gGENERIC: Closing GENERIC with the Entropy Variational Law (EVL)

GENERIC (General Equation for Non-Equilibrium Reversible--Irreversible Coupling) separates reversible and irreversible dynamics via a Poisson operator $L$ and a dissipative operator $M$. In applications, $M$ is often the main modelling freedom: many dissipative operators are compatible with the same invariants and entropy inequality. We introduce a closure principle, the \emph{Entropy Variational Law} (EVL), that selects the irreversible \emph{velocity} locally in state space as the unique minimizer of a strictly convex constrained problem. The modelling input is a \emph{dissipation geometry} $G$ (a symmetric positive definite operator) and a set of linear admissibility constraints $A v=0$ encoding conservation/degeneracy requirements. EVL then yields an explicit, symmetric positive semidefinite operator which maps thermodynamic forces to admissible velocities via a $G$-orthogonal projection onto $\Ker A$, and automatically enforces $M_{\mathrm{EVL}}\,\grad E=0$ when the energy constraint is included in $A$. Embedding $M_{\mathrm{EVL}}$ into GENERIC produces \emph{gGENERIC}: a structurally consistent reversible--irreversible dynamics where the dissipative bracket is not postulated but computed. We provide a proof-oriented theorem set (existence/uniqueness, projection structure, entropy production, and a sharp gap identity), discuss structure-preserving splitting integrators and the precise conditions needed for discrete energy control, and work out canonical examples (thermo-mechanical damping via an internal reservoir; mass-constrained kinetics; and a gradient-flow limit). We also give a convex-analytic / monotone-operator lift of EVL, outlining how generalized (non-quadratic) dissipation potentials and semigroup-level splitting results fit into the same closure module. Finally, we provide a PDE-grade well-posedness template in a Gelfand triple / maximal-monotone setting and verify it on a mass-conserving diffusion model, clarifying exactly which hypotheses are checkable in practice.