Any rigorously specified problem determines an admissible-output relation $R$, and exact correctness depends only on the induced decision quotient relation $s \sim_R s' \iff \operatorname{Adm}_R(s)=\operatorname{Adm}_R(s')$. 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.

翻译：任何严格定义的问题都确定一个可容许输出关系$R$，而精确正确性仅取决于导出的决策商关系$s \sim_R s' \iff \operatorname{Adm}_R(s)=\operatorname{Adm}_R(s')$。精确相关性认证探讨哪些坐标能恢复这些类别。决策、计数、搜索、近似、PAC/遗憾/风险、随机输出保证、随时或有限时域保证以及分布保证，均归结为此商恢复问题。通用精确语义归约将可容许输出商恢复识别为标准对象。优化器商可实现性是最大的，因此商形状本身无法标记易处理性边界。轨道间隙是闭包律不变量结构谓词分类的精确障碍。当目标在闭包轨道上为常数时，通过闭包律不变量谓词进行的精确分类才能成功；在闭包封闭域上，等价地，当正负轨道凸包不相交时，存在一个最小的精确闭包不变量分类器。在四个自然候选结构易处理性准则中，统一的成对目标仿射见证产生同轨道不一致性，并排除了全二元成对域上的精确结构分类。由于该见证类已存在于通用语义框架内，相同的障碍适用于任何对严格定义问题进行的通用精确认证刻画。限制域仅通过移除轨道间隙来提供帮助。若无显式边界控制，任意小的效用扰动都可能翻转相关性与充分性。