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May 22, 2026Open Access

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

Key Points

This research aims to rigorously demonstrate the fundamental ontological structures of the universe, focusing on the nature of dimensions and geometric forms.
Systematic demonstration of four fundamental ontological structures of the universe without assumptions.
Introduction of eight independent pathways proving the necessity of an 11D structure.
In-depth analysis of the required properties for geometric stability and dimensional evolution.
Established that the universe must be a closed-loop space, not isolated.
Identified the minimal space dimension for life and mathematics as four-dimensional.
Proved that the 45° coaxial double cone is the only stable structure meeting required geometrical properties.

Abstract

Within the ontological framework and ultimate axioms of PFUSRC zero-reconstruction, this paper systematically demonstrates four fundamental ontological structures of the universe in a rigorous, step-by-step, assumption-free manner. First, the universe is necessarily a closed-loop space rather than an isolated space. Second, the minimal space dimension that supports field propagation, time generation, evolutionary unfolding, and the emergence of mathematics and life is four-dimensional. Third, the only primitive geometry that satisfies rotational self-stabilization, flux conservation, freedom from distortion, tearing and singularities, 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 pathways rigorously prove 11D 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. This paper ontologically clarifies mainstream viewpoints such as compactified extra dimensions and Calabi–Yau manifolds, proving that 11D is not an external addition but a rigid nested structure of 4D space, and that the only stable cosmic structure is one large double cone coaxially nesting two small double cones. With complete definitions, rigorous argumentation, and logical closure, this paper maintains the same academic depth as PFUSRC‑01.

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