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 scalar quadrature residual appears before the regularized inverse

Example

Assume Countable Choice. Model a quadrature or collocation approximation by the scalar maps $A = 2$ and $A_{h} = 1.9$ , exact target $x^{†} = 1$ , shared data noise $e = 0.1$ , and $λ = 1$ . Then the exact forward datum is $y = 2$ , the perturbed datum is $y_{δ} = 2.1$ , the discretized perturbed datum is $y_{h, δ} = 2.0$ , and the forward residual discarded before inversion is $d_{h} = 0.1$ . The refined error chain keeps its propagated residual $R_{λ} d_{h} = 0.04$ as a separate term.

Facts & Assumptions

Given: Countable Choice, the scalar exact and discretized operators, target, shared noise, and $λ$ above; verify the pre-inversion residual and full error chain.

[F1]

Tikhonov error separates bias, data noise, discretization and solve residual: For $y_{h, δ} = A_{h} x^{†} + e$ and $d_{h} = (A - A_{h}) x^{†}$ , the exact chain contains the separate term $R_{λ} d_{h}$ before the discretized solve.

Verification

1.1givenF1algebra

The scalar regularized maps are $R_{λ} = 2 / (2^{2} + 1) = 2 / 5$ and $R_{λ, h} = (19 / 10) / ((19 / 10)^{2} + 1) = 190 / 461$ . Also $d_{h} = 2 - 1.9 = 1 / 10$ , so [F1] gives the propagated pre-inversion residual $R_{λ} d_{h} = (2 / 5) (1 / 10) = 1 / 25$ .

2.1step 1.1algebra∎

With the exact discrete solve ${\tilde{x}}_{h} = R_{λ, h} y_{h, δ} = 380 / 461$ , the five chain terms are respectively $1 / 5$ , $- 1 / 25$ , $1 / 25$ , $4 / 5 - 380 / 461 = - 56 / 2305$ , and $0$ . Their sum is $461 / 2305 - 56 / 2305 = 405 / 2305 = 81 / 461 = 1 - {\tilde{x}}_{h}$ , verifying the telescoping identity with the forward residual counted once.

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 4: Regularization Methods (standard reference, not scraped)
Guido Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)