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.

A small singular value makes the Tikhonov noise bound sharp

Example

Assume Countable Choice. Take $A = diag (1, 10^{- 3})$ on $R^{2}$ , exact target $x^{†} = 0$ , noise $e = (0, 10^{- 6})$ , observed data $b = e$ , and $λ = 10^{- 6}$ . The regularized inverse sends this noise to a vector of norm $5 \cdot 10^{- 4}$ , exactly the general bound $∥ e ∥ / (2 \sqrt{λ})$ .

Facts & Assumptions

Given: Countable Choice, the diagonal matrix, target, noise and parameter above; verify the regularized noise amplification and the claimed sharp bound.

[F1]

Hilbert-space Tikhonov amplifies data by at most $1 / (2 \sqrt{λ})$ : For every data vector $g$ , the Tikhonov map satisfies $∥ R_{λ} g ∥ \leq ∥ g ∥ / (2 \sqrt{λ})$ .

Verification

1.1givenalgebra

Here $A^{*} A + λ I = diag (1 + 10^{- 6}, 2 \cdot 10^{- 6})$ and $A^{*} e = (0, 10^{- 9})$ , so $R_{λ} e = (0, 5 \cdot 10^{- 4})$ and $∥ e ∥ = 10^{- 6}$ .

2.1givenF1step 1.1algebra∎

Since $\sqrt{λ} = 10^{- 3}$ , [F1] gives the upper bound $10^{- 6} / (2 \cdot 10^{- 3}) = 5 \cdot 10^{- 4}$ , equal to the computed output norm. Thus the constant is attained in this example.

Depends on

Hilbert-space Tikhonov amplifies data by at most $1/(2\sqrt{\lambda})$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Per Christian Hansen, Introduction to Inverse Problems, Chapter 4: Regularization Methods (standard reference, not scraped)
Guido Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)