ExampleConstruction: AI-adaptedVerification: 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.

A certified residual enclosure remains valid for an ill-conditioned 2 by 2 solve

Example

Let $A = (\begin{array}{cc} 1 & 0 \\ 0 & 2^{- 20} \end{array}), b = (\begin{array}{c} 1 \\ 2^{- 20} \end{array}), R = A^{- 1} = (\begin{array}{cc} 1 & 0 \\ 0 & 2^{20} \end{array}), \hat{x} = (\begin{array}{c} 1 \\ 1 + 2^{- 10} \end{array}) .$ These point data have a usable residual certificate with $q = 0$ and $r = 2^{- 10}$ . Its certified error radius equals the actual error even though the spectral condition number of $A$ is $2^{20}$ .

Facts & Assumptions

Given: The exact dyadic matrix, right-hand side, inverse candidate, and approximate solution displayed above.

[F1]

A usable residual certificate gives a unique solution within $r / (1 - q)$ (A uniform approximate inverse and residual enclose every linear solution).

[F2]

The point case of the residual certificate is specified by $∥ I - R A ∥_{2} \leq q$ and $∥ R (b - A \hat{x}) ∥_{2} \leq r$ (Verified point and uniform linear residual certificate data).

Verification

1.1givenF2algebra

Direct multiplication gives $R A = I$ , so $∥ I - R A ∥_{2} = 0$ . Also $A \hat{x} = (1, 2^{- 20} + 2^{- 30})^{T}$ and $R (b - A \hat{x}) = (0, - 2^{- 10})^{T}$ , whose Euclidean norm is $2^{- 10}$ . Thus [F2] holds with $q = 0, r = 2^{- 10}$ .

2.1givenF1step 1.1algebra∎

The exact solution is $x = (1, 1)^{T}$ , so $∥ x - \hat{x} ∥_{2} = 2^{- 10}$ . The singular values of this positive diagonal matrix are $1$ and $2^{- 20}$ , hence its condition number is $2^{20}$ . Applying [F1] gives the same error radius $r / (1 - q) = 2^{- 10}$ , which contains the solution with equality.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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 (standard reference, not scraped)