Topological Discrete Universe (TDU v4.9): A Combinatorial Lattice Framework for Effective Late-Time Cosmic Acceleration, Horizon Freezing, and Graph-Dynamic Stability

# Abstract We present version 4.9 of the **Topological Discrete Universe (TDU)**, an effective lattice-inspired theoretical framework describing late-time cosmic phenomenology through the combinatorial dynamics of an underlying discrete graph space. Spacetime at macro-scales is modeled as a macroscopic manifestation of a Poisson-Delaunay simplicial complex subject to a strict capacity budget Eₜₒₜ = Eₘ(t) + Eᵥ(t) = const., partitioning degrees of freedom between localized gravitating matter condensates and the topological vacuum graph. In this iteration, the macroscopic relaxation parameter β ≡ Γrewire/H is derived from first principles using Fermi's golden rule applied to a discrete link-flipping Hamiltonian on the Fock space of Delaunay simplices in an adiabatic regime. Exact analytical relationships are established for the topological tension functional Σ(u), its critical saturation threshold uc = 1 / (1 + √δEMT), and the non-perturbative geometric energy density ρgeom(u). Under the vacuum rigidity condition, gauge-invariant sub-horizon geometric perturbations vanish (δρgeomN → 0) and the anisotropic stress tensor is zero (πᵢⱼᵗᵒᵖ ≡ 0), enforcing the equality of Newtonian potentials Φ = Ψ in the cosmic matter rest-frame. Furthermore, we explicitly formulate a dedicated section detailing open theoretical limitations—including the Euclidean Poisson-Delaunay mean-field embedding, the absence of a fully covariant action functional, and the transition from fixed benchmarks to complete Markov Chain Monte Carlo (MCMC) likelihood exploration. Finally, we provide the canonical mathematical mapping connecting the topological tension formalism to generalized flow capacity conservation and predictive collapse prevention in distributed physical networks.