CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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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.

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Finite support can make a higher-degree moment Gram matrix singular

Statement refuted

Finite moments through degree $2 p$ guarantee a positive-definite degree- $p$ monomial Gram matrix for every probability law.

Facts & Assumptions

Given: The two-point probability law and the degree-two Gram criterion of Finite product-law polynomial-chaos input data.

[F1]

The degree-two Gram matrix is positive definite exactly when every nonzero coefficient vector has strictly positive squared polynomial norm (Finite product-law polynomial-chaos input data).

Counterexample

1.1givenF1algebra

Let $ρ = (δ_{- 1} + δ_{1}) / 2$ and take $p = 2$ . All moments are finite. The coefficient vector of $q (x) = x^{2} - 1$ in the ordered monomials $(1, x, x^{2})$ is $(- 1, 0, 1) \neq 0$ . Yet $q (- 1) = q (1) = 0$ , so $\int q (x)^{2} d ρ (x) = 0$ .

2.1F1step 1.1algebra∎

Therefore the degree-two Gram quadratic form vanishes on a nonzero vector and is singular despite finiteness of every moment. The stated sufficiency claim is false.

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