CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Zero training loss can fail under an operator shift

Statement refuted

Let $X = Y = R$ and consider the one-step learned update $F_{θ} (a, y) = θ a y$ on the declared domain of all scalar operator/data pairs. For a Borel point rule $g$ and a joint input-target probability law $P$ , define squared risk by $R_{P} (g) = \int ∣ g (a, y) - x^{†} ∣^{2} P (d ((a, y), x^{†}))$ . Let $P_{S}$ be the probability law assigning mass one to $((a, y), x^{†}) = ((1, 1), 1)$ , and let $Q_{t}$ assign mass one to $((a, y), x^{†}) = ((t, t), 1)$ for each shift $t > 1$ . Define the training risk $L_{S} (θ) = R_{P_{S}} (F_{θ})$ and the test risk $R_{Q_{t}} (θ) = R_{Q_{t}} (F_{θ})$ . The forward residual is $r_{t} = ∣ t F_{θ} (t, t) - t ∣$ . The assertion refuted is that some finite constant $C$ bounds both $r_{t}$ and $R_{Q_{t}} (θ)$ for every $t > 1$ whenever $L_{S} (θ) = 0$ .

Statement references

The one-step unrolled update convention.

Borel point-estimator and finite learned-update interface.

Expected loss under joint train and test laws.

Zheng et al., dataset-scoped out-of-distribution experiments (empirical context only)

Daras et al., finite-computation and method-specific asymptotic discussion (context only)

Facts & Assumptions

Given: A scalar one-step architecture with initialization $x_{0} = 0$ , identity update block, trainable layer step $θ$ , training sample $(1, 1, 1)$ , and test laws $Q_{t}$ for $t > 1$ .

[F1]

Unrolled regularization architecture: for one layer, the update applies its declared block to the data-consistency-corrected input, and the deployed output is the finite one-step composition.

[F2]

Learned inverse estimators and reconstruction targets: a point reconstruction rule is Borel, and a finite unrolled learned update uses a Borel initialization, Borel layer maps, and has a point-valued output.

[F3]

Inverse reconstruction risk under train and test laws: a deterministic Borel point rule has risk equal to its extended nonnegative loss integral under a joint input-target law.

[F4]

A continuous map has Borel preimages of Borel sets: a continuous map between topological spaces has Borel preimages of Borel sets.

Counterexample

technique · direct

Given: The data in the Statement refuted.

1.1F1F2F4givenconstructalgebra

Choose the initialization $I_{θ} (a, y) = 0$ , identity block $B_{θ_{0}} ((a, y), v) = v$ , and layer parameter $τ_{0} = θ$ . By [F1], this is an admissible one-step architecture and $F_{θ} (a, y) = θ a (y - a \cdot 0) = θ a y$ . This polynomial map is continuous and hence Borel by [F4], so it satisfies the learned point-rule interface [F2].

2.1F3step 1.1givenalgebra

At the training point, $F_{θ} (1, 1) = θ$ . The law $P_{S}$ assigns mass one to that point, so [F3] gives $L_{S} (θ) = R_{P_{S}} (F_{θ}) = ∣ θ - 1 ∣^{2}$ . It is zero exactly at $θ = 1$ .

3.1step 2.1givenalgebra

For that zero-loss parameter and any $t > 1$ , $F_{1} (t, t) = t^{2}$ , so $r_{t} = ∣ t t^{2} - t ∣ = t (t^{2} - 1)$ .

4.1F3step 2.1step 3.1algebra∎

The test law $Q_{t}$ assigns mass one to its displayed point, so [F3] gives $R_{Q_{t}} (1) = ∣ t^{2} - 1 ∣^{2} = (t^{2} - 1)^{2}$ . Both $r_{t}$ and $R_{Q_{t}} (1)$ tend to $+ \infty$ as $t \to + \infty$ . For every finite $C$ , choosing $t$ sufficiently large violates at least one claimed bound (in fact both), while the training risk remains zero by step 2.1. Therefore the refuted assertion is false.

Remark

In the separate empirical InverseBench study, Figure 5(a) and §4.3 report that most tested plug-and-play diffusion methods on black-hole digit inputs exhibit a ring feature characteristic of the diffusion prior. Figure 5(b) shows inverse-scattering sources with nine cells while the prior was trained on images with one to six cells; §4.3 presents this case as one where the methods recover effectively when the source is close to the prior. These observations concern those methods and datasets only; they are not premises in the scalar proof above. The cited diffusion survey describes certain posterior-sampling procedures as asymptotically exact only in method-specific infinite-computation limits and notes that practical scaling remains unresolved; this sampling discussion also supplies no premise to the counterexample.

Remark references

Zheng et al., InverseBench, out-of-distribution tests and discussion

Daras et al., survey, asymptotically exact methods

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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.

Sources

Hongkai Zheng et al., InverseBench: Benchmarking Plug-and-Play Diffusion Priors for Inverse Problems in Physical Sciences (standard reference, not scraped)
Giannis Daras et al., A Survey on Diffusion Models for Inverse Problems (standard reference, not scraped)