ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-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.

Binary entropy and its extremizers

Example

For a Bernoulli law $q = (p, 1 - p)$ , the entropy in nats is $h (p) : = H (q) = - p \ln p - (1 - p) \ln (1 - p), 0 \ln 0 : = 0.$ It is zero exactly at $p = 0, 1$ and has unique maximum $\ln 2$ at $p = 1 / 2$ .

Facts & Assumptions

Given: $p \in [0, 1]$ and the Bernoulli probability vector $q = (p, 1 - p)$ .

[F1]

Finite distributions, entropy, divergence, and stochastic matrices: $H (q) = - \sum_{q_{i} > 0} q_{i} \ln q_{i}$ , with zero terms set to $0$ .

[F2]

Finite KL divergence is nonnegative with equality exactly at equal laws: for finite probability vectors, $D (r ∥ s) \geq 0$ , with equality iff $r = s$ .

[F3]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\ln$ is strictly increasing and $\ln 1 = 0$ .

Verification

1.1givenF1F3algebra

Substituting $q = (p, 1 - p)$ into [F1] gives the displayed formula. If $p = 0$ or $p = 1$ , the nonzero coordinate is $1$ and [F3] gives $\ln 1 = 0$ , so $h (p) = 0$ .

1.2F1F2givenalgebra

Let $u = (1 / 2, 1 / 2)$ . The finite KL formula gives $D (q ∥ u) = p \ln (2 p) + (1 - p) \ln (2 (1 - p)) = \ln 2 - h (p)$ , also at $p = 0, 1$ because zero terms are omitted. By [F2], this value is nonnegative and vanishes exactly when $q = u$ , i.e. $p = 1 / 2$ . Thus $h (p) \leq \ln 2$ with a unique maximizer.

2.1F3step 1.1givenalgebra∎

If $0 < p < 1$ , then $\ln p < 0$ and $\ln (1 - p) < 0$ by [F3], so both terms in $h (p)$ are positive. Along with step 1.1 this proves that the only zeros are the endpoints.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Robert M. Gray, Entropy and Information Theory, corrected first edition (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, author prepublication (standard reference, not scraped)