A Sharp Preperiod and Explicit Nim-Words for the Four-Move Subtraction Sets {3,7,9,c}
Year: 2026
Type: preprint
Abstract: Research No.63. Ryutaro Yonezu, Independent Researcher. Preprint v1.0, dated 2026-11-11. For the normal-play one-heap subtraction game with move set {3,7,9,c}, where c>9 is odd, this limited mathematical preprint determines the complete Sprague-Grundy nim-word and the least preperiod and eventual period. For c=13 these are (0,16); for every other admissible c they are (c+5,2). An explicit block formula and induction handle all odd c>=23, and six finite recurrence certificates handle c=11,13,15,17,19,21. No result for even c, general a>3, or arbitrary subtraction sets is asserted. The period-two phenomenon was already reported in Zhang (2024), Remark 7, as a finite computational observation. It is not claimed as a new discovery here. The prospective contribution is the proof for this restricted infinite family, the exact least preperiod and the explicit Grundy words. Four-move nim-word construction as a method is known from Flammenkamp. Relevant passages of the original dissertation were compared, but the enlarged book remains unverified. Moriwaki Theorem 2 and Theorem 4 have distinct assumptions; the stated direct constructions do not instantiate to this all-odd target. Indirect implications have not been exhaustively excluded. Targeted full-text reading of Inoue-Inazu did not identify a direct predecessor. AI-assisted research. MODEL/PROOF/COMPUTATION/SCOPE PASS and the subsequent v0.4 differential PASS are inherited within the author-supplied AI-review reports and follow-up evidence. These are not institutional certificates. Author-side primary-source comparisons are separately attributed from reviewer checks. Windows PowerShell 5.1 finite reproduction PASS is inherited from the author-provided sanitized summary for unchanged code. All three frozen verifiers were rerun on Linux during publication preparation, reproducing Python 490 cases and 1,286,250 positions, large C++ 3,406,220 positions, and small C++ 230 certificate positions. A new v1.0 Windows execution was not performed. The separately reported 996-parameter AI recurrence enumeration was not rerun here. Finite checks do not replace the infinite proof. NOVELTY INCONCLUSIVE. REFERENCES PARTIAL. Human peer review and full formal verification: NOT PERFORMED. Copyright (c) 2026 Ryutaro Yonezu. All rights reserved. The mathematical TeX range is byte-identical to v0.4: 6656 bytes; SHA256 3314bc24a76be60369880f1ce03bd69cb3c5845e59406b23b17cad656aa05431. All three verifier sources are unchanged. All ten prepublication gates passed. The accompanying PDF, TeX source, reproducibility ZIP and SHA256 manifest correspond to GitHub Release v1.0. The PDF has four pages. The archive includes portable instructions, unchanged sources, certificates, clean expected output, provenance and scoped review/Windows summaries. No raw Windows logs, personal workstation paths, private account information, credentials, binaries, or third-party copyrighted papers are redistributed. GitHub repository: https://github.com/yonezaemon1-hub/no63-four-move-subtraction-sharp-preperiodGitHub Release v1.0: https://github.com/yonezaemon1-hub/no63-four-move-subtraction-sharp-preperiod/releases/tag/v1.0
Author: Yonezu, Ryutaro
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225780286
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