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. Version 1.1 — core verification supplement, 28 October 2026. A fresh verification passed 18/18 build targets and audited 40 public theorems. The verified result concerns all five real-valued elliptic lattice KdV equations on the full integer lattice and the original nonzero-denominator domain. It includes the actual full-lattice first variation, domain preservation for the stated variations, and exact projection of extended stationary configurations onto the complete original solution set. The auxiliary fields have finite support; the original fields need not be periodic or decay. The current supplement does not construct the natural three-field Lagrangian discussed by the original authors and does not cover complex-valued fields or parameters. It is an extended variational representation, rather than a claim that the original natural-Lagrangian question has been settled. It does not certify every supplementary search statement in the manuscript, physical degrees-of-freedom equivalence, or bibliographic priority. Use Elliptic_KdV_Core_Verification_Scope_20261028.pdf and Elliptic_KdV_Core_Lean_Verification_20261028.zip for the current formal scope. The original manuscript and earlier source package are retained. AI tools were used to assist Lean code development and verification, translation, and to accelerate multiplier searches and tests.