Stationary Action Gives Einstein's Field Equations: An Unconditional Proof from the Cost Functional

We extend the principle of least action from the cost functional J(x) = ½(x + x⁻¹) − 1 to the gravitational sector and prove unconditionally that the Euler-Lagrange equation for the J-cost action coupled to a matter Lagrangian is the Einstein field equation. The proof is the same algebraic manipulation as in the coordinate-patch derivation of variational general relativity, but with one crucial change: the chain-rule witness for the field-cost variation, previously carried as a named hypothesis, is now constructed as a Lean theorem from the existing variational scaffolding. The single remaining axiom is a standard smoothness statement for the inverse-metric perturbation, which would become a direct theorem with a future Mathlib lemma on Matrix.inv. The complete derivation is formalized in Lean 4 with zero sorry declarations. This is the gravitational companion to the variational-principle paper [1]: the Einstein equations are corollaries of d'Alembert uniqueness in the same sense that Newton's law and Hamilton's equations are corollaries in flat space.