Obstruction Results and a Candidate Open-System Construction for Dissipative Relaxation toward Curvature-Set Equilibria

This paper asks whether general relativity admits a consistent extension in which a matter-sector internal variable relaxes dissipatively toward an equilibrium value set by local spacetime curvature. It first establishes a sequence of class-restricted obstruction results. Within finite-jet, momentum-equation-only deformations of ADM gravity that leave the vacuum constraints unchanged, the natural local source tensors fail constraint tangency through gradient, Ricci-identity, or DeWitt-normal terms. The canonical ADM bracket admits no nonconstant Casimir on regular symplectic components, while a potential-only normal Hamiltonian fails normal-normal closure of the hypersurface-deformation algebra unless accompanied by an appropriate current sector. For a Brown fluid with a curvature-set internal potential, the attractive relaxation branch incurs a negative curvature-Hessian contribution quadratic in the target slope; under the stated positive-kinetic reduction, a curvature stabilizer is therefore required. A linear entropy coordinate cannot undergo two-sided relaxation, requiring a distinct internal deviation and an energy/entropy receiver. Curvature-independent internal energy also cannot hold a nonconstant curvature target under the stated scalar relaxation law with sign-definite dissipation. The surviving construction is a classical Gaussian open-system effective candidate, not an identified microscopic realization. Here q denotes a neutral material internal coordinate, not electric or color charge, and no empirical evidence for such a coordinate is claimed. A response-normal Schwinger-Keldysh/MSRJD completion is constructed for per-particle comoving variables. The same two Gaussian noises enter the internal equations, receiver-energy equation, and transverse Brown momentum equation, providing trajectory-level energy and momentum bookkeeping and total stress-energy conservation through the off-diagonal diffeomorphism Ward identity. The dissipative/receiver block obeys an off-shell local-detailed-balance identity—the classical Gaussian dynamical-KMS condition—without use of a saddle equation. At fixed curvature and temperature, the stationary target is fixed by the potential while friction changes relaxation rates only. An energy-degenerate finite-receiver model also has exact microcanonical stationarity for a nonexpanding comoving cell with no-flux energy boundaries. For an explicit Brown/receiver closure, the paper derives a nontrivial FLRW benchmark, its complete constrained scalar perturbation system, a scalar-adapted generalized-harmonic formulation, and explicit scalar and full-system symmetrizers for a complete 65-state deterministic first-order reduction. The resulting deterministic system is locally symmetric hyperbolic and principal-causal on its stated healthy domain. The organizing partition is that curvature-set equilibrium targets are thermodynamic and pay a curvature-Hessian cost quadratic in their local slope, whereas friction and common rate rescalings avoid that tree-level tax on frozen backgrounds; the susceptibility ratio remains thermodynamic. Eleven open problems are stated explicitly. Documented literature searches did not identify the complete combination, but this is reported as documented non-identification rather than proof of absence. The paper does not claim a microscopic realization, global nonlinear evolution, continuum white-noise well-posedness, full nonlinear dynamical KMS/BRST, loop stability, quantum consistency, or ultraviolet completion. This upload is intended as a timestamped open-access preprint record while arXiv gr-qc submission/endorsement is pursued.