ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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.

Exact data can converge without a finite discrepancy stop

Example

Take $X = Y = R$ , $U = R$ , $F (x) = x / 2$ , initial point $x_{0} = 1$ , exact solution $x^{†} = 0$ , exact datum $y = 0$ , and exact data $y^{δ} = y$ with $δ = 0$ . The derivative norm is $1 / 2$ and the uniform TCC holds with $η = 0$ on every ball. For any $τ > 2$ and any finite update budget $M$ , the ideal exact-data iterates are $x_{n} = (3 / 4)^{n}$ and have strictly positive residual at every finite index. They converge to the exact solution, but the zero discrepancy threshold is never met finitely, so the finite-budget algorithm returns budget_exhausted.

Example references

Exact-data discrepancy test and budget status

Exact-data convergence and the absence of a finite-stop guarantee

Facts & Assumptions

Given: $X = Y = R$ , $U = R$ , $F (x) = x / 2$ , $x_{0} = 1$ , $x^{†} = 0$ , $y = y^{δ} = 0$ , $δ = 0$ , $τ > 2$ , and an arbitrary finite $M \in N_{0}$ .

[F1]

Nonlinear Landweber iteration with discrepancy stopping: the procedure returns stopped only when the current residual is at most $τ δ$ ; after $M$ updates it returns budget_exhausted if that test has not succeeded.

[F2]

Nonlinear Landweber Fejér decrease and discrepancy stopping: for exact data under the stated local hypotheses, the ideal Landweber sequence converges to a solution; the theorem does not claim a finite exact-data stop.

Verification

1.1givenalgebra

The derivative is $F^{'} (x) h = h / 2$ , so $∥ F^{'} (x) ∥ = 1 / 2 \leq 1$ , and the linearization remainder is identically zero. Thus the uniform TCC holds with $η = 0$ . With $r = 1$ , $x^{†} = 0 \in {\overline{B}}_{r} (1)$ and ${\overline{B}}_{2 r} (1) \subset R$ , so the local hypotheses hold.

2.1givenF1F2algebrainduction∎

The update is $x_{n + 1} = x_{n} + (1 / 2) (- x_{n} / 2) = (3 / 4) x_{n}$ ; induction gives $x_{n} = (3 / 4)^{n} > 0$ and residual $∣ F (x_{n}) - y ∣ = (1 / 2) (3 / 4)^{n} > 0$ for every finite $n$ . Hence the discrepancy test with threshold $τ δ = 0$ never succeeds at a finite index. By [F2], the exact-data sequence converges to a solution; the only solution of $F (x) = 0$ is $0$ . Therefore any finite-budget run reaches $M$ with positive residual and returns budget_exhausted by [F1].

Depends on

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