Abstract This paper develops a three-layer cohomological framework for groupoid spaces arising from algebraic equations. The outer groupoid space records coefficient migration under permutation of coefficient positions; its cohomology \ (H^*_\) measures obstructions to external coefficient rearrangement. The inner groupoid space records root-ratio generation and layered root dynamics; its cohomology \ (H^*_\) captures history lifts and value compression. The perfect groupoid space is the fiber product of the outer and inner groupoids over a common equation value, and its cohomology \ (H^*_\) records simultaneous compatibility of outer and inner data. The central innovation is the matching-defect cohomology \ (H^*_\), defined as the shifted mapping cone of the comparison map between outer/inner data and perfect data. It measures the obstruction to matching an outer cohomology class and an inner cohomology class over a shared equation-value record. The theory is developed from standard groupoid cohomology foundations: derived functors, Ext descriptions, classifying spaces, spectral sequences, multiplicative structures, and nonabelian \ (H¹\) with torsor interpretations. Structural theorems establish Morita invariance, compression spectral sequences, acyclic-fiber collapse, finite-stabilizer vanishing, common-value descent, and matching-obstruction torsor principles. The framework interfaces with homotopy theory via homotopy-pullback models for perfect classifying spaces, with algebraic geometry via Vieta quotient-stack descent, and with Galois theory via nonabelian Galois-label packages and central obstruction classes. The paper distinguishes history groupoids, compressed-value groupoids, layered cohomology, and the outer–inner–perfect matching problem, keeping structural cohomological results separate from arithmetic or analytic realization hypotheses. Computations reduce to stabilizers of connected components, associated graded pieces of filtered complexes, or finite-stage approximations. --- Keywords groupoid cohomology; coefficient migration; root-ratio generation; outer groupoid; inner groupoid; perfect groupoid; matching-defect cohomology; Vieta map; classifying space; nonabelian cohomology; Galois torsor; spectral sequence; homotopy pullback
Jianming Wang (Thu,) studied this question.