Poisson-GENERIC Neural Operators: Exact Metriplectic Structure in Function Space via Casimir Entropies

Existing thermodynamically consistent neural operators enforce GENERIC degeneracy by projecting the reversible operator, introducing state dependence without guaranteeing Jacobi. We instead use a compatible Lie--Poisson pencil L=α D+λ(uD+Du) for nonlinear transport and constant Poisson multipliers otherwise. Augmenting the observed field u with latent entropy density s makes S=int s a Casimir, so L,δ S/δ z=0 without projection. Energy combines a fixed mechanical quadratic, learned gauge-free potential, and convex internal energy. The friction operator M=AA^top enforces M,δ E/δ z=0 pointwise and supports diffusion and damping without an independent reversible transport generator. Skewness, positivity, and both degeneracies hold exactly; the continuum pencil satisfies Jacobi, verified numerically on the alias-free band. The parameterization represents the closed-form heat and damped-wave friction operators within its admissible range. Physical energy is bounded under coercivity assumptions, while a discrete-gradient integrator preserves both laws when solved to convergence. Across four PDEs in 1D and 2D and three backbones (FNO, Transolver, CNO), the model wins 61/72 seed-level comparisons against matched unconstrained baselines, recovers exact transport and wave symbols, matches heat and Burgers dissipation within 13%, and exhibits negligible advection dissipation. A constant-L ablation exposes the cost of nonlinear transport, while a learned-energy ablation shows that conserving learned energy need not constrain physical energy.