LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

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The stated chaos index sets are finite and contain zero

Statement

For every admissible cutoff vector $p \in N_{0}^{d}$ , order $s \geq 0$ , and threshold $N \geq 1$ , the sets $Λ_{s, p}^{T D}$ and $Λ_{N, p}^{H C}$ of Total-degree and hyperbolic-cross chaos index sets are finite and contain $0$ .

Facts & Assumptions

Given: $d \geq 1$ , an admissible cutoff vector $p \in N_{0}^{d}$ , $s \geq 0$ , and $N \geq 1$ , with the two index-set definitions above.

[F1]

Total-degree and hyperbolic-cross chaos index sets defines the sets by coordinate sums and products of positive integers.

Proof

1.1givenF1algebra

If $α \in Λ_{s, p}^{T D}$ , then every coordinate satisfies $0 \leq α_{j} \leq p_{j}$ by the cutoff condition in [F1]. Hence this set is a subset of the finite box $\prod_{j} {0, \dots, p_{j}}$ .

2.1givenF1step 1.1algebra∎

The same cutoff condition places $Λ_{N, p}^{H C}$ in the finite box $\prod_{j} {0, \dots, p_{j}}$ . Since $0 \leq s$ and $1 \leq N$ , zero satisfies both defining inequalities. Thus both sets are finite and contain zero.

Depends on

Total-degree and hyperbolic-cross chaos index sets

Used by

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

Cited to discharge well-definedness by Total-degree and hyperbolic-cross chaos index sets.

Dependency tree · two levels

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Sources

Bruno Sudret, Polynomial Chaos Expansions and Stochastic Finite Element Methods, §3.3.2 (standard reference, not scraped)
Dongbin Xiu, Fast Numerical Methods for Stochastic Computations, §3 (standard reference, not scraped)