ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 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.

A diagonal inverse has an explicit Fourier truncation error

Example

Write $ℓ^{2} = ℓ^{2} (N_{> 0}; R)$ , where $N_{> 0} = {1, 2, \dots}$ , so all sequence coordinates below have positive indices. Let $p, s, R > 0$ , let $m \geq 1$ be an integer, let $θ \in ℓ^{2}$ satisfy $\sum_{j \geq 1} j^{2 s} θ_{j}^{2} \leq R^{2}$ , and let $(A_{p} θ)_{j} = j^{- p} θ_{j}$ . Given data $y = A_{p} θ + η$ with $∥ η ∥_{ℓ^{2}} \leq δ$ , the mesh- $m$ truncated inverse is $g_{m} (y) = (j^{p} y_{j})_{j \leq m}$ with zero output coordinates for $j > m$ . It satisfies the continuum error bound $∥ g_{m} (y) - θ ∥_{ℓ^{2}} \leq m^{p} δ + R (m + 1)^{- s} .$ This separates measurement-noise amplification from the mesh truncation tail; it makes no statistical or neural-operator claim.

Example references

A mesh-transfer example on the compact diagonal inverse-map domain. ‡

Facts & Assumptions

Given: $p, s, R > 0$ , an integer $m \geq 1$ , $\sum_{j \geq 1} j^{2 s} θ_{j}^{2} \leq R^{2}$ , data $y = A_{p} θ + η$ , and $∥ η ∥_{ℓ^{2}} \leq δ$ .

[A1]

$g_{m} (y)$ retains the first $m$ coordinates $j^{p} y_{j}$ and sets all later coordinates to zero.

Verification

1.1givenA1algebra

For $j \leq m$ , $j^{p} y_{j} = θ_{j} + j^{p} η_{j}$ . The norm of the first- $m$ noise vector is at most $m^{p} ∥ η ∥_{ℓ^{2}} \leq m^{p} δ$ .

1.2givenA1algebra

The omitted truth tail satisfies $\sum_{j > m} θ_{j}^{2} \leq (m + 1)^{- 2 s} \sum_{j > m} j^{2 s} θ_{j}^{2} \leq R^{2} (m + 1)^{- 2 s}$ . Its norm is at most $R (m + 1)^{- s}$ .

2.1A1step 1.1step 1.2algebra∎

The error vector is the sum of the retained-coordinate noise error and the omitted truth tail. The triangle inequality with steps 1.1 and 1.2 gives $∥ g_{m} (y) - θ ∥_{ℓ^{2}} \leq m^{p} δ + R (m + 1)^{- s}$ .

Used by

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Dependency tree · 0 levels

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