Complete Nim-Words and Sharp Preperiods for the Four-Move Subtraction Family {5,11,15,c}
Year: 2026
Type: preprint
Abstract: AI-assisted mathematical preprint, No.64, version 1.0. Ryutaro Yonezu — Independent Researcher. Published 11 November 2026 (JST). For the normal-play one-pile subtraction family S_c={5,11,15,c}, odd c>15, this paper classifies the complete Sprague–Grundy sequences and sharp eventual periods and preperiods. For every odd c>=59, a uniform explicit nim-word has least T=c+29 and least p=2. The remaining 21 odd c=17..57 are handled by finite recurrence certificates. The general all-odd-a quadratic formula T=c+2a^2-4a-1 is an unproved computational conjecture. External independent AI audit: MODEL/PROOF/COMPUTATION/MINIMALITY/PACKAGE PASS within the reviewed scope, reported in author-supplied review text. Reviewer raw code and logs were not provided or independently rerun by the publication editor. Author-run Windows PowerShell 5.1 reproduction PASS on canonical frozen v0.1; not independently witnessed. No new Windows execution is claimed for the editorial v1.0 package. Linux Python/C++17 rerun and regenerated CSV byte checks PASS during publication preparation. Finite numerical tests supplement the analytical infinite-family proof. NOVELTY INCONCLUSIVE. REFERENCES PARTIAL. Human peer review NOT PERFORMED. Formal proof-assistant verification NOT PERFORMED. All rights reserved. Known prior results: Ho (2015), Theorem 5, gives the base triple {5,11,15} with T=44,p=2. Zhang (2024), Remark 7, previously reports bounded observations, including c=21 and c=29. Those results are not claimed as new. Relevant portions of Flammenkamp’s original dissertation were compared in the reported audit; the expanded 1997 book ISBN 3-932136-10-1 remains unread. Moriwaki (2026), Theorems 2 and 4, fail as direct applications under the compared conditions; no exhaustive exclusion of related work is claimed. Public assets: preprint PDF, corresponding LaTeX, reproducibility ZIP, README and SHA256 checksums. Mathematical core, frozen Python/C++17/PowerShell sources and finite CSV certificates are unchanged. No raw Windows log, local absolute path, private audit archive or third-party full text is redistributed. GitHub repository: https://github.com/yonezaemon1-hub/no64-four-move-subtraction-nimwordsGitHub Release v1.0: https://github.com/yonezaemon1-hub/no64-four-move-subtraction-nimwords/releases/tag/v1.0 Copyright (c) 2026 Ryutaro Yonezu. All rights reserved.
Author: Yonezu, Ryutaro
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225835700
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