ExampleConstruction: AI-generatedVerification: 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.

Counting bounds at zero, one, half, and all marked

Example

Take $N = 16$ , $M = 4$ , and $k = 1$ . Evaluate the generic raw count error expression at $t = 0, 1, 8, 16$ and compare it with the endpoint certainty claims.

Facts & Assumptions

Given: $N = 16$ , $M = 4$ , $k = 1$ , and counts $t \in {0, 1, 8, 16}$ .

[F1]

The raw count error expression is $2 π k \sqrt{t (N - t)} / M + π^{2} k^{2} N / M^{2}$ (Quantum counting error and boundary cases).

[F2]

Counts $t = 0$ and $t = N$ are returned exactly when $M$ is even (Quantum counting error and boundary cases).

Verification

1.1givenF1algebra

Substituting $N = 16, M = 4, k = 1$ into [F1] gives the generic expression $E (t) = \frac{π}{2} \sqrt{t (16 - t)} + π^{2}$ .

2.1givenF1F2step 1.1algebra

At $t = 0$ , this expression is $π^{2}$ but [F2] sharpens the actual error to zero. At $t = 1$ , it is $\frac{π \sqrt{15}}{2} + π^{2}$ . At $t = 8$ , it is $4 π + π^{2}$ . At $t = 16$ , the expression again equals $π^{2}$ , while [F2] gives exact output $16$ .

3.1F1F2step 2.1∎

The first-order term is zero at both endpoints and largest here at the half-count $t = 8$ ; the second-order uniform term alone does not encode the exact endpoint behavior. All four substitutions agree with the theorem, and the endpoint cases use its explicit certainty clause.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Brassard, Høyer, Mosca, and Tapp, Quantum Amplitude Amplification and Estimation, complete article, §§4–5, Theorems 11–18 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, complete lecture notes, §§19.1–19.3 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, complete 239-page notes, §7.3 (standard reference, not scraped)