 0$ , and its AC hypothesis is [A1].

2.1A1F1F2givenalgebra∎

Since $(a - b)^{2} \geq 0$ and $2 τ^{2} > 0$ , the displayed value is nonnegative and equals zero when $a = b$ . In that case the two Gaussian laws coincide, so the equality clause is consistent with [F2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Example 4.7 (standard reference, not scraped)