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.

Gaussian coefficient sequence

Example

Let $H = ℓ^{2} (N)$ with standard basis $e_{j}$ , take a single parameter with signal $θ = (θ_{j}) \in ℓ^{2} (N)$ , and fix $ϵ > 0$ . The coordinate statistic $Y_{j} = Y (e_{j}) = θ_{j} + ϵ Z (e_{j})$ has independent coordinates with laws $N (θ_{j}, ϵ^{2})$ . Each finite truncation $(Y_{1}, \dots, Y_{N})$ therefore has the product of its $N$ scalar normal laws. The coordinate statistic takes values in $R^{N}$ ; this description does not identify the observation with an $ℓ^{2}$ -valued vector.

Example references

White-noise spectral sequence representation

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Gaussian Sequence Space Model, equation (1.8), PDF pp. 21–22

Ding and Mathé, Minimax rates for statistical inverse problems under general source conditions, §1, equations (1)–(3), PDF pp. 11–12

Facts & Assumptions

[F1]

White-noise spectral sequence representation: In a fixed complete finite or countable orthonormal basis, the observation coordinates are independent with laws $N (⟨ K x, u_{j} ⟩_{H}, ϵ^{2})$ .

Given: The signal $θ \in ℓ^{2} (N)$ , the standard basis of $H$ , and $ϵ > 0$ in the Example.

Verification

1.1givenF1

Apply [F1] with $K x = θ$ and $u_{j} = e_{j}$ to obtain the independent coordinate laws $N (θ_{j}, ϵ^{2})$ for every $j$ .

2.1givenF1algebra∎

Independence of every finite subfamily means that for each $N$ the joint law of $(Y_{1}, \dots, Y_{N})$ is the product of those $N$ marginal normal laws. The sample-space assertion is the coordinate-statistic map into $R^{N}$ , and no step identifies it with an element of $ℓ^{2}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, Gaussian Sequence Space Model, equation (1.8), PDF pp. 21–22 (standard reference, not scraped)
Ding and Mathé, Minimax rates for statistical inverse problems under general source conditions, §1, equations (1)–(3), PDF pp. 11–12 (standard reference, not scraped)