PFUSRC-031: Complete Derivation and Nonlinear Dynamics of PFUSRC Equations From 45° Biconical Geometry to a Unified Mathematical Framework for Modified Gravity

The PFUSRC framework has established that the 45° triple coaxial bicone is the unique steady-state geometry of the universe. However, the temporal evolution of the biconical breath—expansion, contraction, exclusion, return—has not yet been fully described mathematically. Based on the PFUSRC axiomatic system, this paper rigorously derives a set of nonlinear dynamical equations. The core contributions are as follows: (1) constructing nonlinear dynamical equations in terms of the normalized expansion–contraction difference x = Δ A/Δ Aₘₐₓ and the waist-ring accumulated stress y = Σwaist/Σₘₐₓ, with all parameters determined by geometric constraints rather than free fitting; (2) defining the exclusion order parameter within a nonequilibrium topological phase transition framework, deriving the critical exponent βcrit = 1/2, predicting the ring-like density profile of dark matter halos with radius ratio Rₒᵤₜₑᵣ/Rᵢₙₙₑᵣ = 12/11; (3) strictly deriving the Einstein field equations in the weak-field limit as a special case of PFUSRC dynamics, providing strong-field corrections, and proving that singularities are eliminated by the topological exclusion mechanism; and (4) proposing three quantifiable, falsifiable predictions with explicit observational thresholds and exclusion criteria. This work completes the transition of the PFUSRC framework from static topology to dynamical evolution.