What is mathematics? Where does it come from? What is its skeletal structure? What drives its inherent evolution? Using the Collatz conjecture (the hailstone sequence) as a concrete analytical sample, and grounded in the PFUSRC topological unified framework, this paper provides a unified answer to these four fundamental ontological questions of mathematics.This paper first presents the iterative rules of the Collatz conjecture and the topological path data of key numbers, then analyzes path evolution patterns to identify the underlying topological dynamic mechanisms. This leads to four core theses: First, the ontological starting point of mathematics is “1,” not “0”—0 is merely a phase-reference interface, not absolute nothingness. Second, the overall topological architecture of mathematics follows the principle of Axis Unification—all number-theoretic iterative paths originate from the primordial unit 1, unfold outward along the 45° triple coaxial central axis, and after cyclic evolution, all converge back to 1. Third, the complete skeletal structure of mathematics consists of the three-point topological base: 1-5-11. The product of these three base elements equals 55—the total number of global steady-state anchor points in the PFUSRC system. Fourth, the fundamental driver of all mathematical structural evolution is topological broken symmetry.This paper further demonstrates that mathematics is neither a human-invented computational tool nor an independent world of ideas floating outside objective reality—it is the structural fingerprint left by the Topological Cosmos on the number-theoretic projection layer. The parity differentiation of numbers is the phase distinction of topological growth sequences. Geometric points, lines, surfaces, and volumes are solidified slices of four-dimensional flow variables projected onto the 3D projection layer. Special constants and identities—Euler’s identity, π, ζ(-1)=-1/12, Ramanujan’s modular forms—are natural manifestations of the topological ontology on different projection sections. All effective explanatory power of mathematics is grounded in the objective steady-state structure of the Topological Cosmos. No mathematical structure exists at Level 0 (the Universal Universe). Discussion of mathematics apart from the Topological Cosmos has no ontological foundation. The world is fundamentally the entity “1”—there is no absolute nothingness.This paper concludes that mainstream mathematics suffers from a fundamental cognitive limitation: using only 3D solidified points, lines, surfaces, and volumes as the foundational paradigm for interpreting mathematical ontology is equivalent to using shadows to infer the light source—one can observe only the projection appearance, never reaching the topological origin behind mathematics.
Zhenmin Wang (Fri,) studied this question.