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 KV = W LD⁺ WT. The hidden sector undergoes a spectral phase transition — the Dark Big Bang — when its edge count crosses mc ~ 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 LVᵉff = LV − ε²·KV on the visible sector. We establish two lemmas: Lemma A shows that the binary TCGE covariance satisfies covD = (1/4) (I+βLD) ⁻¹ → β⁻¹LD⁺ for β·λ₂ (D) ≫1 (Monte Carlo validation r=0. 993) ; Lemma B shows εc ≥ 112 for all simulated networks (n∈30, 100), giving a safety margin ε/εc < 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.