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.

Constant BBHT stage success beyond the threshold scale

Statement

Let $0 < a = t / N < 3 / 4$ , $θ = \arcsin \sqrt{a}$ , and let $J$ be uniform on ${1, \dots, M}$ . If the positive integer $M$ satisfies $M \geq 4 / \sqrt{a}$ , then $E [\sin^{2} ((2 J + 1) θ)] \geq \frac{3}{8} .$

Facts & Assumptions

Given: $0 < a < 3 / 4$ , $\sin θ = \sqrt{a}$ , $θ \in (0, π / 2)$ , and an integer $M \geq 4 / \sqrt{a}$ .

[F1]

Exact success probability after an integer number of Grover iterates: after $j$ integer iterates, the marked probability is $\sin^{2} ((2 j + 1) θ)$ .

[F3]

The addition formulas for sine and cosine: sine and cosine obey their addition formulas.

[F4]

Double-angle and quadratic power-reduction identities: $\sin (2 θ) = 2 \sin θ \cos θ$ and $\sin^{2} x = (1 - \cos 2 x) / 2$ .

[F5]

Principal inverse sine and inverse cosine and Signs, monotonicity intervals, and ranges of sine and cosine: the principal angle is in $[- π / 2, π / 2]$ , sine is strictly increasing there, and cosine is positive on $(0, π / 2)$ .

[F6]

Parity and the Pythagorean identity for sine and cosine: $\sin^{2} θ + \cos^{2} θ = 1$ .

[F7]

Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ : nonnegative reals have unique nonnegative square roots.

[F8]

The derivatives of sine and cosine are cosine and minus sine: $\sin 0 = 0$ .

[F9]

Quarter-turn values and shifts by pi/2 and pi: $\sin (π / 2) = 1$ .

[F10]

Squaring is monotone on the nonnegatives: square roots preserve order on nonnegative reals.

Proof

1.1givenF4F5F6F7F8F9F10algebra

Since $0 < a < 3 / 4$ , [F7] and [F10] give $0 < \sqrt{a} < 1$ . From [F4], $\sin θ = \sqrt{a}$ and $θ \in [- π / 2, π / 2]$ . By [F8], $\sin 0 = 0$ , so strict increase [F5] gives $θ > 0$ . Since $\sin (π / 2) = 1$ by [F9], strict increase and $\sin θ < 1$ give $θ < π / 2$ ; then $\cos θ > 0$ by [F5]. The Pythagorean identity [F6] gives $\cos θ = \sqrt{1 - a}$ .

1.2givenF1F4algebra

By [F1] and [F4], the average equals $\frac{1}{2} - \frac{1}{2 M} \sum_{j = 1}^{M} \cos ((4 j + 2) θ)$ .

2.1step 1.2F3algebra

By [F3], for every $j$ , $2 \sin (2 θ) \cos ((4 j + 2) θ) = \sin ((4 j + 4) θ) - \sin (4 j θ)$ . Summing from $j = 1$ to $M$ and using $\sin u - \sin v = 2 \sin ((u - v) / 2) \cos ((u + v) / 2)$ gives $\sum_{j = 1}^{M} \cos ((4 j + 2) θ) = \sin (2 M θ) \cos ((2 M + 4) θ) / \sin (2 θ)$ . Thus the average in step 1.2 is $\frac{1}{2} - \frac{\sin (2 M θ) \cos ((2 M + 4) θ)}{2 M \sin (2 θ)}$ .

3.1givenstep 1.1step 2.1F4F5F6F7F10algebra∎

Since $a < 3 / 4$ , $1 - a > 1 / 4$ , so [F10] gives $\sqrt{1 - a} > 1 / 2$ . Steps 1.1 and [F4]–[F7] then give $\sin (2 θ) = 2 \sqrt{a} \sqrt{1 - a} > \sqrt{a}$ . Therefore $M \sin (2 θ) > 4$ and the absolute value of the correction in step 2.1 is less than $1 / 8$ . The average is greater than $3 / 8$ , hence at least $3 / 8$ .

Depends on

Used by

Dependency tree · two levels

45 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

Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp, Quantum Amplitude Amplification and Estimation, §2 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §7.2 (standard reference, not scraped)