TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-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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The finite disk cover validator returns a uniform resolvent bound on success

Statement

If the validator's input has $n \geq 1$ and Finite complex contour resolvent validator with cover and inconclusive statuses returns certified-resolvent-cover with bound $m$ , then every $A$ in the input family and every $z$ on the supplied contour segments satisfy that $z I - A$ is invertible and $∥ (z I - A)^{- 1} ∥_{\infty, C} \leq m$ . The result does not count eigenvalues inside the contour.

Facts & Assumptions

Given: The finite contour, disk assignments, matrix family and a successful validator return.

[F1]

Outward interval operations enclose all exact matrix and scalar quantities used by the validator (Outward-rounded operations are inclusion-isotone and enclose DAG evaluations).

[F2]

Certified disk bounds $e_{j}, δ_{A}, s_{j}$ with $e_{j} + s_{j} (r_{j} + δ_{A}) < 1$ imply a uniform resolvent bound on the covered contour (Diskwise Neumann certificates bound a complex matrix resolvent).

[F3]

The algorithm checks endpoint coverage and reports its maximum disk bound only after all strict inequalities succeed (Finite complex contour resolvent validator with cover and inconclusive statuses).

Proof

1.1givenF1F2F3algebra

By [F1], each reported $e_{j}, δ_{A}, s_{j}$ bounds the exact norm or family supremum it is intended to bound. The algorithm checks the disk endpoint conditions. Each whole assigned segment lies in its disk by convexity, so the disks cover every point of the contour. Its computed $q_{j} < 1$ and $m \geq \max_{j} s_{j} / (1 - q_{j})$ therefore satisfy all hypotheses and the bound comparison in [F2].

2.1F2step 1.1∎

Apply [F2] to obtain invertibility and the diskwise resolvent bound at every contour point and for every family member. Since each disk bound is at most $m$ , the returned uniform inequality follows. The theorem invoked in [F2] makes no eigenvalue-count assertion, so none is concluded here.

Depends on

Used by

Cited to discharge well-definedness by Finite complex contour resolvent validator with cover and inconclusive statuses.

Dependency tree · two levels

14 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

Siegfried M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica 19 (2010), §§10.5–10.6 (standard reference, not scraped)
Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §5.1 (standard reference, not scraped)