PFUSRC-132L-Digital Root as Decimal Projection of Living Calculus

Abstract

Digital root is a basic operation in elementary number theory, which iteratively sums the digits of any natural number until a single-digit integer between 1 and 9 is obtained. Based on the Living Calculus ontological system proposed in PFUSRC-130, this paper rigorously proves that digital root operation serves as the direct three-dimensional solidified projection of the 11-dimensional triple coaxial 45° biconical topological ontology. Its finite iterative reduction fully satisfies three core axioms of Living Calculus: the Discrete Anchor Axiom, the Cyclic Normalization Axiom and the Phase Origin Axiom. The forced termination of repeated digit summation corresponds to the sign-off rule of discrete curvature anchors; the closed 1–9 numerical cycle reflects the normalization property of a complete 720° biconical breathing cycle converging to Topological One; the modulo-9 constraint derived from base-10 decimal system (10−1=9) acts as the observational reading of the native 12/11 phase difference between upper and lower cones mapped onto artificial counting layers. Within this projection framework, remainder 0 modulo 9 lacks independent single-digit representation, hence digit 9 functions as the closed-loop equivalent marker of Topological Zero on the decimal observation layer. This paper further compares topological projection phenomena across multiple fields including calendrical solar terms, atomic electron shell structures and market periodic cycles. The comparison verifies that digital root is not an isolated arithmetic trick but a universal manifestation of unified biconical ontology on different dimensional observation slices. As a manually computable case, it provides a low-threshold entry for readers without advanced topological background to understand the core logic of Living Calculus and the layered projection mechanism of PFUSRC ontology.