ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

An exact normal-equation residual gives an a posteriori solution bound

Example

Assume Countable Choice. Let $A = 2$ on $R$ , $λ = 1 / 2$ , and $y_{δ} = 1$ . The exact regularized solution is $4 / 9$ . For the approximate value $\tilde{x} = 4 / 9 + 1 / 100$ , the exact normal-equation residual is $9 / 200$ , so the bound $∣ \tilde{x} - 4 / 9 ∣ \leq ∣ r ∣ / λ$ holds. This uses an exact residual; it does not certify how floating-point residual evaluation error should be bounded.

Facts & Assumptions

Given: Countable Choice, $A = 2$ , $λ = 1 / 2$ , $y_{δ} = 1$ , and $\tilde{x} = 4 / 9 + 1 / 100$ ; compute the exact residual and compare it with the error bound.

[F1]

The shifted Hilbert-space normal operator is invertible with inverse norm at most $1 / λ$ : The inverse of $A^{*} A + λ I$ exists and has norm at most $1 / λ$ .

Verification

1.1givenalgebra

Here $A^{*} A + λ I = 4 + 1 / 2 = 9 / 2$ , while the exact regularized solution is $A^{*} y_{δ} / (A^{*} A + λ I) = 2 / (9 / 2) = 4 / 9$ . Thus $∣ \tilde{x} - 4 / 9 ∣ = 1 / 100$ .

2.1givenF1step 1.1algebra∎

The exact residual is $r = (9 / 2) (4 / 9 + 1 / 100) - 2 = 9 / 200$ . Since $\tilde{x} - 4 / 9 = (9 / 2)^{- 1} r$ , [F1] bounds the error by $∣ r ∣ / λ = (9 / 200) / (1 / 2) = 9 / 100 \geq 1 / 100$ . The residual here is exact, not a floating-point evaluation certificate.

Depends on

The shifted Hilbert-space normal operator is invertible with inverse norm at most $1/\lambda$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Guido Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)
Per Christian Hansen, Introduction to Inverse Problems, Chapter 5: Regularization Parameter Choice (standard reference, not scraped)