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.

Increasing the Tikhonov parameter trades smaller data response for more bias

Example

Assume Countable Choice. In the scalar problem $A = 1$ , $x^{†} = 1$ , and $e = 0.1$ , the data are $b = 1.1$ . At $λ = 0.01$ , the regularization bias is $1 / 101$ and the data-noise contribution has magnitude $10 / 101$ . At $λ = 1$ , the corresponding magnitudes are $1 / 2$ and $1 / 20$ . This pair of parameter choices shows the tradeoff; it does not assert a universally optimal parameter rule.

Facts & Assumptions

Given: Countable Choice, $A = 1$ , $x^{†} = 1$ , $e = 0.1$ , $b = A x^{†} + e = 1.1$ , and the two positive parameter values; verify the regularization bias and data-noise terms.

[F1]

Tikhonov error separates bias, data noise, discretization and solve residual: For finite matrices, $x^{†} - A_{λ}^{#} b = (I - A_{λ}^{#} A) x^{†} - A_{λ}^{#} e$ with $A_{λ}^{#} = (A^{*} A + λ I)^{- 1} A^{*}$ .

Verification

1.1givenF1algebra

In this scalar case $A_{λ}^{#} = 1 / (1 + λ)$ , so the bias magnitude is $∣ 1 - A_{λ}^{#} A ∣ = λ / (1 + λ)$ and the noise magnitude is $∣ A_{λ}^{#} e ∣ = 0.1 / (1 + λ)$ by [F1].

2.1step 1.1algebra∎

At $λ = 0.01$ these values are $0.01 / 1.01 = 1 / 101$ and $0.1 / 1.01 = 10 / 101$ ; at $λ = 1$ they are $1 / 2$ and $0.1 / 2 = 1 / 20$ . The two components move in opposite directions for these choices, with no parameter-choice guarantee asserted.

Depends on

Tikhonov error separates bias, data noise, discretization and solve residual

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 5: Regularization Parameter Choice (standard reference, not scraped)
Guido Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)