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We present a self-consistent, operator-level unification framework governed by the equation: \[ \dot{y} = L\nabla H + M\nabla \mathcal{S}, \] with degeneracy constraints \(L^\top = -L\), \(M \succeq 0\), \(L\nabla \mathcal{S} = 0\), and \(M\nabla H = 0\). The formulation introduces a stochastic controller whose deterministic limit yields the renormalization-group (RG) flow under a logarithmic entropy clock \(\sigma = \ln(\mu / \mu_c)\). The thermodynamic entropy corresponds directly to the gauge-theoretic \(\beta\)-functions, and the one- and two-loop structures---including threshold corrections---emerge as physically valid limits of the controller dynamics. This demonstrates that the \(\beta\)-loop assumption in Gauge unification is not merely empirical but arises from entropy production. A corrective entropy term is identified for full cosmological integration into the framework. The framework, therefore, provides an operator-level proof of entropy-driven unification across thermodynamics, gauge theory, and cosmology, and identifies the additive entropy term as a potential source of undiscovered physical structure at both microscopic and macroscopic scales. The self-consistency of this framework is validated in both control-noise theory and through fundamental principle relations. It stands out from other GUT frameworks because it re-defines not just what unification means, but also informs us that Einstein's equations are simply a non-dissipative limit of foundational principles. Lastly, we provide an additional realization to show how quantum gravity may be best understood through discrete quanta. Ultimately, it is a paradigm shift in how we view symbolic modeling by clarifying the physical meaning of geometric structure.