identity
plain-language theorem explainer
The identity least-action completion leaves every virtue-action unchanged. It is the valid projector precisely when every proposed action is already σ=0-feasible, so no correction is needed. Anyone wiring the RS→RL propose-then-project loop cites it as the trivial baseline completion. The fields are filled by the identity map and two trivial proofs.
Claim. The identity least-action completion is the projector $\mathrm{project}(a)=a$ on virtue-actions. It satisfies the feasibility postcondition and the least-added-cost condition whenever every proposed action is already $\sigma=0$-feasible (so no correction is required).
background
The RS→RL bridge treats Recognition Science as control theory: states are MoralState (ledger, bonds, skew, energy), admissible moves are 14 virtue generators packaged as VirtueAction, and hard physics is enforced by projecting onto the σ=0 feasible set. The module separates creativity (policy proposes) from physics (completion projects).
LACompletion is that projector: given an unconstrained proposed direction, it maps onto the σ=0 manifold while minimizing added J-cost (the Recognition cost $J$, with $J(1)=0$). The structure carries a project map plus two guarantees: feasibility is preserved after projection, and the completion is least-cost among feasible corrections.
The identity instance is the baseline case of that interface: when the proposal already lies on the feasible set, the cheapest feasible correction is to do nothing.
proof idea
Pure structure instance. The projection field is the ordinary identity function on virtue-actions. Both proof obligations (preserves_feasibility and minimizes_cost) are discharged by trivial, matching the placeholder shape of those fields in the structure (the full dynamics statement is still a True stub). No upstream lemmas are invoked.
why it matters
Gives the trivial end of the propose-then-project axis in the RS→RL bridge: a completion that is correct exactly when the policy already respects σ=0. Downstream machinery that needs an LACompletion value (energy-cost checks, temperance bounds, Gibbs policies over 8-tick windows) can plug this in as the no-op baseline before nontrivial projectors (e.g. φ-scaling) are swapped in.
It sits under the module's hard-constraint design principle and the T6/eight-tick cadence story: physics is enforced by projection, not by baking constraints into the explorer. It does not itself force D=3 or the octave; it only supplies the identity point of the completion API those layers consume.
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