Locality, Consistency, and the Tractability Frontier

Rice's theorem shows that nontrivial extensional properties of partial recursive functions are undecidable. For finite weighted Boolean optimization/CSP-style slices, a Rice-style structural analogue holds for tractability classification: correctness forces invariance under theorem-forced presentation moves, and orbit gaps are exactly the obstruction to exact classification by closure-invariant predicates. The scope is universal for exact specifications. Any rigorously specified problem determines an admissible-output relation, and exact certification depends only on the induced equivalence relation $s \sim_R s' \iff \operatorname{Adm}_R(s)=\operatorname{Adm}_R(s')$. Decision, search, approximation, randomized-output, statistical, and distributional guarantees all enter through this admissible-output quotient. On closure-closed domains with polynomial-time-computable transports, every correct tractability classifier must be constant on closure orbits. Exact closure-invariant classification is possible exactly when positive and negative orbit hulls are disjoint; in that case the closure hull is a closure operator giving the least exact classifier. The finite structural regime is a basic-local first-order fragment over extracted pairwise syntax. Four binary-pairwise obstruction families--dominant-pair concentration, margin masking, ghost-action support, and action-specific offsets--witness same-orbit disagreement for natural finite structural predicates, while the hull-separation theorem gives the positive criterion when classification is possible. Without explicit margin control, arbitrarily small utility perturbations can flip relevance and sufficiency.