ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 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 zero-noise boundary of truncated risk

Example

Assume Countable Choice for the upstream sequence-model coordinates. Take $ϵ = 0$ and $Q > 0$ . In the zero-noise sequence model, $Z_{j} = θ_{j}$ exactly, so the cutoff error is the deterministic squared tail $\sum_{j > D} θ_{j}^{2}$ . For weights $a_{j} = j^{r}$ with $r > 0$ , its worst case over $Θ_{a} (Q)$ is $Q^{2} / (D + 1)^{2 r}$ and tends to $0$ as $D \to \infty$ . For the equally weighted ellipsoid $a_{j} = 1$ , the worst case is $Q^{2}$ for every finite $D$ , attained by $θ = Q e_{D + 1}$ . This is a deterministic boundary of the sequence-risk formula; it makes no claim about arbitrary bounded perturbations of the original inverse data.

Example references

Gaussian sequence inverse model and its zero-noise extension

Weighted ellipsoid class

Exact cutoff-risk theorem

Facts & Assumptions

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is used upstream to choose the compact SVD coordinates and identify the complete-basis coefficient space.

[F1]

Gaussian sequence inverse model: At $ϵ = 0$ , $Z_{j} = θ_{j}$ deterministically.

[F2]

Weighted ellipsoid class for sequence inversion: For any positive nondecreasing weights, $Θ_{a} (Q)$ is defined by $\sum_{j} a_{j}^{2} θ_{j}^{2} \leq Q^{2}$ .

[F3]

Exact worst-case risk of the truncated estimator: The zero-noise worst-case squared error is $Q^{2} / a_{D + 1}^{2}$ , attained by a truth supported only at coordinate $D + 1$ .

Given: Countable Choice, $Q > 0$ , a finite cutoff $D \in N_{0}$ , and either weights $a_{j} = j^{r}$ with $r > 0$ or $a_{j} = 1$ for all $j$ .

Verification

1.1A1F1F2F3algebra

Under [A1], the sequence model coordinates are fixed. Using [F1] and substituting $ϵ = 0$ and $a_{D + 1} = (D + 1)^{r}$ in [F3] gives $Q^{2} / (D + 1)^{2 r}$ ; since $r > 0$ , this tends to $0$ as $D \to \infty$ .

2.1A1F2F3algebraconstruct∎

Under [A1], [F3] applies to the unit-weight ellipsoid. The ellipsoid condition in [F2] is $∥ θ ∥_{2}^{2} \leq Q^{2}$ , so the squared tail is at most $Q^{2}$ . The vector $θ = Q e_{D + 1}$ belongs to the ellipsoid and its entire squared norm lies beyond $D$ , so it attains $Q^{2}$ . This is also the $ϵ = 0$ , $a_{D + 1} = 1$ case of [F3].

Depends on

Used by

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Sources

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, Lemma 1 (standard reference, not scraped)
Iain M. Johnstone, Gaussian estimation: Sequence and wavelet models, §3.2 (standard reference, not scraped)