F1/辩证主义逻辑之下的新数学:卟学之超数学基本理论—— 将经典规则延拓入禁区(zh)/New Mathematics under Dialectical Logic: Basic Theory of the Bu Theory's Supra‑Mathematics : Extending Classical Rules into the Forbidden Zone(en)}

本文提出一种基于辩证逻辑的全新公理体系——卟学(Buxue),通过在公理层面引入“点胚”结构,将经典算术延拓至包含奇点(零与无穷)的全域。三条不可再分的原始公理构成全部根基:公理 I 定义点胚 ϑ=(0,1),在莫比乌斯环内部实现 0=1 的拓扑等同;公理 II 撤除“零不能作除数”的禁令,规定 1/0=∞,从而打开禁区;公理 III 将数轴闭合为拓扑圆(S 1 ,射影直线),使 +∞ 与 −∞ 在 ∞ 处连通。基于此,本文建立补断微积分,证明点胚完美解析定理(所有阶导数存在且唯一)、残留阶数定理(精确给出极点处各阶导数的闭式值),并构造遗忘投影 π 实现与经典数学的严格对接——在经典定义域上 π∘B=id,经典结果是点胚数学的第一投影层,而点胚内部的“革命”信息(如分支缠绕、无穷阶导数谱)在投影下受控泄露或归零。数系经 Clifford 代数扩充至全域,由四同态代数推出模 8 周期。本文为奠基性纲领,以宣言式、解释性文本呈现核心洞见,所有结论均在卟学公理体系内通过几何与拓扑推导得出,经典数学的域公理与极限过程作为特例被完整保留。 This paper proposes a new axiomatic system based on dialectical logic—Buxue (the study of "bud" or "embryonic" points)—which extends classical arithmetic into a global domain containing singularities by introducing the structure of "point-embryo" at the axiomatic level. Three indivisible primitive axioms constitute the entire foundation: Axiom I defines the point-embryo ϑ=(0,1), realizing the topological identification 0=1 inside a Möbius ring; Axiom II removes the prohibition that "zero cannot be a divisor", stipulating 1/0=∞, thereby opening the forbidden zone; Axiom III closes the number line into a topological circle (S 1, the projective line), connecting +∞ and −∞ at ∞. Based on this, we establish the calculus of "Bu-duan" (supplementing breaks/discontinuities), prove the Point-Embryo Perfect Analyticity Theorem (all-order derivatives exist and are unique), and the Residue Order Theorem (giving closed-form values for all-order derivatives at poles). We also construct a forgetful projection π to strictly interface with classical mathematics: on classical domains of definition, π∘B=id, so classical results are the first projection layer of point-embryo mathematics, while the "revolutionary" information inside point-embryos (e.g., branch windings, infinite-order derivative spectra) is either controlled-leaked or reduced to zero under projection. The number system is extended via Clifford algebras to the global domain, and the mod 8 periodicity is derived from the four-homomorphism algebra. This is a foundational manifesto, presenting core insights in a declarative and explanatory style; all conclusions are derived geometrically and topologically within the Buxue axiomatic system, with classical field axioms and limit processes fully preserved as special cases.