Exact Fault-Robust Stretch of Period-Three Temporal Cycles

Research Note No.37, unrefereed public preprint v1.0. Ryutaro Yonezu, Yonezu Research Lab. We classify the exact robust fastest-duration/static-distance stretch of period-three undirected temporal cycles when an adversary may replace the recurring phase of at most one edge. The optimal values are 1 for n=3; 3/2 for n=4; 2 for n=5,7 and n>=6 divisible by 3; and 7/3 for n>=8 not divisible by 3. Duration is last traversal minus first traversal plus one, with strict traversal and free departure. The no-fault case is included. The proof uses a forward/backward path identity, a three-edge obstruction, general periodic constructions and an exceptional seven-cycle word. The mathematical sections are unchanged from the audited v0.1 source. Bundled author-side verification includes complete normalized enumeration n=3..11, an independent time-dependent clock oracle on all normalized words n=3..9, and construction checks through n=36. Fresh Linux/Python 3.12.14 release checks passed; Windows execution is unconfirmed. The external AI reviewer reports independent C++ enumeration through n=16 and constructions through n=39, but those additional raw sources/logs were not supplied. The external AI report gives PROOF/COMPUTATION PASS, REFERENCES PARTIAL, NOVELTY INCONCLUSIVE and PUBLICATION HOLD. Following the report and a scoped primary-literature model comparison, the author explicitly instructed this qualified public release on 24 October 2026. This is not reviewer release approval or journal/conference peer review. No historical novelty or priority is claimed. The residual reference and model audit is included; full journal-level comparison remains incomplete. Files: English paper PDF and evidence ZIP containing LaTeX source, offline Python verifiers, finite evidence, historical audit materials and SHA256 manifests. All rights reserved. Research inquiries and earlier overlapping work: yonezaemon1+research@gmail.com. Repository: https://github.com/yonezaemon1-hub/no37-fault-robust-period-three-temporal-cycles