Defect Local–Global A: Gluing Obstructions and Weighted Dictionaries in the Local–Global Passage

The local–global principle exhibits two failure modes of different nature in descent theoryand arithmetic geometry: the first is distortion, where local objects exist and the descentdata are effective, but the global object is distorted relative to the local data; the second isabsence, where the local objects themselves may fail to exist, and the absence of the globalobject is measured by a second-order obstruction or by the kernel of the local–global map.This paper establishes an exact counting identity on the finite groupoid layer: the groupoidcardinality in the isomorphism-class sense equals a function of the local data, the coarse countequals the cardinality of first-order non-abelian cohomology, and the ratio of the two—thedefect factor—is completely determined by the moment family of the conjugacy-class-sizelist. This closed form shows that, on the first-order working layer, the defect factor measuresthe deviation of the counting prediction caused by nontriviality of automorphisms, and notthe absence of objects. On the arithmetic side, the Ono–Sansuc formula is positioned asa proven instance of the same skeleton at the measure-theoretic endpoint, where a Shatype kernel carries the failure semantics of the existence layer. Exact enumeration over fourfamilies of finite groups and closed-form reconciliation close all inputs via two independentroutes, and yield the complete shape of the spectrum: endpoints, monotonicity, and limit areread directly from the conjugacy-class-size distribution. The realization problem for fusionsystems on the group-theoretic side is registered as a conditional framework; its defect indexis computable at the toy layer, but the category-level construction of the realization functorremains an open debt.