Proper Hat Guessing on Friendship Graphs: The Hourglass Value and the Three-Petal Threshold

This unrefereed, AI-assisted preprint studies proper hat guessing on friendship graphs F_r = K_1 ∨ (r K_2), with arbitrary deterministic vertex-specific local strategies and one simultaneous guess per vertex. It gives HG_P(F_r) = 5 for r = 1, 6 for r = 2, and 7 for every r ≥ 3. The triangle value five is already known. The claimed finite contribution is the exact hourglass value six and the sharp three-petal threshold for attaining seven colors in the established model. The manuscript proves a pair-cover completion criterion, an eight-color exclusion for every number of petals, and a seven-color single-petal degree bound that excludes the hourglass. Explicit cyclic tables provide the lower constructions. Fresh standard-library checks enumerate 60, 2,400, and 189,000 proper colorings with zero failures; a separate bitmask checker has zero ambiguous views and detects both negative controls. The finite computations verify the constructions; the universal upper bounds rely on the mathematical arguments. Assessment: PROOF PASS (conditional, in a user-supplied external-AI differential report); COMPUTATION PASS for the checked constructions; REFERENCES PARTIAL; historical NOVELTY INCONCLUSIVE. No human peer review, formal proof-assistant certification, priority, or first-discovery claim is made. The external reviewer’s raw program and solver logs were not supplied. The originating assistant’s reference checks are not presented as independent human checks. No Windows execution on the author’s machine is claimed. The author authorized public dissemination on 23 October 2026 as a transparently qualified preprint. The five-page PDF typesets the fixed v0.2 mathematical body with updated publication-status notes. The evidence ZIP preserves all 41 original audit files and the original No33_Audit_v0_2.zip byte for byte, alongside the PDF and source, the received differential-audit record, a publication-status addendum, rights notice and SHA-256 manifest. Historical HOLD and pending labels in those originals describe the preparation stage; the later authorization changes dissemination status without upgrading REFERENCES or NOVELTY. GitHub: https://github.com/yonezaemon1-hub/no33-proper-hat-guessing-friendship-graphs Research Inquiries — Yonezu Research LabContact: yonezaemon1+research@gmail.com © 2026 Ryutaro Yonezu / Yonezu Research Lab. All rights reserved.