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.
Finite reversible Markov chain with positive stationary law
Definition
A finite reversible Markov chain is a finite row-stochastic kernel together with a supplied probability vector such that for every , , , and
The stationary law and detailed balance are explicit input hypotheses; the definition does not invoke a theorem that produces a stationary law. No irreducibility or aperiodicity is assumed. In particular, a periodic reversible chain is included.
Depends on
Used by
- Absolute spectral gap of a finite reversible chain Definition
- Marked sets, stationary mass, and walk-search tasks Definition
- The two-vertex deterministic swap has absolute gap zero Example
- Two-state reversible walk and its endpoint phases Example
- A finite reversible transition operator is self-adjoint on weighted L2 Lemma
- Finite stationary Markov operators contract in weighted L2 Lemma
- Spectrum of the lazy Johnson walk Lemma
- Absolute gap equals one minus the largest nonconstant eigenvalue modulus Theorem
- The Szegedy discriminant is self-adjoint and contractive Theorem
Dependency tree · two levels
8 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, Chapter 18 (standard reference, not scraped)
- Ronald de Wolf, Quantum Computing: Lecture Notes, Chapter 8 (standard reference, not scraped)