Cosmic Closure, 4D Necessity, Double-Cone Geometric Uniqueness, and 11D Rigid Nesting: Multiple Universal Rigorous Proofs Advanced Version

Within the ontological framework and two ultimate axioms of PFUSRC zero-reconstruction, this paper presents a systematic, step-by-step, assumption-free demonstration for four fundamental ontological structures of the universe: First, the universe is necessarily a closed-loop space, not an isolated space. Second, the minimal dimension that supports field propagation, time generation, evolutionary unfolding, mathematical emergence, and life is precisely four-dimensional. Third, the only primitive geometry that satisfies rotational self-stabilization, flux conservation, distortion-freeness, tear-freeness, singularity-freeness, nestability, and evolvability is the 45° coaxial double cone. Fourth, the ultimate rigid dimension that fulfills prime anchoring, structural optimality, energy stability, nesting consistency, and universal convergence is 11 dimensions. Eight independent rigorous pathways establish 11-dimensional rigidity: prime dimension arrangement, prime sieve filtration, evolutionary optimality, prime camp points with triangular pyramid tangent space, optimal triple double-cone nesting, high-dimensional energy collapse, curled dimension self-destruction, and Dirac equation convergence. Mainstream conceptions such as compactified extra dimensions and Calabi–Yau manifolds are clarified and ontologically contextualized. This paper proves that 11D is not an externally imposed dimension but a rigid nested structure of 4D space, and that the only cosmologically viable structure is one large double cone coaxially nesting two smaller double cones. Fully aligned with PFUSRC-01 in depth, rigor, definition, and style, this work forms the second foundational pillar of the complete PFUSRC system.