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

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 Ishigami function's only possible interaction is X1-X3

Example

Let $X_{1}, X_{2}, X_{3}$ be independent uniform random variables on $[- π, π]$ and let $f (X) = \sin X_{1} + a \sin^{2} X_{2} + b X_{3}^{4} \sin X_{1},$ where $a, b \in R$ . Its only possibly nonzero nonconstant ANOVA components are $f_{{1}}$ , $f_{{2}}$ , and the interaction $f_{{1, 3}}$ . For every $a, b$ , the output variance is positive and $V = \frac{1}{2} + \frac{a^{2}}{8} + \frac{b π^{4}}{5} + \frac{b^{2} π^{8}}{18} .$

Facts & Assumptions

Given: The independent uniform inputs, real parameters $a, b$ , and $f$ in the Example.

[F1]

Product-law Hoeffding ANOVA components: components are finite alternating sums of coordinate averages.

[F2]

Independent product inputs have an orthogonal Hoeffding decomposition: the components reconstruct $f$ , are orthogonal, and their squared norms sum to the variance.

[F3]

First-order and total Sobol indices: first-order and total indices divide their variance-component numerators by positive $V$ .

[F4]

Independent random elements and Independent random elements have product joint law: the input vector has the product joint law.

[F5]

Fubini's theorem for L^1 functions on a sigma-finite product: integrable product functions may be integrated coordinatewise.

[F6]

Moments, variance, and covariance on a probability space: variance is the expected squared deviation from the mean.

Verification

Given: The model and product law in the Example.

1.1F4F5F6givenalgebra

The inputs are independent by assumption [F4], so Fubini [F5] factors products of coordinate functions. Symmetry gives $E \sin X_{1} = 0$ , $E \sin^{2} X_{1} = 1 / 2$ , and $E \sin^{4} X_{1} = 3 / 8$ . Direct integration gives $E X_{3}^{4} = π^{4} / 5$ and $E X_{3}^{8} = π^{8} / 9$ . Also $E \sin^{2} X_{2} = 1 / 2$ and $Var (\sin^{2} X_{2}) = 3 / 8 - (1 / 2)^{2} = 1 / 8$ by [F6].

2.1F1step 1.1givenalgebra

Set $m_{4} = π^{4} / 5$ . Integrating the defining function over complementary independent coordinates gives $f_{{1}} = (1 + b m_{4}) \sin X_{1}$ , $f_{{2}} = a (\sin^{2} X_{2} - 1 / 2)$ , and $f_{{1, 3}} = b \sin X_{1} (X_{3}^{4} - m_{4})$ . Integrating any other nonempty subset's alternating projection sum gives zero, since the function is a sum of these three terms and each displayed factor is centered in every coordinate in its support. This is exactly the component convention in [F1].

3.1F2F6step 1.1step 2.1algebra

The component variances are $V_{1} = (1 + b m_{4})^{2} / 2$ , $V_{2} = a^{2} / 8$ , and $V_{13} = \frac{b^{2}}{2} (π^{8} / 9 - π^{8} / 25) = 8 b^{2} π^{8} / 225$ by [F6]. They are mutually orthogonal by [F2]. Adding them yields $V = 1 / 2 + a^{2} / 8 + b π^{4} / 5 + b^{2} π^{8} / 18$ . This variance is positive: if $b \neq 0$ then $V_{13} > 0$ , and if $b = 0$ then $V_{1} = 1 / 2$ ; in either case $V_{2} \geq 0$ .

4.1F2F3step 3.1∎

By [F3], $S_{1} = V_{1} / V$ , $S_{2} = V_{2} / V$ , and $S_{3} = 0$ , while $S_{T_{1}} = (V_{1} + V_{13}) / V$ , $S_{T_{2}} = V_{2} / V$ , and $S_{T_{3}} = V_{13} / V$ . Thus the interaction contributes to the total effects of inputs 1 and 3 but not to either first-order effect.

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Sources

Thierry Klein, Introduction to Sensitivity Analysis: The Statistical Study of Sobol Indices, complete lecture notes, §2 (standard reference, not scraped)
Art B. Owen, Variance Components and Generalized Sobol' Indices, §2.1 (standard reference, not scraped)