Three-Layer Weak Graph Groupoids: Matrix Permutation Classes, Node-Expanded Groupoid Spaces, and Graph Groupoid Cohomology

This paper develops a three-layer weak group theory for graphs based on the atom–slot–pairing principle: vertices are movable labels (atoms), vertex positions are fixed slots, and adjacency is first defined on the slot set. A label pairing between positions and labels determines a realized graph; a weak label permutation changes this pairing without moving positions, thereby generating a class of adjacency matrices on the label frame. The original graph embeds as the identity member of this matrix-permutation class, but the full adjacency-matrix permutation orbit is also used as input data to generate derived subisomorphism classes through a feedback operator. The first layer defines label-induced matrix permutation classes and their weak orbits. The second layer promotes these matrices and realized graphs to objects of a graph groupoid space, where label permutations become arrows, stabilizers recover automorphism groups, and graph groupoid cohomology is defined through nerve complexes and local coefficient systems. The third layer replaces each node of the original groupoid by a local graph groupoid generated from the adjacency matrix attached to that node, producing a node-expanded graph groupoid space with its own cohomology. The framework includes realization-fiber formulas, matrix stabilizers, Burnside counts, compression spectral sequences, comparison cones, matching-defect cohomology, feedback dynamics, descent and transfer results, torsor interpretations of first cohomology, obstruction classes for expansion sections, gluing principles, and fixed-frame reconstruction. Directed graphs, weighted graphs, hypergraphs, simplicial complexes, and filtered graph systems fit into the same relational formalism. The theory gives exact orbit, stabilizer, groupoid, expansion, and cohomological objects; it does not by itself provide a polynomial-time solution to graph isomorphism, nor does it turn incomplete graph invariants into complete classifications. Keywords weak group; graph theory; label permutation; adjacency matrix; matrix permutation class; graph groupoid space; node-expanded groupoid; graph groupoid cohomology; matching defect; descent; torsor; feedback obstruction; subisomorphism class; position-slot framework