Alphabeta Math
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 a plane wave

Example

Assume Countable Choice. For b∈Rn, the positive-frequency plane wave satisfies

F(e2πib⋅x)=δb

in S′(Rn).

Facts & Assumptions

Given: Countable Choice and b∈Rn.

[F1]

The elementary-transform theorem gives Fδa=e−2πia⋅ξ and F(e2πib⋅x)=δb (Fourier transform of delta constants plane waves and polynomials).

Verification

technique · check the sign against a translated delta and apply the supplied plane-wave formula
1.1

Set a=−b in [F1]. Then Fδ−b=e2πib⋅ξ; the positive exponential sign therefore corresponds to a delta initially placed at −b.

F1
2.1

The plane-wave clause of [F1] directly gives F(e2πib⋅x)=δb. Step 1.1 checks the sign by locating the pre-transform delta at −b. For b=0, the formula reduces to F1=δ0. 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