{New Mathematics under Dialectical Logic: Basic Theory of the Bu Theory's Supra‑Mathematics— Extending Classical Rules into the Forbidden Zone

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.