An Extended Action for the Full Elliptic Lattice KdV System with Exact Recovery of the Solution Set

We construct a local extended action for all five relations of the three-field elliptic lattice KdV system. Independent auxiliary variables are introduced after compatibility searches in explicitly bounded classes of original-field multipliers. The stationary configurations project exactly onto all original solutions, on the original nonzero-denominator domain, without periodicity or decay assumptions on the original fields. The full-lattice action uses finitely supported auxiliary fields. This applies the established adjoint-variable construction; it does not establish the missing natural three-field Lagrangian, a Lagrangian multiform, or equality of physical degrees of freedom. This deposit includes the manuscript PDF, LaTeX sources, Lean proofs, build and axiom-audit logs, and exact symbolic search scripts and certificates. The manuscript and theorem map distinguish Lean-verified derivatives, domain and projection results from separately checked symbolic searches.