ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

The Dirichlet kernel at zero and away from zero

Example

For each $N \geq 1$ ,

$D_{N} (0) = 2 N + 1,$

while

$D_{N} ⁣ (\frac{1}{4 N + 2}) = \frac{1}{\sin (π / (4 N + 2))} > 0, D_{N} ⁣ (\frac{3}{4 N + 2}) = - \frac{1}{\sin (3 π / (4 N + 2))} < 0.$

So the Dirichlet kernel has a removable peak at $0$ and already changes sign at two explicit nearby points.

Facts & Assumptions

Given: An integer $N \geq 1$ .

[L1]

The Dirichlet kernel is $D_{N} (t) = \sum_{∣ k ∣ \leq N} e_{k} (t)$ (Dirichlet and Fejer kernels).

[L2]

For $t \notin Z$ , $D_{N} (t) = \frac{\sin ((2 N + 1) π t)}{\sin (π t)},$ and $D_{N} (0) = 2 N + 1$ (Closed form and size bounds for the Dirichlet kernel).

Verification

technique · direct

1.1

At $t = 0$ , [L1] gives $D_{N} (0) = \sum_{∣ k ∣ \leq N} e_{k} (0) = \sum_{∣ k ∣ \leq N} 1 = 2 N + 1,$ which agrees with the removable value recorded in [L2].

L1L2algebra

2.1

The numbers $(4 N + 2)^{- 1}$ and $3 (4 N + 2)^{- 1}$ are not integers, so [L2] applies. Since $(2 N + 1) π / (4 N + 2) = π / 2$ and $3 (2 N + 1) π / (4 N + 2) = 3 π / 2$ , the numerator is respectively $1$ and $- 1$ . The denominators are positive because their angles lie in $(0, π)$ . Hence the two displayed signs follow.

L2algebra∎

Depends on

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

Richard S. Laugesen, Harmonic Analysis Lecture Notes (standard reference, not scraped)
Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (standard reference, not scraped)