ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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 Riesz product for three powers of three

Example

For unimodular $ε_{1}, ε_{3}, ε_{9}$ , the product

$P (x) = \prod_{n \in {1, 3, 9}} (1 + Re (ε_{n} e_{n} (x)))$

is a nonnegative mass-one trigonometric polynomial, with $\hat{P} (- n) = \overline{ε_{n}} / 2$ for $n = 1, 3, 9$ .

Facts & Assumptions

Given: Three complex numbers $ε_{1}, ε_{3}, ε_{9}$ of modulus one.

Verification

technique · expand the three ratio-three factors

1.1

Each factor equals $∣ 1 + ε_{n} e_{n} ∣^{2} / 2$ , so $P \geq 0$ . In an [given, algebra] expansion, each chosen frequency is a signed sum of $1, 3, 9$ with coefficients in ${- 1, 0, 1}$ .

givenalgebra

2.1

Such a signed sum is zero only when all three coefficients are zero: [step 1.1, algebra] the largest nonzero term has magnitude larger than the sum of all smaller ones. Thus the constant coefficient is $1$ , so $\int_{0}^{1} P = 1$ .

step 1.1algebra

3.1

The same observation shows that frequency $- n$ has only the first-order [step 2.1, algebra] contribution $\overline{ε_{n}} e_{- n} / 2$ . Therefore $\hat{P} (- n) = \overline{ε_{n}} / 2$ , agreeing with Riesz-product witnesses for a Hadamard-lacunary set.

step 2.1algebra∎

Depends on

Riesz-product witnesses for a Hadamard-lacunary set

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Loukas Grafakos, Classical Fourier Analysis, 3rd ed., Definition 3.6.5 (standard reference, not scraped)