ON THE CLOSED-LOOP GLOBAL DYNAMICAL EVOLUTION: PERTURBATION INITIATION, BICONE CONVERGENCE, AND COSMIC SELF-STABILIZATION

As the fourth foundational paper of the Pythagorean Frustum Unified System (PFUSRC), this work establishes the unique, complete, and mathematically closed theory of the closed-loop global dynamical evolution of the cosmos. It is rigorously constructed upon the ontological axioms (Paper 01), the geometrically unique 45° coaxial double-cone structure (Paper 02), and universal empirical consistency (Paper 03). Taking the ontologically necessary primordial perturbation as the sole dynamical origin, the coupling interface 0 as the flux-exchange hub, the global dynamical operator β₁ and intrinsic scale constant π₁=12/11 as the uniquely determined regulatory core, and the topologically mandatory expansion–convergence cycle of the 45° coaxial bicone as the master process, this paper systematically elucidates the only possible dynamical flow of cosmic evolution: starting from unavoidable primordial perturbation, proceeding along the rigid prime-dimensional path 1 ⇒5 ⇒11, and realizing structural growth, dimensional nesting, spacetime emergence, flux compensation, and steady-state topological closure. Three core axioms—the perturbation axiom, valuation axiom, and number–matter issuance axiom—are uniquely and rigorously determined by ontological closure. The unification of certainty and uncertainty, fault tolerance and redundancy, the helical nature of time, and photon-mediated flux transfer are necessary consequences, not auxiliary assumptions. Three pivotal formulas are formally presented: the number–matter issuance formula, the bicone convergence formula, and the cosmic irreversibility formula. Empirical evidence for the helical nature of time is included. The entire framework is self-consistent, topologically closed, and dynamically complete without singularities, external assumptions, free parameters, or alternative structures. This paper completes the fourth and final pillar of PFUSRC: Axiom · Geometry · Empiricism · Dynamics. The only possible unified, self-contained, and ontologically final theory of the cosmos is hereby realized.