Exact Parallel Propagation and Throttling with One Leak on a Cycle

Research Note No.41. Unrefereed, AI-assisted public preprint v1.0, 26 October 2026. Ryutaro Yonezu, Yonezu Research Lab. For standard zero forcing on a simple undirected cycle C_n (n >= 3), one fixed arbitrary vertex leak is chosen after the initial blue set; it cannot force but can become blue. Updates are synchronous. The note gives an exact worst parallel propagation time: retain maximal initial blue runs of length at least two, and count initially white vertices on each intervening arc, including any intervening isolated blue vertices in the arc. For a non-full set containing a blue edge, the worst time equals the maximum such white count. It is infinite when there is no initial blue edge, and zero for the full blue set. The proof uses known robustness of the two endpoints of a path plus elementary supplementary observations O1/O2. The fixed-budget optimum, minimum initial budget at a deadline, and additive throttling minimum ceil(sqrt(8n)) - 2 follow arithmetically. The contribution is an explicit timing formula with an elementary reproof. No new forcing principle, historical priority, or first-discovery claim is made. Historical NOVELTY remains INCONCLUSIVE and REFERENCES remains PARTIAL. Stored author-side exhaustive verification covers n=3..16: 131,064 initial sets, 1,966,072 leak scenarios, 147 budget and 147 deadline entries, no mismatches. Supplied self-audit reports describe a separate C implementation through n=22, but that source and its raw logs were not supplied or authenticated. Repeated self-audits are not independent third-party or human peer reviews. No new dynamics run was performed for publication, and Windows execution is unconfirmed. Files: six-page English PDF and companion evidence ZIP with matching LaTeX, the complete byte-preserved v0.4 audit snapshot, offline standard-library Python verifier, evidence, source comparisons, audit history and SHA-256 manifests. Historical HOLD labels in the frozen snapshot describe the preparation stage; the author subsequently authorized this public preprint. This does not upgrade novelty or reference status. Copyright (c) 2026 Ryutaro Yonezu / Yonezu Research Lab. All rights reserved. No general reuse or redistribution license is granted. Research inquiries and earlier overlapping work: yonezaemon1+research@gmail.comRepository and release: https://github.com/yonezaemon1-hub/no41-one-leak-cycle-propagation/releases/tag/v1.0