LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generated

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.

A CAID's information-density variance equals its conditional dispersion

Statement

For every CAID $P \in Π (W)$ , let $Q^{*} = P W$ and let $(X, Y) \sim P (x) W (y ∣ x)$ . For every $x$ with $P (x) > 0$ , $E [\ln (W (Y ∣ x) / Q^{*} (Y)) ∣ X = x] = C (W),$ and $Var [\ln (W (Y ∣ X) / Q^{*} (Y))] = V (P, W) .$

Facts & Assumptions

Given: A finite DMC and a capacity-achieving input law $P$ .

[F1]

Finite DMC information capacity is attained: $P$ is a maximizer of $I (P, W)$ and $I (P, W) = C (W)$ .

[F2]

The capacity-achieving output law is unique: every CAID induces the same output law $Q^{*}$ .

[F3]

Capacity-achieving output laws bound every row divergence: each row is supported on $Q^{*}$ and has divergence $d_{x} : = D (W (\cdot ∣ x) ∥ Q^{*}) \leq C (W)$ .

[F4]

Finite mutual information: $I (P, W)$ is the KL divergence of the joint law $P (x) W (y ∣ x)$ from its product marginals.

[F5]

Conditional information variance and DMC dispersion: $V (P, W) = \sum_{x} P (x) {Var}_{W_{x}} \ln [W (Y ∣ x) / (P W) (Y)]$ .

Proof

1.1F1F2F3F4givenalgebra

Since $Q^{*} = P W$ , [F4] expands to $I (P, W) = \sum_{x} P (x) D (W (\cdot ∣ x) ∥ Q^{*}) = \sum_{x} P (x) d_{x}$ . By [F1], this average is $C (W)$ , while [F3] gives $d_{x} \leq C (W)$ for every $x$ . Therefore each $x$ with $P (x) > 0$ has $d_{x} = C (W)$ ; otherwise the weighted average would be strictly smaller.

2.1F3F5step 1.1givenalgebra∎

Under the joint law, the conditional mean of $Z = \ln [W (Y ∣ X) / Q^{*} (Y)]$ given $X = x$ is $d_{x}$ . Step 1.1 makes this mean the constant $C (W)$ on every positive-mass input. Hence $E Z = C (W)$ and $Var (Z) = E [(Z - C (W))^{2}] = \sum_{x} P (x) {Var}_{W_{x}} (Z) = V (P, W),$ where the last equality is [F5]. All terms are finite because the alphabets and charged supports are finite.

Depends on

Used by

Nondegenerate normal approximation for finite unconstrained DMCs Theorem

Dependency tree · two levels

24 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

Yury Polyanskiy, H. Vincent Poor, and Sergio Verdú, Channel Coding Rate in the Finite Blocklength Regime (standard reference, not scraped)
Masahito Hayashi, Information Spectrum Approach to Second-Order Coding Rate in Channel Coding (standard reference, not scraped)