eulerLagrange_status

The module formalizes Euler-Lagrange equations for two action functionals on the cost manifold. The cost-rate action integrates the pointwise J-cost, so its EL equation reduces to the constant path at the minimum. The Hessian-energy action integrates half the second derivative of J times squared velocity, producing the geodesic equation in the metric g(x) = 1/x^3 with Christoffel symbol -3/(2x). Upstream, the cost-rate equivalence states that among admissible positive continuous paths the constant ground state is the unique solution of the cost-rate EL equation. The geodesic equation is defined pointwise as the second derivative plus Christoffel times velocity squared equals zero.

This is a sorry stub that directly constructs the status string by listing the cost-rate equivalence theorem, the geodesic equation definition, and the ground-state uniqueness theorem, then appending the note of zero sorries and axioms.

This status marker records completion level for the Euler-Lagrange section, which shows the cost-rate variational principle and the Hessian geodesic principle agree on the unique ground state. It supports the Recognition Science framework that derives physics from the J-functional equation, Recognition Composition Law, and eight-tick octave. The paper companion is RS_Least_Action.tex, Section on the Euler-Lagrange Equation as the Geodesic Equation. It touches the open question of removing all scaffolding from the Action module.