Part 33: Effective Field Description of Dual–H4 Geometry A Closed Dynamical System for Topological Growth and Cosmic Evolution

Overview Previous Parts of the Origin Geometry program developed a cosmological branch in which dark-sector topological relaxation, bulk stress propagation, nonlinear localization, void-enhanced relaxation, and topological network growth provide a possible microscopic interpretation of effective cosmic expansion [1–8]. Part 29 introduced lattice proliferation as a mechanism for emergent expansion. Part 30 identified observable signatures of void-dependent geometric growth. Part 31 developed the spectral and detectability framework for high-frequency bulk stress modes. Part 32 introduced order-of-magnitude parameter constraints linking microscopic relaxation parameters to cosmological scales. The present Part takes the next step by formulating a closed effective field description of the dual–H4 cosmological branch. Coupled Evolution Variables The central goal is to describe the coupled evolution of three coarse-grained dynamical variables: ρ_stress(x, t): the effective stress energy density stored in topological obstruction, phase mismatch, pinning, and dark-sector geometric compression. ρ_bulk(x, t): the effective energy density carried by collective bulk stress modes. n(x,t): the realized topological network count within a coarse-grained reference domain [15–32]. These variables are not independent. Stress is released through topological relaxation, bulk modes transport and redistribute the released energy, and sufficiently localized bulk stress may activate additional topological degrees of freedom. This creates a feedback cycle: ρ_stress → ρ_bulk → n → H_eff Effective Dynamical System We introduce a minimal phenomenological field system: ∂t ρ_stress = D_s ∇²ρ_stress + S_topo − λ_relax (ρ_stress)² ∂t ρ_bulk = D_b ∇²ρ_bulk + ε_bulk λ_relax (ρ_stress)² − Γ_damp ρ_bulk − L_loc(ρ_bulk) ∂t n = D_n ∇²n + γ_n n P_act(ρ_bulk) − Γ_n (n − n₀) Here D_s, D_b, and D_n describe coarse-grained redistribution; λ_relax (ρ_stress)² represents pair-like topological relaxation; ε_bulk is the fraction of released stress energy transferred into bulk modes; Γ_damp is an effective damping rate; L_loc describes nonlinear localization or trapping of bulk energy; and P_act is an activation probability for topological growth. Dimensional consistency requires that threshold activation compare energy density with energy density. Therefore, instead of writing a threshold of the form Θ(ρ_bulk − ΔE), we define: P_act(ρ_bulk) = [ 1 + exp( -(ρ_bulk − ρ_act) / Δρ ) ]⁻¹ where ρ_act is an activation energy density and Δρ controls the transition width. The effective expansion rate is connected to topological growth through: a_eff ∝ n^(1/3) when n represents a volumetric realized network count. Consequently: H_eff = (∂t a_eff) / a_eff ~ (1/3) (∂t n / n) This factor 1/3 is essential for consistency with the node-growth scaling established in Part 32. Scope and Limitations The framework does not claim to replace Einstein’s equations, derive precision cosmological data, eliminate the cosmological constant, or prove that dark energy is unnecessary [9–14]. Rather, it provides a closed effective dynamical system through which the Origin Geometry cosmological branch can be studied, simulated, constrained, and eventually compared with observations.