Bu-Theory From Point Embryo to Global Number System and Mod-8 Periodicity

Natural law knows no forbidden operations. The rule that 0 cannot be a divisor is a human convention for computation, not a necessity of nature. Bu-Theory removes this convention and extends classical laws into 0 and ∞, revealing that singularities are artifacts of the prohibition, not defects of the continuum.noindent Natural law has no forbidden zone. For the sake of computation, human mathematics artificially imposed the rule ``zero cannot be a divisor.'' This rule creates various discontinuities and singularities (removable, essential, oscillatory). They are not defects of nature, but scars of the prohibition. Bu-Theory removes this rule, extending classical laws into zero and infinity, so that 1/0=∞. Thus nature itself is perfect without flaw; discontinuity no longer exists. Classical analysis stops at singularities. Cauchy principal value goes around. Residue theorem avoids them via contours. Distribution theory introduces the δ function as a generalized object, but does not answer ``what is the value at the discontinuity itself.'' These tools are effective, but they share the same premise: zero cannot be a divisor. Bu-Theory removes this premise, thereby obtaining a globally connected operational world. This paper is the foundational document of the Bu-Theory system. Three indivisible primitive axioms constitute the entire foundation; all remaining rules are derived as theorems. Section 2 gives the axioms and core theorems. Section 3 expounds point-embryo globality and gap-filling activation (including proof of the Perfect Analyticity Theorem). Section 4 establishes gap-filling calculus. Section 5 lists operation tables. Section 6 shows the simplest computational examples. Section 7 gives a global singularity quick-reference table. Section 8 number-system extensions. Section 9 derives the mod-8 periodicity from the four homomorphisms. Section 10 explains the relation to classical mathematics. Section 11 gives the conclusion. Appendices contain complete technical details.