TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-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.

The uniform interval residual certifier is sound on success

Statement

If Uniform interval linear residual certificate with inconclusive status returns certified with $(R, \hat{x}, q, r)$ , then every $(A, b)$ in the admitted families has a unique solution and its distance from $\hat{x}$ is at most $r / (1 - q)$ . Inconclusive output asserts no such conclusion.

Facts & Assumptions

Given: Nonempty matrix and right-hand-side families enclosed by the supplied rectangles and a successful algorithm return with $q < 1$ .

[F1]

Every exact family value in a finite interval expression DAG is enclosed by its outward result (Outward-rounded operations are inclusion-isotone and enclose DAG evaluations).

[F2]

Frobenius norm bounds the induced Euclidean operator norm (Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison).

[F3]

The reported tuple is uniform residual certificate data (Uniform interval linear residual certificate with inconclusive status, Verified point and uniform linear residual certificate data).

[F4]

Usable uniform residual data give a unique solution and the stated error radius (A uniform approximate inverse and residual enclose every linear solution).

Proof

1.1F1F2F3algebra

By [F1], the interval DAGs for $I - R A$ and $R (b - A \hat{x})$ enclose these expressions simultaneously for every admitted $A, b$ . The component modulus bounds, sums of squares and square roots therefore bound their Frobenius or Euclidean norms uniformly. By [F2], the first Frobenius bound also bounds $∥ I - R A ∥_{2}$ . Thus the returned $q, r$ satisfy the inequalities in the uniform residual certificate definition.

2.1givenF3F4step 1.1∎

The run returns certified only with $q < 1$ . Apply [F4] to the uniform certificate: for every admitted pair there is a unique solution $x_{A} (b)$ and $∥ x_{A} (b) - \hat{x} ∥_{2} \leq r / (1 - q)$ .

Depends on

Used by

Cited to discharge well-definedness by Uniform interval linear residual certificate with inconclusive status.

Dependency tree · two levels

22 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), §§5.1–5.5 and 10.5 (standard reference, not scraped)
Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §§1 and 5.1 (standard reference, not scraped)