TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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.

Finite abelian Fourier inversion

Statement

For every $f : G \to C$ and $x \in G$ , $f (x) = N_{R}^{- 1 / 2} \sum_{χ \in \hat{G}} (F_{G} f) (χ) χ (x), N_{R} : = ι_{R} (∣ G ∣) .$ Thus the inverse uses the positive character rather than its conjugate.

Facts & Assumptions

Given: A finite abelian group $G$ , the transform $F_{G}$ , and a function $f : G \to C$ .

[F1]

The coordinate matrix $U$ of $F_{G}$ satisfies $U^{*} U = I_{G}$ and $U U^{*} = I_{\hat{G}}$ (Finite abelian Fourier transform is unitary).

[F2]

Its entries are $U_{χ, x} = N_{R}^{- 1 / 2} \overline{χ (x)}$ (Normalized Fourier transform on a finite abelian group).

Proof

1.1F1given

Write $F_{G} f = U f$ in the point-mass coordinate bases. From [F1], $U^{*}$ is a two-sided inverse of $U$ , so $f = U^{*} (F_{G} f)$ .

2.1F2step 1.1algebra∎

The $x$ coordinate of $U^{*} (F_{G} f)$ is $\sum_{χ \in \hat{G}} \overline{U_{χ, x}} (F_{G} f) (χ)$ . By [F2], $\overline{U_{χ, x}} = N_{R}^{- 1 / 2} χ (x)$ , which gives exactly the displayed formula.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

Ronald de Wolf, Quantum Computing: Lecture Notes, version 6 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)
John Preskill, Quantum Information, Chapter 6: Quantum Algorithms (standard reference, not scraped)