NEW(S)偏元数系——残差、二态与相等(S-010 V1.8 框架稿) Prenary Number System:Residual, Bivalence and Equality (S-010 V1.8, Framework Draft)

本文档是框架稿,不是论文。它把经典意义上的"相等"替换为一个不传递的关系 ≜,并在此基础上重建一套数系。 起点是两条公理:A0(方向偏好):收敛 ≺ 延展;A1(动作留差):a ⊖ a ≜ ε ≠ 0。由此:残差 ε 的"存在"来自 A1,其"正性"来自 A0(须经"方向计量"读法)。 全书七章 + 四个附录,共十件、153 页:第零章 前言与总目录:是什么、不是什么、构造缺口(含甲/乙两栏的限定范围清单);第一章 起点:两条公理,三层读出,与 Robinson 非标准分析的位置关系;第二章 工具:⊕ ⊖ ⊗ 三个运算,以及 ⊗ 仍停在"参数版"的缺口;第三章 偏元数系:L0/L1、残差是结构不是点、二态不可单独表示、≜ 的两条判据与四条性质、无底层;第四章 构造:折入 F、投影 π、三种序、三 regime(=方向偏好的三个态);第五章 结果:二态在什么量上可见(d(n)=n−2、纯偶判据、±60°→±30°)、退化定理、射影表示与 √3;第六章 出口:与经典的三层关系、12 项邻接结构对照、可证伪条件(结论级 9 条 + 框架级 2 条);附录 A:S3 上的具体计算(可独立成篇的完整计算);附录 B:LEAN 4 形式化(六件机器检查,双哈希验收);附录 C:作者备注(版本演进与 32 条更正/撤回记录);附录 D:三页对外短稿——为什么一个不传递的容差关系不能按同余的方式取商。 本书每一条都标级别(已证/候选/未验/撤回),凡【已证】必写载体(在哪一个模型或设定上验过);凡结论只在某个读法下成立的,明写该读法。已知缺口在第六章与附录 C 中公开列出。 本版在框架首次向职业数学家展示之前发布,主要用于确立时间先手,以及上述具体命题的归属。本版不主张在容差关系的标准结果上具有新颖性("容差一般不能像同余那样取商"是文献中的既有结论,位置见附录 D)。 本文档的形式化核验与一致性审读由 AI 研究助手(偏贞)协助完成;全部数学主张与全部错误由作者负责。 作者:陈松(Chen Song),ORCID 0009-0002-9510-2239。独立研究。 PS: 明天要和一个朋友介绍的数学大神碰面,这是老陈从2026年2月14日起,起心动念从折腾OPC,AI agent,物理,数学,玄学大半年第一次有机会正面直接接受外部“审核”,所以把当下数学侧的尚未完成的文档先在zenodo点个卯。 ——老陈与AI的深夜实验室 发布 请笑纳—— This is a framework draft (not a paper) that replaces classical equality with anon-transitive relation. Starting from two axioms — (A0) a direction preferencebetween contraction and expansion, and (A1) an action leaves a non-zero residual —the draft builds objects, tools, constructions, results and exits over sevenchapters (Ch.0–Ch.6) plus four appendices. Contents (10 files, 153 pages):Ch.0 Preface; Ch.1–2 Axioms and Tools; Ch.3 The prenary number system(L0/L1, residual as structure, bivalence not separately representable, ≜ with itstwo criteria); Ch.4 Constructions (fold-in F, projection π, three orders, threeregimes); Ch.5 Results (bivalence visibility, degeneracy theorem, projectiverepresentation); Ch.6 Exits (relation to classical, falsifiability conditions);Appendix A: computations on S3 (self-contained, readable as a standalone paper);Appendix B: LEAN 4 formalisation (six machine-checked files, double-hash verified);Appendix C: author notes (version history and 32 corrections/withdrawals);Appendix D: a 3-page external note on why a non-transitive tolerance cannot bequotiented in the usual way. Every claim in the draft carries an explicit grade — proved / candidate /unverified / withdrawn — and a named carrier (the model or setting in which it waschecked). Conditions that were over-claimed in earlier versions have beendowngraded; where a statement holds only under a reading, the reading is named.Known gaps are listed openly in Ch.6 and Appendix C. This version is timestamped before the first presentation of the framework toprofessional mathematicians. It is published primarily to establish priority ofthe idea and of the specific propositions listed above; it does not claim noveltyover standard results on tolerance relations, and it cites the relevant literaturein Appendix D. Developed with the assistance of an AI research assistant (ima.copilot) forformalisation checking and consistency review; all mathematical claims and allerrors remain the author's.