An Extended Variational Formulation of Schouten-Codazzi Field Equations

We construct a local covariant extended action for the Schouten-Codazzi field equations, using independent adjoint tensor fields. Their variations recover the original Schouten-source equation and the complete Codazzi condition. Every original solution has a stationary lift with vanishing adjoint fields; conversely, every stationary configuration projects to an original solution. The original algebraic admissibility prescriptions for the additional geometric tensor are retained. The construction applies the established formal-Lagrangian method and does not assert equality of physical degrees of freedom or an auxiliary-free gravitational action. The deposit contains the manuscript, reproducible Lean proof sources, and a statement of the precise formalization scope.