Exact Structural Abstraction and Tractability Limits

Any rigorously specified problem determines an admissible-output relation R, and exact correctness depends only on the induced decision quotient relation s∼_Rs′ ⇔ Adm_R(s) = Adm_R(s′). Exact relevance certification asks which coordinates recover those classes. Decision, counting, search, approximation, PAC/regret/risk, randomized-output guarantees, anytime or finite-horizon guarantees, and distributional guarantees all reduce to this quotient-recovery problem.

Universal exact-semantics reduction identifies admissible-output quotient recovery as the canonical object. Optimizer-quotient realizability is maximal, so quotient shape alone cannot mark a tractability frontier. Orbit gaps are the exact obstruction to classification by closure-law-invariant structural predicates.

Exact classification by closure-law-invariant predicates succeeds exactly when the target is constant on closure orbits; on a closure-closed domain, equivalently, when the positive and negative orbit hulls are disjoint, in which case there is a least exact closure-invariant classifier. Across four natural candidate structural tractability criteria, a uniform pair-targeted affine witness produces same-orbit disagreements and rules out exact structural classification on the full binary pairwise domain. Because that witness class already sits inside the universal semantic framework, the same obstruction applies to any universal exact-certification characterization over rigorously specified problems. Restricting the domain helps only by removing orbit gaps. Without explicit margin control, arbitrarily small utility perturbations can flip relevance and sufficiency.