DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Circular phase distance modulo one

Definition

For a real $x \in [- 1, 1]$ , define its circular distance to the nearest integer by $d_{T} (x) : = \min {∣ x ∣, 1 - ∣ x ∣} .$ Here $∣ x ∣$ is the real absolute value and $\min$ is the minimum of the displayed finite set (Order on the reals, Maximum and minimum of a set); existence of this finite minimum follows from Every nonempty finite set of reals has a maximum and a minimum. Both entries lie in $[0, 1]$ , so the value lies in $[0, 1 / 2]$ . If $∣ x ∣ \leq 1 / 2$ , the nearest integer is $0$ ; if $x > 1 / 2$ , it is $1$ ; if $x < - 1 / 2$ , it is $- 1$ , so the formula is the distance to the nearest integer. For phases $θ, ϕ \in [0, 1)$ , their circular distance is $d_{T} (θ - ϕ)$ ; the difference belongs to $(- 1, 1)$ , the stated domain.

Depends on

Used by

Nearest-grid and tail estimates in phase estimation Example
Circular-distance tail bound in phase estimation Lemma
Nearest-grid probability in phase estimation Lemma
Amplitude-estimation additive error and success probability Theorem
Correctness of phase-sample rational reconstruction Theorem
Finite reversible-walk finding cost Theorem
Phase-estimation nearest-grid and tail bounds Theorem
Walk detection correctness and cost Theorem

Dependency tree · two levels

9 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

Lin Lin, Lecture Notes on Quantum Algorithms for Scientific Computation, §§16.1–16.3 (standard reference, not scraped)