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

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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.

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A unit VSC coefficient need not force Bregman convergence

Statement refuted

For $X = Y = R$ with the usual norm and pairing, $D (F) = R$ , $F (x) = 0$ , $R (x) = \max {0, - x}$ , $x^{†} = 0$ , $y = 0$ , $ξ^{†} = - 1 \in \partial R (0)$ , $R_{0} = 1$ , $p = 2$ , and $ψ (t) = \sqrt{t}$ , the boundary source inequality with coefficient $β_{1} = 1$

$- ξ^{†} (x - x^{†}) \leq D_{R}^{ξ^{†}} (x, x^{†}) + ψ ⁣ (∣ F (x) - F (x^{†}) ∣^{2}) (R (x) \leq R_{0}),$

holds for every $x$ with $R (x) \leq R_{0}$ , where $D_{R}^{ξ^{†}} (x, 0) : = R (x) - R (0) - ξ^{†} x$ . For every $δ \in (0, 1]$ , set $y^{δ} = 0$ , $α (δ) = δ$ , and $J_{δ} (x) : = \frac{1}{2} ∣ F (x) - y^{δ} ∣^{2} + α (δ) R (x)$ . A global minimizer means a point $x_{δ}$ with $J_{δ} (x_{δ}) \leq J_{δ} (z)$ for every $z \in R$ . The false claim is that every family of such global minimizers satisfies $D_{R}^{ξ^{†}} (x_{δ}, 0) = O (δ)$ as $δ ↓ 0$ , meaning $D_{R}^{ξ^{†}} (x_{δ}, 0) \leq C δ$ for some fixed $C$ and all sufficiently small $δ > 0$ .

Statement references

Banach subgradient inequality

Bregman distance

Variational source condition

Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, Theorem 2.20 and equation (2.26)

Facts & Assumptions

Given: The scalar model and source inequality in the Statement.

[F1]

Banach-space convex subgradients and dual pairings: $ξ \in \partial R (x)$ exactly when $R (z) \geq R (x) + ⟨ ξ, z - x ⟩$ for every $z$ .

[F2]

Bregman distances for convex regularizers: $D_{R}^{ξ} (x, x^{†}) = R (x) - R (x^{†}) - ⟨ ξ, x - x^{†} ⟩$ .

[F3]

Variational source conditions on a regularizer sublevel: the rate theorem's VSC requires a coefficient strictly below one; this item tests the same source inequality at its excluded boundary value one.

Counterexample

1.1givenF1F2F3algebra

The function $R$ is finite and nonnegative, hence proper. For $u, v \in R$ and $t \in [0, 1]$ , $R ((1 - t) u + t v) = \max {(1 - t) 0 + t 0, (1 - t) (- u) + t (- v)} \leq (1 - t) \max {0, - u} + t \max {0, - v}$ , so $R$ is convex. For every $z \in R$ , $R (z) \geq - z = R (0) + (- 1) (z - 0)$ , so [F1] gives $ξ^{†} = - 1 \in \partial R (0)$ . By [F2], $D_{R}^{- 1} (x, 0) = R (x) + x$ , which equals $x$ for $x \geq 0$ and $0$ for $x < 0$ . Since $F$ is zero, the source inequality has left side $x$ and right side $D_{R}^{- 1} (x, 0)$ ; it holds for all $x$ , hence on $R (x) \leq 1$ , with boundary coefficient $β_{1} = 1$ .

2.1givenstep 1.1algebra∎

For every $δ > 0$ , $J_{δ} (x) = δ R (x) \geq 0$ , and it equals zero for every $x \geq 0$ . Thus $x_{δ} = 1$ is a global minimizer for every $δ$ , but $D_{R}^{- 1} (1, 0) = 1$ . No fixed constant bounds $1$ by $C δ$ for all sufficiently small $δ$ , so the claimed rate fails.

Depends on

Used by

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Dependency tree · two levels

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Sources

Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, Theorem 2.20 and equation (2.26) (standard reference, not scraped)