ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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 two-vertex deterministic swap has absolute gap zero

Example

Let $X = {0, 1}$ and let

$P = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}), π = (1 / 2, 1 / 2) .$

The chain alternates deterministically between the two states. The absolute spectral gap is defined by Absolute spectral gap of a finite reversible chain on the mean-zero subspace.

Facts & Assumptions

Given: The finite reversible chain above, the period convention Period of a state, and the absolute-gap convention Absolute spectral gap of a finite reversible chain.

[F1]

Finite reversible Markov chain with positive stationary law: a reversible chain has a positive stationary law satisfying row-stationarity and detailed balance.

[F2]

Period of a state: the period is the gcd of the positive return times with positive transition probability.

[F3]

Absolute spectral gap of a finite reversible chain: $δ_{a b s} (P) = 1 - ∥ P ∣_{H_{0}} ∥_{2, π}$ .

[F4]

Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism: eigenvalues have nonzero eigenvectors and form the spectrum.

Verification

1.1F1givenalgebra

Both rows of $P$ sum to one, $π P = π$ , and $π_{0} P_{01} = π_{1} P_{10} = 1 / 2$ ; the diagonal detailed-balance equations are automatic. Hence $(X, P, π)$ is a finite reversible chain.

1.2F2givenalgebra

Starting from either state, a return occurs exactly at positive even times, so the return-time set is ${2, 4, 6, \dots}$ and its gcd is $2$ . The chain is period two.

2.1F3F4step 1.1algebra∎

The constant vector $1 = (1, 1)$ has eigenvalue $1$ , while $h = (1, - 1)$ spans $H_{0}$ and satisfies $P h = - h$ . Thus the restriction to $H_{0}$ has norm $1$ , so [F3] gives $δ_{a b s} (P) = 0$ . The eigenvalues of $P$ are exactly $1$ and $- 1$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

Andrew M. Childs, Lecture Notes on Quantum Algorithms, §18.4 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §§8.1–8.2 (standard reference, not scraped)