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.

翻译：Rice定理表明，部分递归函数的非平凡外延性质是不可判定的。对于有限加权布尔优化/CSP风格切片而言，可处理性分类存在一个Rice风格的结构类比：正确性迫使性质在定理作用下保持表示变换下的不变性，而轨道间隙正是闭包不变谓词精确分类的唯一障碍。该范围对精确规范具有普适性。任何严格定义的问题都确定一个可接受输出关系，而精确认证仅取决于诱导等价关系$s \sim_R s' \iff \operatorname{Adm}_R(s)=\operatorname{Adm}_R(s')$。决策、搜索、近似、随机输出、统计及分布保证均通过该可接受输出商集纳入框架。在具有多项式时间可计算传递的闭包封闭域上，每个正确的可处理性分类器必须在闭包轨道上保持常值。精确的闭包不变分类仅当正轨道凸包与负轨道凸包不相交时成立；此时闭包凸包即为给出最小精确分类器的闭包算子。有限结构范畴是通过提取成对语法得到的基本局部一阶片段。四类二元成对障碍族——优势对集中、边界掩蔽、幽灵动作支持及动作特定偏移——为自然有限结构谓词见证了同轨道分歧性，而凸包分离定理给出了分类可能时的正判据。若缺乏显式边界控制，任意微小收益扰动即可翻转相关性与充分性。