CounterexampleConstruction: 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.

Finite KL divergence is directional and may be infinite

Statement refuted

For every pair of finite probability vectors $p, q$ on the same alphabet, $D (p ∥ q) = D (q ∥ p)$ .

Facts & Assumptions

Given: The finite KL convention and $0 < ε < 1$ .

[F1]

Finite distributions, entropy, divergence, and stochastic matrices: $D (p ∥ q) = \sum_{i : p_{i} > 0} p_{i} \ln (p_{i} / q_{i})$ , with $+ \infty$ if $p_{i} > 0 = q_{i}$ for some coordinate.

[F2]

Finite KL divergence is nonnegative with equality exactly at equal laws: finite KL is nonnegative and is zero exactly for equal probability vectors.

Counterexample

Given: Take the alphabet ${0, 1}$ , $p = δ_{0}$ , $q = (ε, 1 - ε)$ , and $0 < ε < 1$ .

1.1givenF1F2algebra

The support of $p$ is contained in that of $q$ because $q_{0} = ε > 0$ . By [F1], $D (p ∥ q) = 1 \cdot \ln (1 / ε) = - \ln ε$ , a finite real number. Since $p \neq q$ , [F2] makes this value strictly positive, consistent with the formula.

2.1givenF1step 1.1cases

The second coordinate has $q_{1} = 1 - ε > 0 = p_{1}$ . Thus [F1] gives $D (q ∥ p) = + \infty$ . The two directions are unequal, which refutes the statement.

3.1F1construct∎

For the separate disjoint point masses $p = δ_{0}$ and $q = δ_{1}$ , coordinate $0$ has $p_{0} > 0 = q_{0}$ and coordinate $1$ has $q_{1} > 0 = p_{1}$ ; applying [F1] in each direction gives $D (p ∥ q) = D (q ∥ p) = + \infty$ . This records the disjoint-support boundary.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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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)