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.

A constant output has no normalized Sobol indices

Example

On any finite product probability space, let $f (X) = c$ be constant. Then every nonempty ANOVA component vanishes and $Var (f) = 0$ . The normalized Sobol indices are undefined by convention; setting them to zero would introduce an additional convention not present in their definition.

Facts & Assumptions

Given: A finite product probability law and a constant real output $f = c$ .

[F1]

Independent product inputs have an orthogonal Hoeffding decomposition: the ANOVA components are orthogonal and sum to $f$ .

[F2]

First-order and total Sobol indices: normalized indices are defined only when $V = Var (f) > 0$ .

[F3]

Product-law Hoeffding ANOVA components and Product-coordinate averaging operators: components are finite alternating sums of the product-law coordinate averages $P_{J}$ .

Verification

Given: The constant-output model in the Example.

1.1F1F3givenalgebra

For every $J \subseteq [d]$ , the coordinate average [F3] gives $P_{J} f = c$ . The finite alternating sum in [F3] for a nonempty component therefore has coefficient sum $(1 - 1)^{∣ u ∣} = 0$ , so $f_{u} = 0$ for $u \neq \emptyset$ .

2.1F2step 1.1algebra∎

The variance is $E [(c - c)^{2}] = 0$ , and all nonempty component variances vanish. Since the denominator condition in [F2] fails, neither $S_{i}$ nor $S_{T_{i}}$ is defined.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Thierry Klein, Introduction to Sensitivity Analysis: The Statistical Study of Sobol Indices, complete lecture notes, Definition 2.1 (standard reference, not scraped)