ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

One finite kolmogorov frequency block

Example

For $P = 1 + e_{1}$ and $Q = e_{5} P$ , the support of Q is ${5, 6}$ , $A_{4} Q = 0$ , $A_{5} Q = e_{5}$ , $A_{6} Q = Q$ , and $∣ Q ∣ = ∣ P ∣$ .

Facts & Assumptions

[F1]

Analytic sums and the finite coefficient convention are fixed Kolmogorov analytic partial sum maximal function.

Verification

Given: $P (x) = 1 + e^{2 π i x}$ and $Q (x) = e^{10 π i x} P (x)$ .

1.1F1

Multiplication gives $Q (x) = e^{10 π i x} + e^{12 π i x} = e_{5} (x) + e_{6} (x)$ . Both coefficients are one and all others zero by the character-integral calculation in F1, so its support is exactly ${5, 6}$ . The cutoff through four is the empty supported sum, through five is $e_{5}$ , and through six is $e_{5} + e_{6} = Q$ .

2.1F1step 1.1∎

Since $∣ e_{5} (x) ∣ = 1$ , $∣ Q (x) ∣ = ∣ 1 + e^{2 π i x} ∣$ . Factoring gives $1 + e^{2 π i x} = 2 e^{π i x} \cos (π x)$ , hence both magnitudes are $2 ∣ \cos (π x) ∣$ . At x=0 they equal two, and at x=1/2 they vanish. This concrete shift changes the frequencies while preserving every pointwise magnitude.

Depends on

Kolmogorov analytic partial sum maximal function

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Grafakos, Classical Fourier Analysis, third edition (standard reference, not scraped)