TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Termination and soundness of the corrected stationary residual certificate

Statement

For every exact rational input in the domain of Correct and certify a finite-chain stationary candidate, the procedure terminates with one of its specified statuses. If it returns certified with corrected candidate $x$ , correction distance $c = ∥ y - x ∥_{1}$ , residual $r = P x - x$ , and $q = δ (P^{m}) < 1$ , then P has a unique stationary probability $π$ and

$∥ y - π ∥_{1} \leq c + \frac{m ∥ r ∥_{1}}{1 - q} .$

Facts & Assumptions

Given: An exact rational input to the candidate-correction algorithm.

[F1]

Correct and certify a finite-chain stationary candidate: the procedure has finite loops over $n$ coordinates, $m$ matrix powers and finitely many column pairs, uses exact rational operations, and returns one of its specified statuses. A successful output has $x = y^{+} / s$ , correction $c = ∥ y - x ∥_{1}$ , residual $r = P x - x$ , and $q = δ (P^{m}) < 1$ .

[F2]

m-step Dobrushin certificate for a stationary probability: if $q = δ (P^{m}) < 1$ , the stationary probability $π$ is unique and $∥ x - π ∥_{1} \leq m ∥ P x - x ∥_{1} / (1 - q)$ .

Proof

Given: P, y, m, and a successful certificate returned by the stated algorithm.

Proof technique: apply the residual theorem to the corrected probability and add the explicit correction distance.

1.1F1givenalgebra

The correction loop visits the $n$ coordinates once; computing $P^{m}$ uses the finite loop $k = 0, \dots, m - 1$ , and each matrix product and coefficient scan uses finitely many rational operations. The branches for invalid P, $s = 0$ , $q \geq 1$ , and $q < 1$ each return a specified status or certificate. Thus the procedure terminates.

1.2F1F2given

If the procedure returns certified, [F1] gives a column probability vector x, the exact residual $r = P x - x$ , correction $c = ∥ y - x ∥_{1}$ , and $q = δ (P^{m}) < 1$ . By [F2], $∥ x - π ∥_{1} \leq m ∥ r ∥_{1} / (1 - q)$ .

2.1F1step 1.2algebra∎

Since $y - π = (y - x) + (x - π)$ , the triangle inequality and [F1] give $∥ y - π ∥_{1} \leq ∥ y - x ∥_{1} + ∥ x - π ∥_{1} \leq c + m ∥ r ∥_{1} / (1 - q)$ .

Depends on

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Sources

Ilse C. F. Ipsen and Teresa M. Selee, Ergodicity Coefficients Defined by Vector Norms (standard reference, not scraped)
Yuanyuan Liu, Perturbation Bounds for the Stationary Distributions of Markov Chains (standard reference, not scraped)