ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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 $Q (x) = e^{10 π i x} P (x)$ .

1.1

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 $e_{5} + e_{6} = Q$ .

2.1

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.

F1step 1.1∎

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)