CounterexampleConstruction: 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 Landweber can stall without a local tangential-cone condition

Statement refuted

Exact solvability by itself is enough for exact-data nonlinear Landweber to converge from every initial point; no local tangential-cone or initial-ball hypothesis is needed.

Statement references

The local hypotheses and exact-data conclusion

Facts & Assumptions

Given: The real Hilbert line $X = Y = R$ , $U = R$ , $F (x) = x^{2} - 1$ , exact solution $x^{†} = 1$ , exact datum $y = 0$ , and initial point $x_{0} = 0$ .

[F1]

Nonlinear Landweber iteration with discrepancy stopping specifies the exact-data update $x_{n + 1} = x_{n} + F^{'} (x_{n})^{*} (y - F (x_{n}))$ . Here we study that formula as a bare recurrence, not as a run of the defined procedure: step 2.1 shows this instance violates its admissible local TCC hypothesis. A zero derivative gives a zero update.

[F2]

Local tangential cone condition: at a base point $x$ the local TCC requires $∣ F (x + h) - F (x) - F^{'} (x) h ∣ \leq η ∣ F (x + h) - F (x) ∣$ for every increment in its radius.

[F3]

Nonlinear Landweber Fejér decrease and discrepancy stopping: its exact-data convergence conclusion assumes the specified outer ball, derivative bound, and uniform TCC on that ball.

Counterexample

1.1givenF1algebrainduction

For every $x, h \in R$ , $F (x + h) - F (x) - 2 x h = h^{2}$ , so $F$ is Fréchet differentiable with $F^{'} (x) h = 2 x h$ . Thus $F (1) = 0 = y$ is an exact solution and $F^{'} (x_{0}) = F^{'} (0) = 0$ . The Landweber update in [F1] gives $x_{1} = 0$ ; induction gives $x_{n} = 0$ and residual $∣ y - F (x_{n}) ∣ = 1$ for every $n$ . The exact-data iterates therefore do not converge to a solution.

2.1givenF2F3choosealgebra∎

For any proposed local radius $δ > 0$ , choose $h = δ / 2 > 0$ . At $x_{0} = 0$ , $F (h) - F (0) = h^{2}$ and $F^{'} (0) h = 0$ , so [F2] would require $h^{2} \leq η h^{2}$ . Since $h^{2} > 0$ , this forces $η \geq 1$ . Thus no local TCC ball at this initial point has a constant $η < 1 / 2$ , violating the theorem's retained local hypothesis [F3].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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.