ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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 additive response-evaluation tolerance enters the fixed-point residual certificate

Example

This scalar calculation isolates the additive response-error term in the SCF residual certificate; it is not itself an eigensolve.

For the scalar contraction $Φ (x) = x / 2 + 1 / 4$ on $[0, 1]$ , evaluate at $x = 0$ . The exact response is $1 / 4$ . Suppose the computed response is $\tilde{Φ} (0) = 6 / 25$ with certified additive error $η = 1 / 100$ .

Facts & Assumptions

Given: The contraction $Φ (x) = x / 2 + 1 / 4$ , its fixed point $x_{*} = 1 / 2$ , $x = 0$ , computed value $\tilde{Φ} (0) = 6 / 25$ , and $∣ \tilde{Φ} (0) - Φ (0) ∣ \leq 1 / 100$ .

[F1]

An inexact update's error bound uses the exact response residual plus its certified evaluation error (A contractive self-consistent map converges and admits residual and inexact-solve bounds).

Verification

1.1givenalgebra

The computed residual is $∣ 0 - 6 / 25 ∣ = 24 / 100$ . By the triangle inequality, $∣ 0 - Φ (0) ∣ \leq ∣ 0 - \tilde{Φ} (0) ∣ + ∣ \tilde{Φ} (0) - Φ (0) ∣ \leq 24 / 100 + 1 / 100 = 25 / 100.$

2.1step 1.1F1algebra∎

Here $q = 1 / 2$ and $∣ 0 - x_{*} ∣ = 1 / 2$ . The correct certified error bound from [F1] is $(25 / 100) / (1 - 1 / 2) = 1 / 2$ , exact here. Omitting $η$ would give $(24 / 100) / (1 / 2) = 48 / 100 < 1 / 2$ , which is false.

Depends on

A contractive self-consistent map converges and admits residual and inexact-solve bounds

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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