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

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A Cauchy input has no L2 degree-one polynomial chaos

Statement refuted

Every real probability law supports a degree-one orthonormal polynomial chaos basis.

Facts & Assumptions

Given: The Cauchy probability law and the finite moment-Gram requirements of Finite product-law polynomial-chaos input data.

[F1]

Degree one requires the finite second moment and positive-definite monomial Gram matrix of Finite product-law polynomial-chaos input data.

Counterexample

1.1givenF1constructalgebra

Take the standard Cauchy law, whose density is $ρ (x) = 1 / (π (1 + x^{2}))$ on $R$ . Its second moment integral is $\int x^{2} d ρ (x) = π^{- 1} \int_{R} x^{2} / (1 + x^{2}) d x .$ For $∣ x ∣ \geq 1$ , $x^{2} / (1 + x^{2}) \geq 1 / 2$ , so the integral over $[- R, - 1] \cup [1, R]$ is at least $(R - 1) / π$ .

2.1F1step 1.1algebra∎

Letting $R \to \infty$ shows the second moment is infinite. Hence $x \notin L^{2} (ρ)$ , the degree-one monomial Gram entry is not finite, and [F1] fails; the finite Gram--Schmidt construction cannot supply a degree-one chaos basis for this law.

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