ExampleConstruction: AI-generatedVerification: 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.

Exact evaluation of a one-use BSC dependence-testing bound

Example

For a one-use BSC with crossover $p = 0.1$ , uniform input, and $M = 2$ , the dependence-testing upper bound on achievable average error is exactly $0.35$ .

Facts & Assumptions

Given: The BSC transition probabilities $W (y ∣ x) = 0.9$ for $y = x$ and $0.1$ for $y \neq x$ , uniform input, and $M = 2$ .

[F1]

Finite-DMC dependence-testing achievability bound: a deterministic code exists whose average error is at most $E \exp (- [ı (X; Y) - \ln ((M - 1) / 2)]_{+})$ .

[F2]

Finite discrete memoryless channels: a BSC row is a probability vector, and its output law under an input vector is the matrix product $P W$ .

Verification

1.1F2givenalgebra

Under the uniform input, the output is uniform. Therefore $ı (X; Y) = \ln (0.9 / 0.5) = \ln 1.8$ with probability $0.9$ and $ı (X; Y) = \ln (0.1 / 0.5) = \ln 0.2$ with probability $0.1$ .

2.1F1step 1.1casesalgebra∎

The threshold is $\ln ((M - 1) / 2) = \ln (1 / 2)$ . On the correct-output event, the integrand is $\exp [- (\ln 1.8 - \ln (1 / 2))] = 1 / 3.6 = 5 / 18$ , contributing $0.9 (5 / 18) = 1 / 4$ . On the crossover event, $\ln 0.2 < \ln (1 / 2)$ , so the positive part is zero and the integrand is one, contributing $0.1$ . Thus the expectation is $1 / 4 + 1 / 10 = 7 / 20 = 0.35$ , and [F1] supplies a code with average error at most this value.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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)