PFUSRC-130 Ontological Foundation of Living Calculus: Complete Mathematical System Rooted in 45° Triple Coaxial Biconical Topology

Abstract

Classical slice calculus established by Newton and Leibniz relies on four fictitious transcendental presuppositions: absolute continuous real lines, objective infinitesimals, executable infinite iteration, and globally smooth spacetime geometry. It artificially truncates the bidirectional expansion-convergence breathing cycle of the 11D biconical topology, resulting in widespread theoretical divergences including quantum singularities and Zeno’s paradoxes. Previous PFUSRC documents only deliver scattered local geometric conclusions without an integrated, self-consistent mathematical base. As the exclusive foundational mathematical monograph of the whole system, this paper takes PFUSRC-00’s 45° steady-state bicone axiom as the sole logical starting point and unifies all scattered topological constants and partial derivations. Dual geometric-algebraic deductions are carried out: this work rigorously proves the Topological One via the 12/11 gauge ratio and puts forward the Global Normalization Theorem that all complete rheological cycles converge to 1 instead of zero; it topologically interprets the imaginary phase operator i and the identity i² = -1, stripping them of pure algebraic convention; three core living calculus operators are reconstructed: topological sign-off limit, four-dimensional phase-difference derivative, and discrete hierarchical integral, alongside complete arithmetic and composition rules. A ΔA cross-boundary topological correction system is constructed to bridge classical approximation and ontological mathematics. Unified biconical resonance numerical comparisons demonstrate the applicable scope of two calculus systems, and a three-layer mathematical classification standard is defined. All derivations are anchored by biconical geometry, algebraic equations, inherent topological constants and astronomical predictions, eliminating purely speculative reasoning. This paper acts as the mandatory prerequisite for all PFUSRC cosmology, unified field theory and dimensional ascent research.