Part 34: Numerical Realization of Topological Growth in Dual–H4 Geometry Simulation of Stress Localization and Emergent Cosmic Expansion

Overview Previous Parts of the Origin Geometry program developed a relaxation-driven cosmological branch in which topological obstruction, geometric stress, collective bulk propagation, nonlinear localization, and effective topological network growth may provide a microscopic interpretation of emergent cosmic expansion. Part 29 introduced lattice proliferation as a mechanism for effective expansion. Part 30 identified void-dependent observational signatures. Part 31 developed the spectral language of bulk gravitational wave-like stress modes. Part 32 introduced order-of-magnitude parameter constraints. Part 33 formulated a closed effective field system for the coupled variables ρₛtress, ρbulk, and n 1–5. The present Part develops the first numerical realization framework for this closed system. The objective is not to reproduce the real Universe, fit precision cosmological data, solve Einstein’s equations numerically, or replace standard ΛCDM simulations. Instead, the goal is narrower and more structural: to determine whether the local rules implied by the Part 33 effective system can generate, in simplified discrete networks, the qualitative ingredients required by the OG cosmological branch. Discrete Network Model We construct a discrete network model in which each cell i carries three dynamical variables: ρₛtress, i (t): the local stored geometric stress energy density; ρbulk, i (t): the local collective bulk stress energy density; and nᵢ (t): the local realized topological network count within a fixed coarse-grained reference cell. The update rules are chosen as finite-difference analogues of the closed effective system: ρₛtress, i (t+Δt) = ρₛtress, i (t) + Δt Dₛ Σⱼ Aᵢj (ρₛtress, j (t) - ρₛtress, i (t) ) + Sᵢ (t) - λᵣelax (ρₛtress, i (t) ) ² ρbulk, i (t+Δt) = ρbulk, i (t) + Δt Db Σⱼ Aᵢj (ρbulk, j (t) - ρbulk, i (t) ) + εbulk λᵣelax (ρₛtress, i (t) ) ² - Γdamp ρbulk, i (t) - Lₗoc (ρbulk, i (t) ) nᵢ (t+Δt) = nᵢ (t) + Δt Dₙ Σⱼ Aᵢj (nⱼ (t) - nᵢ (t) ) + γₙ nᵢ (t) Pₐct (ρbulk, i (t) ) - Γₙ (nᵢ (t) - n₀) Here Aᵢj is the adjacency weight of the simulation graph, Dₛ, Db, and Dₙ are redistribution coefficients, λᵣelax controls stress relaxation, εbulk controls transfer into bulk modes, Γdamp is a damping rate, and Pₐct controls activation of realized topology. The activation function is written in dimensionally consistent form: Pₐct (ρbulk, i) = 1 + exp (- (ρbulk, i - ρₐct) / Δρ) ⁻¹ Nonlinear localization is modeled through: Lₗoc (ρbulk, i) = βₗoc (ρbulk, i - ρₗoc) Θ (ρbulk, i - ρₗoc) or through a smoothed activation factor. Emergent Scale Factor and Expansion The effective scale factor is defined from the average realized topological count: aₑff (t) = ⟨n (t) ⟩ / ⟨n (t₀) ⟩ ^ (1/3) and the corresponding Hubble-like rate is: Hₑff (t) ≈ (1/Δt) ln aₑff (t+Δt) / aₑff (t) The framework therefore translates the closed effective field model of Part 33 into a concrete numerical algorithm. Scope and Falsifiability This Part does not claim that the simulations already constitute evidence for Origin Geometry. Rather, it defines the numerical architecture required to test whether stress localization, bulk propagation, activation thresholds, and topological network growth can robustly generate expansion-like behavior, void-enhanced growth, and large-scale coarse-grained regularity.