ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

Independent non-identical finite sources can have no information-rate limit

Example

Let $N_{0} = 0$ and $N_{k} = 2^{k^{2}}$ for $k \geq 1$ . For $N_{k - 1} < i \leq N_{k}$ , let $p_{i}$ be the fair Bernoulli law when $k$ is even, and the point mass at $0$ when $k$ is odd. For each $n$ , let $P_{n} (x^{n}) = \prod_{i = 1}^{n} p_{i} (x_{i})$ be the resulting finite product law on ${0, 1}^{n}$ , and on its support put $I_{n} (x^{n}) = - n^{- 1} \ln P_{n} (x^{n})$ . These block laws are consistent under coordinate projection. The coordinates are independent within each block law, but their marginals are not identical. The normalized information densities have no limit in probability to a constant, where this means that $P_{n} (∣ I_{n} - c ∣ > ε) \to̸ 0$ for some $ε > 0$ and every constant $c$ : at even block endpoints they tend to $\ln 2$ , and at odd endpoints they tend to $0$ . This example uses only finite product laws; no infinite-product extension is assumed.

Facts & Assumptions

Given: The endpoint sequence and finite product laws in the Example.

[F1]

Self-information and surprisal variance for a finite law: For a finite law $q$ , its self-information on the support is $- \ln q (x)$ .

[F2]

Finite-alphabet IID information AEP with a variance bound: The AEP conclusion is for an identically distributed independent sequence.

Verification

Given: The block laws $P_{n}$ in the Example.

1.1givenalgebraconstruct

Each $p_{i}$ is a probability law on ${0, 1}$ . The finite product $P_{n} (x^{n}) = \prod_{i = 1}^{n} p_{i} (x_{i})$ is nonnegative and sums to $\prod_{i = 1}^{n} \sum_{x_{i} \in {0, 1}} p_{i} (x_{i}) = 1$ , so it is a well-defined finite law. Summing over the last coordinate gives $P_{n - 1}$ , proving consistency.

2.1F1step 1.1algebra

Let $F_{n}$ be the number of fair coordinates among $1, \dots, n$ . At every positive-probability word, each fair coordinate contributes $\ln 2$ to $- \ln P_{n} (x^{n})$ and every point-mass coordinate contributes $0$ . Hence $I_{n} (x^{n}) : = - n^{- 1} \ln P_{n} (x^{n}) = F_{n} \ln 2 / n$ on the entire support, so its law under $P_{n}$ is a point mass at this value.

3.1step 2.1algebra

If $k$ is even, the newest block is fair and $N_{k} - N_{k - 1} \leq F_{N_{k}} \leq N_{k}$ ; since $N_{k - 1} / N_{k} = 2^{- 2 k + 1} \to 0$ , $F_{N_{k}} / N_{k} \to 1$ . If $k$ is odd, the newest block is deterministic and $0 \leq F_{N_{k}} \leq N_{k - 1}$ , so $F_{N_{k}} / N_{k} \to 0$ .

4.1F2step 2.1step 3.1algebra∎

By step 2.1, $I_{N_{k}}$ is deterministic under $P_{N_{k}}$ . Step 3.1 gives subsequential limits $\ln 2$ and $0$ . For any constant $c$ , at least one of $∣ c ∣$ and $∣ c - \ln 2 ∣$ is at least $\ln 2 / 2$ ; along that subsequence, for $ε = \ln 2 / 3$ , the probability of $∣ I_{N_{k}} - c ∣ > ε$ is eventually $1$ . Thus there is no limit in probability to a constant. The coordinate laws alternate, so the IID hypotheses of [F2] are absent.

Depends on

Used by

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Sources

Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd ed., Chapter 3, §3.1 (standard reference, not scraped)
MIT OpenCourseWare 6.441 Information Theory, Spring 2010, Lecture 3: Convergence and Asymptotic Equipartition Property (standard reference, not scraped)