From Admissibility to Envelope Cost: Why Verifier-Governed Systems Require Projection-Aware Defect

This work establishes a structural bridge between verifier-governed execution systems and projection-induced cost models. We prove that any verifier enforcing admissibility over observable traces, without access to hidden realizations, must bound defect using a supremum envelope over the corresponding fiber of hidden executions: D(R) ≥ supξ ∈ F_R W(ξ) This result demonstrates that envelope cost is not a design choice, but a necessary consequence of limited observability under projection. The paper closes the operational framework introduced in prior work (Coh Category / governed execution systems) by identifying the structural origin of admissibility-based behavior. It further prepares the transition to projection-induced oplax cost structure, where envelope cost is shown to obey subadditive composition laws. Scope conditions:The theorem is stated for observable records with nonempty fibers and finite/envelope-bounded hidden costs. Executable verifiers may use deterministic conservative envelopes satisfying δ̂(R) ≥ δ(R). Files included:- Full LaTeX source and compiled PDF- Figures illustrating projection and envelope structure- Minimal Lean alignment sketch for fiber-level soundness This publication represents the formal closure of Layer 1 (admissibility) and establishes the foundation for Layer 2 (projection-induced oplax structure).