ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Fourier transform of delta derivatives and monomials

Example

Assume Countable Choice. In one dimension,

$F (δ_{0}^{'}) = 2 π i ξ, F (x) = - \frac{1}{2 π i} δ_{0}^{'} .$

Facts & Assumptions

Given: Countable Choice and the negative-sign $2 π$ convention.

[F1]

The elementary-transform theorem supplies the derivative and monomial identities in $S^{'}$ with their distributional signs (Fourier transform of delta constants plane waves and polynomials).

Verification

technique · test calculation and the supplied transform table

1.1

Evaluate the transform of $δ_{0}^{'}$ on an arbitrary $φ \in S (R)$ .

$⟨ F δ_{0}^{'}, φ ⟩ = - {\frac{d}{d x} \hat{φ} (x) ∣}_{x = 0} = 2 π i \int_{R} ξ φ (ξ) d ξ .$

Thus $F δ_{0}^{'} = 2 π i ξ$ . The two minus signs are respectively the derivative of delta and the negative Fourier exponential. [F1]

2.1

The monomial clause of [F1], specialized to the one-dimensional multi-index $1$ , directly gives $F (x) = - (2 π i)^{- 1} δ_{0}^{'}$ , which is the second formula. Together with step 1.1 this records both directions of the delta-derivative/monomial pair. Countable Choice is used only through [F1].

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Semyon Dyatlov, Lecture notes for 18.155 (2022) (standard reference, not scraped)