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.

Periodic iteration can fail while the stationary solve is well-defined

Example

For the two-state periodic chain

$P = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}),$

the unique stationary law is $(1 / 2, 1 / 2)^{T}$ , but power iteration from $e_{1}$ alternates forever. For $m = 1$ , the global overlap coefficient is one and the residual certificate is unavailable. The lazy matrix $L = (I + P) / 2$ has identical columns, coefficient zero, and certifies the stationary law after one step.

Verification

Given: The periodic matrix P, initial vector $e_{1}$ , and lazy matrix $L = (I + P) / 2$ .

[F1] An irreducible finite class has a unique invariant vector: an irreducible finite transition matrix has a unique invariant probability, even when its powers are periodic.

[F2] Column-overlap formula for the zero-sum ℓ1 coefficient: the coefficient is one minus the minimum pairwise column overlap.

[F3] Correct and certify a finite-chain stationary candidate: for exact rational P and raw y with $\sum_{i} y_{i}^{+} > 0$ , a computed coefficient at least one returns certificate-unavailable; a coefficient below one returns a residual bound.

[F4] m-step Dobrushin certificate for a stationary probability: a coefficient below one gives the stationary residual error bound.

Proof technique: direct exact calculation.

1.1givenF1algebra

The matrix P is a row and column transition matrix, is irreducible, and $π = (1 / 2, 1 / 2)^{T}$ satisfies $P π = π$ . Hence [F1] identifies it as the unique stationary law. But $P e_{1} = e_{2}$ and $P e_{2} = e_{1}$ , so $P^{n} e_{1}$ alternates between two distinct vectors and does not converge.

2.1F2F3step 1.1algebra

The columns of P are $e_{2}, e_{1}$ and have overlap zero, so [F2] gives $δ (P) = 1$ ; every power is either P or I and also has coefficient one. With exact raw input $y = e_{1}$ , the algorithm reaches its coefficient branch and [F3] returns certificate-unavailable for the ordinary chain.

3.1F2F3F4step 1.1algebra∎

The lazy matrix has both columns equal to $π$ , hence [F2] gives $δ (L) = 0$ . With raw input $y = e_{1}$ , the corrected algorithm has correction zero, residual $L e_{1} - e_{1} = (- 1 / 2, 1 / 2)^{T}$ of norm one, and bound $1 / (1 - 0) = 1$ . This equals $∥ e_{1} - π ∥_{1}$ , so [F4] certifies the stationary law.

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

Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, Chapter 11 (standard reference, not scraped)
Ilse C. F. Ipsen and Teresa M. Selee, Ergodicity Coefficients Defined by Vector Norms (standard reference, not scraped)