Complete Proof of Necessity and Uniqueness of the Original Steady-State Topology System: Based on the 45° Triple Coaxial Biconic Configuration, the 1→5→11 Dimensional Evolution Sequence, Prime Topology Reference Points, 55 Global Steady-State Datum Points, and 12:11 Universal Gauge Invariance

Most contemporary fundamental physics, higher-dimensional field theory, differential geometry and analytic number theory are constructed on preset spatial frameworks, fixed dimensional specifications and predefined discrete fundamental units, lacking rigorous first-principle derivation for the inherent necessity and uniqueness of cosmic primordial topological geometry. Discarding artificial hypothetical premises entirely, this work adopts steady-state physical restrictions, the principle of least action, topological nesting laws, primitive element combinatorial logic, algebraic topological bijection isomorphism, hierarchical geometric datum statistics and quantitative falsification verification. With the 12: 11 universal gauge invariance set as a core steady-state axiom, two independent deductive routes are built for cross validation of all core structural features. Strict derivation confirms the exclusive primordial stable geometry is the 45° triple coaxial bicone. Via topological fusion of primordial units and complete field domains plus redundant dimension elimination, the only feasible dimensional evolution sequence is proven as 1511. The bijection between prime irreducibility and topological anchor independence verifies prime-numbered nodes constitute the inevitable self-organized steady-state spatial distribution. Under unified orthogonal projection rules, layered counting and counterevidence confirm the global steady-state datum point total is fixed at 55. Three core supplementary propositions are concluded: First, system closure denotes topological boundary closure instead of thermodynamic isolation, permitting regulated energy and flux exchange bounded by the 12: 11 gauge ratio. Second, prime nodes split inherently into fixed boundary anchor primes \2, 3, 5, 7\ and mobile internal flow primes larger than 7, forming a rigid-boundary fluid-interior cosmic fundamental structure. Third, the system accommodates two coupled motion patterns: orbital motion anchored on small prime nodes and fluid motion governed by the asymptotic prime density (n) 1/ n, with their coupling intensity controlled by the universal gauge constant 12/11. Constrained consistently by 12: 11 gauge invariance, all deductions achieve full logical closure across microscopic topological units and macroscopic cosmic scales without arbitrary assumption. Serving as the foundational axiomatic paper of the PFUSRC theoretical framework, its core results can be directly cited for follow-up field derivation and interdisciplinary research without repetitive primordial-structure demonstration.