ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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.

Three-input degree-two chaos uses ten total-degree modes versus twenty-seven tensor modes

Example

For three independent uniform inputs on $[- 1, 1]$ with admissible cutoff $p = (2, 2, 2)$ , a total-degree order-two chaos basis has ten modes, while the full coordinatewise degree-two tensor basis has twenty-seven modes. A tensor Gaussian rule with three nodes in each coordinate also uses twenty-seven node evaluations.

Facts & Assumptions

Given: Three independent uniform inputs on $[- 1, 1]$ , cutoffs $p_{j} = 2$ , total-degree order $s = 2$ , and a three-node Gaussian rule for each marginal.

[F1]

Total-degree and hyperbolic-cross chaos index sets defines the cutoff total-degree set by coordinate sum and the tensor box by coordinatewise cutoffs.

[F2]

The stated chaos index sets are finite and contain zero confirms the specified finite sets are legitimate index sets containing zero.

[F3]

Tensor-product Gaussian quadrature preserves coordinatewise exactness gives exactly $\prod_{j} m_{j}$ distinct nodes for a tensor rule with coordinate orders $m_{j}$ .

Verification

1.1F1F2givenalgebra

For one uniform marginal, the moment Gram matrix through degree two is $(\begin{array}{ccc} 1 & 0 & 1 / 3 \\ 0 & 1 / 3 & 0 \\ 1 / 3 & 0 & 1 / 5 \end{array})$ . Its quadratic form at $(c_{0}, c_{1}, c_{2})$ is $(c_{0} + c_{2} / 3)^{2} + c_{1}^{2} / 3 + 4 c_{2}^{2} / 45$ , which is positive for every nonzero vector; all required moments are finite on $[- 1, 1]$ . Thus the cutoff is admissible. In three coordinates, the total-degree set has one zero multi-index, three degree-one indices, three indices $2 e_{j}$ , and three indices $e_{i} + e_{j}$ with $i < j$ . These are disjoint and exhaust the nonnegative triples with sum at most two, so its cardinality is $1 + 3 + 3 + 3 = 10$ .

2.1F1F3step 1.1algebra∎

The full degree-two tensor box has $3$ choices in each of three coordinates, hence $3^{3} = 27$ multi-indices. The three-node Gaussian rules are defined for the positive uniform measures on $[- 1, 1]$ ; their tensor rule with $m_{1} = m_{2} = m_{3} = 3$ has $3^{3} = 27$ distinct nodes by [F3]. Thus in this example the total-degree basis uses ten modes compared with twenty-seven tensor modes and node evaluations.

Depends on

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