Integer calendars for fixed-turnover replenishment: a 1/ln 2 guarantee on trees, the price of short cycles and the price of constant gaps

Unrefereed research candidate. A depot must revisit every client at least once within that client's turnover time, keeping the long-run average daily routing cost low. Bosman et al. (Algorithmica, 2022) gave a factor-two approximation on trees and asked whether it can be improved. This package shows that it can: rounding the event times of real-valued power-of-two clocks (with a logarithmically chosen common phase) up to whole days gives a feasible calendar costing at most 1/ln 2 ≈ 1.4427 times the optimum, computed deterministically in O(n log n) time, for any monotone submodular daily cost. The calendar repeats within the longest turnover time, and for such calendars 1/ln 2 is the exact worst-case price; the analysis of the algorithm is tight on trees. Individual gaps may alternate between two integers. If every client must keep a constant gap with gaps dividing one another (the form of Bosman et al.'s output), the best worst-case factor lies between 1.6001 and 1.6014 (explicit four-ladder mixture: 48/29). Also: the exact nested two-edge optimum, divisibility and lcm criteria for attaining the frequency lower bound, a pooled lower bound, the cycle-budget profile of the worst-case family, and certified instances where the optimum exceeds the frequency bound by a factor 1.204. The archive contains the manuscript (PDF, Markdown and LaTeX), exact standard-library verifiers that do not rely on assert, 195 exact optimum certificates, exact certificates for the constant-gap bounds, 13 semantic negative controls, CI and SHA-256 sums. Assurance: written proofs and exact rational certificates; internal AI-assisted review only (not peer review, independent reproduction or formal verification). The rounding tools and the constant are classical and credited. Priority is not claimed: a 2025 DTU MSc thesis with a closely matching title could not be obtained. All instances are synthetic. Creator: Anonymous. Text and data CC0-1.0; code MIT.