LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-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.

The reciprocal filter defines valid three-outcome amplitudes

Statement

For every real $z$ and $κ \geq 1$ , $f_{κ} (z)^{2} + g_{κ} (z)^{2} \leq 1 / 4$ . Hence $h (z)$ in Odd-well and even-ill reciprocal filter is real, and the three displayed amplitudes have squared moduli summing to one.

Facts & Assumptions

Given: The piecewise functions in Odd-well and even-ill reciprocal filter.

[F1]

Odd-well and even-ill reciprocal filter and Parity and the Pythagorean identity for sine and cosine: in the transition band the two nonzero functions are one half times sine and cosine of the same angle; on each plateau they are constants or one reciprocal term.

Proof

1.1F1algebra

If $∣ z ∣ \geq 1 / κ$ , then $g_{κ} (z) = 0$ and $∣ f_{κ} (z) ∣ = 1 / (2 κ ∣ z ∣) \leq 1 / 2$ , so their squared sum is at most $1 / 4$ .

2.1F1algebra∎

If $1 / (2 κ) \leq ∣ z ∣ < 1 / κ$ , the identity $\sin^{2} θ + \cos^{2} θ = 1$ gives $f_{κ} (z)^{2} + g_{κ} (z)^{2} = 1 / 4$ . If $∣ z ∣ < 1 / (2 κ)$ , the sum is $0 + (1 / 2)^{2} = 1 / 4$ . Thus all cases give the bound, and $1 - f^{2} - g^{2} \geq 3 / 4 > 0$ .

Depends on

Used by

Dependency tree · two levels

7 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

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, Quantum algorithm for linear systems of equations (standard reference, not scraped)
Andrew M. Childs, Robin Kothari, and Rolando D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)