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 point linear solve certifier returns a sound solution ball on success

Statement

For exact stored dyadic $A, b, R, \hat{x}$ , if Finite precision point linear solve certificate with explicit status returns certified with ball $B (\hat{x}, s)$ , then $A$ is invertible and its unique solution $x$ to $A x = b$ belongs to that ball. A range-failure or inconclusive result makes no such claim.

Facts & Assumptions

Given: The exact input data and a certified return with outward bounds $q < 1$ , $r$ , and $s \geq r / (1 - q)$ .

[F1]

The point algorithm's rounded products and residuals satisfy the stated operationwise error allowances (Finite precision point linear solve certificate with explicit status, Rounded matrix products give certified linear and normal-equation residual allowances).

[F2]

Outward interval DAG evaluation encloses every exact operation and its range is isotone (Outward-rounded operations are inclusion-isotone and enclose DAG evaluations).

[F3]

Usable residual data imply a unique solution within $r / (1 - q)$ (A uniform approximate inverse and residual enclose every linear solution).

Proof

1.1F1F2algebra

On a successful run, the columnwise rounded products enclose every column of $R A$ with Euclidean errors at most $v_{j}$ , by [F1]. The Frobenius error between the rounded product matrix and $R A$ is therefore at most $\sqrt{\sum_{j} v_{j}^{2}}$ . Outward soundness [F2] encloses $I - \hat{R A}$ , so the reported $q$ bounds $∥ I - R A ∥_{2}$ by the triangle inequality and $∥ M ∥_{2} \leq ∥ M ∥_{F}$ .

1.2F1F2algebra

The rounded residual allowance $W$ bounds $∥ \hat{w} - (A \hat{x} - b) ∥_{2}$ and the rounded product allowance $v_{c}$ bounds $∥ \hat{c} - R \hat{w} ∥_{2}$ by [F1]. Thus $∥ R (b - A \hat{x}) ∥_{2} \leq ∥ \hat{c} ∥_{2} + ∥ R ∥_{2} W + v_{c} \leq r$ , where [F2] makes each reported upper bound outward and $∥ R ∥_{2} \leq ∥ R ∥_{F}$ .

2.1givenF3step 1.1step 1.2algebra∎

These are usable point residual certificate data with $q < 1$ . By [F3] the system has a unique solution and $∥ x - \hat{x} ∥_{2} \leq r / (1 - q) \leq s$ . The latter inequality is exactly the returned ball membership.

Depends on

Used by

Cited to discharge well-definedness by Finite precision point linear solve certificate with explicit status.

Dependency tree · two levels

20 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 (standard reference, not scraped)
Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §1 (standard reference, not scraped)