goal_LACompletion_identity_project

The RSA RL module turns the Recognition Stability Audit into an RL-friendly Lean environment: whitelisted simp lemmas, milestone steps, and a library of proved gold goals that reward an external proposer when goals close.

A virtue action is a $14$-coefficient real vector over the DREAM-complete virtue generators; the policy emits these coefficients, not raw moves. An LACompletion is a least-action completion projector: it maps a proposed action onto the $\sigma=0$ feasible manifold while minimizing added $J$-cost (propose-then-project).

The identity LACompletion is the trivial case: its project map is the identity function, valid when every proposed action is already feasible. Its feasibility and cost-minimization obligations are discharged by trivial. Upstream, the foundation also has an identity recognition event at the $J$-cost minimum $x=1$, and a separate observer projection from deterministic states to finite observations; those are not used in this equality.

One-line definitional proof. Unfolding the identity LACompletion gives project := id, so the goal is id a = a, which closes by rfl. No lemmas or tactics beyond reflexivity are required.

This is one of the canonical gold training goals in the RSA RL library: a sorry-free theorem that exercises the pipeline on the trivial projector. It anchors the propose-then-project contract at the identity end of the LACompletion family (before $\varphi$-scaling or nontrivial cost-minimizing projectors). No downstream theorems currently depend on it; its role is checklist closure and reward signal for the RL loop, not a step in the T0–T8 forcing chain. It does not touch RCL, mass ladder, or constants; it only certifies that the identity completion is a pure projector on virtue actions.