An Extended Action for the Full Nonlinear MTEGR Gravitational Equations

We construct a local extended variational formulation of all sixteen nonlinear MTEGR gravitational equations in the local limit of nonlocal gravity. The complete tetrad geometry, local constitutive relation, gravitational energy-momentum terms, arbitrary smooth susceptibility S(x) > -1, and nonzero parity parameter are retained. Sixteen independent response fields are introduced by integrating the original equations with respect to the new variational variables. The full first variation yields every original equation and the complete weak adjoint system. Every stationary extended configuration projects to an original solution, and every original solution has an explicit stationary lift with zero response fields. The Lean 4.19.0 / mathlib development verifies the actual spacetime integral and its full first variation, derives equation regularity from the nonlinear geometric definitions, and proves exact projected recovery. A fresh build passed all nine verification targets and audited all eighty public theorems without unfinished proofs or added nonstandard axioms. Every smooth compactly supported tetrad direction is covered. The formal scope is smooth nondegenerate coframes on the four-coordinate domain R4 with prescribed smooth symmetric matter stress. Auxiliary fields are compactly supported; the original coframe is not required to be periodic or to vanish at infinity. This applies the established adjoint-field variational construction. It does not establish a conventional action using only the original fields, equality of physical degrees of freedom, microscopic matter dynamics, or global manifold and coordinate-covariance formalization. No claim of bibliographic priority is made. The deposit includes the manuscript PDF, LaTeX sources, Lean proofs, reproducible verifier, source hashes, build logs, and axiom audits. AI tools assisted Lean code development and verification, translation, and accelerated multiplier searches and tests.