LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

Deterministic direct search returns within its finite poll budget

Statement

For every integer poll budget $N \geq 0$ , each invocation of Finite-budget deterministic direct search with positive-spanning polling returns after at most $N$ polls with a feasible current center and a best evaluated point.

Facts & Assumptions

Given: An open domain $U$ , finite-valued objective on $U$ , feasible initial point, finite ordered poll list, and the mesh/forcing inputs of Finite-budget deterministic direct search with positive-spanning polling.

[F1]

The initial point is evaluated before polling; each poll scans a finite ordered list, accepts only in-domain points, and otherwise retains its current center (Finite-budget deterministic direct search with positive-spanning polling).

Proof

1.1givenF1induction

The initial center lies in $U$ and has a finite objective value, so the best-point record is initially nonempty. In a poll, each checked direction is either skipped outside $U$ or evaluated at a point in $U$ ; a success moves to that point and an unsuccessful poll keeps the old feasible center. Thus feasibility and a nonempty best record persist.

2.1F1step 1.1induction∎

Each completed poll consumes one unit of the integer budget. After at most $N$ polls the loop returns budget-exhausted; when $N = 0$ it returns immediately after recording the initial point.

Depends on

Finite-budget deterministic direct search with positive-spanning polling

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by Finite-budget deterministic direct search with positive-spanning polling.

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

A. R. Conn, K. Scheinberg, and L. N. Vicente, Introduction to Derivative-Free Optimization, ch. 7 (standard reference, not scraped)
Tamara G. Kolda, Robert Michael Lewis, and Virginia Torczon, Optimization by Direct Search, §3 (standard reference, not scraped)