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.

The initial residual can already meet the discrepancy threshold

Example

Take $X = Y = R$ with the usual real Hilbert structure, $U = R$ , $F (x) = x$ , $x_{0} = x^{†} = 0$ , $y = F (x^{†}) = 0$ , noise level $δ = 1$ , noisy datum $y^{δ} = 1$ , discrepancy parameter $τ > 2$ , and update budget $M = 0$ . The local Landweber assumptions hold with any radius $r > 0$ and $η = 0$ , and the initial residual is $1 \leq τ δ$ . The algorithm therefore returns stopped at index zero with iterate $0$ and residual $1$ , before consulting the zero update budget.

Example references

Landweber update, stopping test, and budget order

Facts & Assumptions

Given: The scalar real Hilbert space $X = Y = R$ , $U = R$ , $F (x) = x$ , $x_{0} = x^{†} = 0$ , $y = 0$ , $δ = 1$ , $y^{δ} = 1$ , $τ > 2$ , and $M = 0$ .

[F1]

Nonlinear Landweber iteration with discrepancy stopping: the algorithm tests $∥ y^{δ} - F (x_{k}^{δ}) ∥_{Y} \leq τ δ$ before testing whether $k = M$ .

[F2]

Local tangential cone condition: the local TCC remainder is $F (x + h) - F (x) - F^{'} (x) h$ and its required bound is relative to $∥ F (x + h) - F (x) ∥$ .

Verification

1.1givenF2algebra

The map is continuously Fréchet differentiable with $F^{'} (x) h = h$ , so $∥ F^{'} (x) ∥ = 1$ . Its TCC remainder vanishes for every pair, hence the uniform TCC holds with $η = 0$ ; ${\overline{B}}_{2 r} (0) \subset R$ and $x^{†} = 0 \in {\overline{B}}_{r} (0)$ for any $r > 0$ . Also $∣ y^{δ} - y ∣ = 1 = δ$ .

2.1givenF1algebra∎

At the initial point, $∥ y^{δ} - F (x_{0}) ∥ = ∣ 1 - 0 ∣ = 1 \leq τ δ$ . By [F1] the procedure returns stopped at $k = 0$ with iterate $x_{0} = 0$ and residual $1$ before the $M = 0$ budget condition is checked.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.