Sharp density thresholds for finite two-visit scheduling
Fabian Fetik · Zenodo (CERN European Organization for Nuclear Research) · 2026
In finite two-visit pinwheel scheduling, each of n labelled tasks with positive integer deadline dᵢ must occur exactly twice in a word of length 2n. Its first occurrence must be among the first dᵢ positions, and the two occurrences must be at most dᵢ positions apart. We prove that every instance of density ∑ᵢ 1/dᵢ ≤ 1 is feasible, including equality. More sharply, for every fixed n ≥ 2, all instances of density at most 1 + 1/[n(n−1)] are feasible except relabellings of (n−1,n,…,n), and this exception is infeasible. A constructive anchor-cluster repair yields a schedule with linear combinatorial work after sorting and an exact density check. The proof combines an ordered reciprocal inequality with a calendar that controls the relative order of the two visits. This record contains the article PDF and a standalone supplementary package with editable sources, a construction algorithm, an independent output-certificate checker, selected exact checks, and reproduction instructions. The article, documentation, example data and figures are licensed under CC BY 4.0; all supplied Python software is licensed under MIT.