Alphabeta MathHow 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.
Rational bounds for pi used in finite schedules
Statement
The circle constant satisfies .
Facts & Assumptions
Given: The principal inverse tangent and the usual positive constant .
and (Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series). The positive terms decrease to zero; the alternating-series test places the sum between consecutive partial sums and bounds the error by the first omitted term (The alternating series test: if is nonincreasing with then converges, the sum lies between any two consecutive partial sums, and the error after terms is at most , Every complete ordered field is Archimedean).
The definition identifies as the first positive zero of cosine and fixes the same circle constant used in the arctangent identity (Pi as twice the smallest positive zero of cosine).
Proof
Group the first eight terms of the alternating series in [F1] into four positive pairs. Their sum is , since . The alternating-series remainder after an even partial sum is positive, so and .
For , the integrand is positive. Polynomial division gives . Its integral over is by [F1]. The integral is strictly positive, hence .
Depends on
- Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series
- Pi as twice the smallest positive zero of cosine
- The alternating series test: if $(b_k)$ is nonincreasing with $b_k \to 0$ then $\sum_{k} (-1)^{k} b_k$ converges, the sum lies between any two consecutive partial sums, and the error after $n$ terms is at most $b_n$
- Every complete ordered field is Archimedean
Used by
- Median confidence amplification for amplitude estimation Algorithm
- Relative approximate counting with unknown marked count Algorithm
- Gaussian and Simpson mass assembly give distinct exact and approximate checks Example
- A finite-grid quantum subgradient from approximate values Lemma
- Coherent spectral minimum and a frozen Gibbs energy shift Lemma
- Dyadic positive-mean estimator under an L2 bound Lemma
- Finite exact comparison of real cyclotomic values Lemma
- Relative stability of the HHL filter on supported eigenvalues Lemma
- Second moment of the signed sine-window phase kernel Lemma
- Signed phase decoding has no alias on the promised spectrum Lemma
- Terminating certified rounding for a counting estimate Lemma
- Two nearest phase labels carry probability at least eight over pi squared Lemma
- Dual reflection evaluator has bounded-error correctness Theorem
- Explicit median confidence guarantee for amplitude estimation Theorem
- Quantum bounded-expectation guarantee and query cost Theorem
- Relative approximate counting success guarantee Theorem
- Relative probability estimation with a charged finite cap Theorem
Dependency tree · two levels
35 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)