The processing of information is fundamentally coupled to the dissipation of thermodynamic heat. While biological myelination and artificial Mixture of Experts (MOE) neural architectures are conventionally analyzed within disparate disciplines, both represent structurally isomorphic adaptations to this physical boundary. In this paper, we propose a unified Time-Dependent Ginzburg-Landau (TDGL) formalism demonstrating that both substrates utilize topological state-erasure to satisfy the Landauer dissipation limit. Defining the macroscopic order parameter $\phi \in [0,1]$ as the topological coupling density, we show that when the steady-state dissipative cost of global routing ($P_{\mathrm{diss}}$) exceeds the substrate's thermal limit ($\dot{Q}_{\mathrm{limit}}$), the accumulated thermal debt explicitly breaks the symmetry of the free energy landscape. This induces a deterministic, non-equilibrium macroscopic relaxation into a sparse, low-entropy ground state ($\phi \to 0$). To preserve spatial and dimensional rigor, we bifurcate the functional to address both continuous Euclidean manifolds (biological phase separation) and discrete computational graphs (silicon MOE), utilizing a Huber-smoothed $\mathcal{L}_1$ norm to enforce rigorous graph sparsity. Furthermore, we formalize an Algorithmic Temperature ($T_{\mathrm{alg}}$) equivalent for \textit{in silico} substrates, proving that computational routing transitions are governed natively by classical fluctuation-dissipation mechanics.