Topological Coupling and Spectral Mediation in Network Models of Dark Matter (TCGE Framework)

We introduce a network-based framework (TCGE — Theory of Emergent Global Constraints) in which a hidden dark sector modifies visible sector geometry through a spectral coupling operator K_V = W L_D⁺ W^T. The hidden sector undergoes a spectral phase transition — the Dark Big Bang — when its edge count crosses m_c ~ c·n·ln(n), transitioning from incoherent (λ₂(D)=0) to coherent (λ₂(D)>0). After this transition, perturbative marginalization of a joint Gibbs measure yields L_V^eff = L_V − ε²·K_V on the visible sector. We establish two lemmas: Lemma A shows that the binary TCGE covariance satisfies cov_D = (1/4)(I+βL_D)⁻¹ → β⁻¹L_D⁺ for β·λ₂(D)≫1 (Monte Carlo validation r=0.993); Lemma B shows ε_c ≥ 112 for all simulated networks (n∈[30,100]), giving a safety margin ε/ε_c < 10⁻³. A structural prediction links hidden sector topology to effective density morphology: scale-free hidden sectors produce significantly more concentrated density distributions (CV=0.68 vs 0.53 for ER; p=0.009, Cohen d=0.63) — a cuspy versus isothermal profile distinction, validated across 90 networks and three topological classes. This framework is positioned as a class of network models with structural analogies to cosmological dark matter phenomenology. It is not a physical dark matter theory: the model is defined on abstract networks without spatial embedding, and makes no quantitative prediction about gravitational wave spectra or CMB observables.