Remarks on Primitive Regulation

Abstract

We prove, and mechanize in Rocq, an abstract obstruction theorem for closure predicates $C : \mathsf{Form} \to \mathsf{Prop}$ over the closed implication-falsity fragment $A,B ::= \bot \mid A \to B$. Evaluation completeness, $\mathsf{Eval}(C)$, says that every formula-valued behavior of codes is represented up to closure equivalence, where $A \simeq_C B$ abbreviates $C(A \to B) \land C(B \to A)$. For any $C$ closed under modus ponens and satisfying consistency, this completeness principle is incompatible with the internal excluded-middle schema $\mathsf{LEM}(C)$, namely $\forall A, C(A)\lor C(\neg A)$. Thus $\mathsf{Eval}(C), \mathsf{MP}(C), \mathsf{Cons}(C)$, and $\mathsf{LEM}(C)$ cannot hold jointly. The proof uses evaluation completeness to obtain a formula $B$ such that $B \simeq_C \neg B$. Applying $\mathsf{LEM}(C)$ to this $B$, either alternative gives $C(\bot)$ by detachment, contradicting consistency. Consequently, any Boolean decision procedure for $C$ induces the obstructed excluded-middle schema. Mere refutation behaves differently: its false branch carries no closure condition, and is therefore inhabited by the always-false classifier.