ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 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.

Two direct white-noise functionals

Example

Take the single-parameter experiment with $H = R^{2}$ , its standard inner product, parameter set ${x}$ , mean signal $K x = (m_{1}, m_{2})$ , and noise level $ϵ > 0$ . Let $e_{1} = (1, 0)$ , $e_{2} = (0, 1)$ , and $g = a e_{1} + b e_{2}$ for $a, b \in R$ . The direct observations $Y_{x} (e_{1})$ and $Y_{x} (e_{2})$ are independent with laws $N (m_{1}, ϵ^{2})$ and $N (m_{2}, ϵ^{2})$ . In $L^{2} (P)$ , $Y_{x} (g) = a Y_{x} (e_{1}) + b Y_{x} (e_{2})$ , with mean $a m_{1} + b m_{2}$ and variance $ϵ^{2} (a^{2} + b^{2})$ ; when $a = b = 0$ , $Y_{x} (g)$ is the constant zero.

Example references

Isonormal white-noise inverse experiment

Kretschmann and Werner, Maximum a posteriori testing in statistical inverse problems, §1.1, PDF pp. 1–2

Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §1.2.2, equations (1.6)–(1.8), PDF pp. 21–22

Facts & Assumptions

[F1]

Isonormal Gaussian white-noise inverse experiment: The process is linear into $L^{2} (P)$ , centered, has covariance $E [Z (g) Z (h)] = ⟨ g, h ⟩_{H}$ , and has independent standard-normal coordinates on finite orthonormal tuples.

Given: The two-dimensional experiment, $e_{1}, e_{2}$ , $m_{1}, m_{2}$ , $a, b$ , and $ϵ > 0$ in the Example.

Verification

1.1givenF1

Since $e_{1}, e_{2}$ are orthonormal, [F1] gives independent standard-normal noises $Z (e_{1}), Z (e_{2})$ . Each observation is an invertible affine function of its noise coordinate because $ϵ > 0$ , so it has the same generated sigma-algebra; the affine-image definition of $N$ gives the two asserted laws and preserves independence.

2.1givenF1algebra∎

Linearity of the process gives $Y_{x} (g) = a Y_{x} (e_{1}) + b Y_{x} (e_{2})$ as an equality of its almost-sure equivalence classes. The mean is $a m_{1} + b m_{2}$ , and the covariance identity in [F1] gives variance $ϵ^{2} ∥ a e_{1} + b e_{2} ∥^{2} = ϵ^{2} (a^{2} + b^{2})$ . If $a = b = 0$ , then $g = 0$ and linearity gives $Y_{x} (0) = 0$ .

Depends on

Isonormal Gaussian white-noise inverse experiment

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Kretschmann and Werner, Maximum a posteriori testing in statistical inverse problems, §1.1, PDF pp. 1–2 (standard reference, not scraped)
Giné and Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §1.2.2, equations (1.6)–(1.8), PDF pp. 21–22 (standard reference, not scraped)