An Extended Action for Gravitational Aether with Exact Recovery of the Classical Solution Set

We give a local covariant extended action for the classical gravitational-aether equations. Independent multiplier fields are paired with the metric equation, matter conservation, aether evolution and four-velocity normalization. Successive integration with respect to these fields determines the action, and a two-direction argument proves that its stationary configurations project exactly onto all solutions of the original equations. No irrotational-flow restriction, cosmological symmetry reduction or periodic boundary condition is imposed. A scalar-matter calculation identifies a failure of direct metric pairing and illustrates why a multiplier that repairs one sector need not repair the full system. The construction applies the established formal-Lagrangian method; it does not establish equality of physical degrees of freedom or provide a quantum completion. Accompanying Lean code verifies the continuous-integral projection theorem with explicit component equations and stated regularity assumptions. This deposit contains the manuscript PDF and the LaTeX source with the Lean proof package, reproduction script, theorem-to-text mapping, source hashes, and actual build and axiom-audit logs. A clean build on October 27, 2026 passed all seven targets and audited 61 public theorems. Section 6 of the manuscript and the ancillary README specify the assumptions and scope of formal verification.