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 Parseval identity

Statement

For $f, g : G \to C$ , $\sum_{χ \in \hat{G}} (F_{G} f) (χ) \overline{(F_{G} g) (χ)} = \sum_{x \in G} f (x) \overline{g (x)} .$ For a finite set $X$ , the counting inner product on $C^{X}$ is $⟨ f, g ⟩_{X} : = \sum_{x \in X} f (x) \overline{g (x)}$ ; it is linear in the first argument. The displayed identity says this inner product is preserved.

Facts & Assumptions

Given: A finite abelian group $G$ and functions $f, g : G \to C$ .

[F1]

The transform matrix $U$ satisfies $U^{*} U = I_{G}$ (Finite abelian Fourier transform is unitary).

[F2]

The transform is represented by the matrix entries $U_{χ, x} = N_{R}^{- 1 / 2} \overline{χ (x)}$ (Normalized Fourier transform on a finite abelian group).

Proof

1.1F1F2algebra

Expanding the left side in point-mass coordinates using [F2] gives $\sum_{χ} (\sum_{x} U_{χ, x} f (x)) \overline{(\sum_{y} U_{χ, y} g (y))} = \sum_{x, y} f (x) \overline{g (y)} (\sum_{χ} U_{χ, x} \overline{U_{χ, y}})$ . The inner finite sum is the conjugate of the $(x, y)$ entry of $U^{*} U$ , so [F1] makes it $1$ if $x = y$ and $0$ otherwise.

2.1step 1.1∎

The diagonal terms remaining from step 1.1 are exactly $\sum_{x} f (x) \overline{g (x)}$ . Taking $g = f$ gives preservation of the squared counting norm as well.

Depends on

Used by

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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)