Support-Graph Reachability and Matrix-Composition Obstructions

This paper studies the difference between reachability in a Boolean support graph and reachability through evaluated projected matrix products. A support graph records whether individual projected operator blocks are nonzero, but it does not retain their images, kernels, or compositional alignment. Consequently, a two-step graph path need not produce a nonzero projected composition. The exact finite-dimensional layer distinguishes the direct support graph, the two-step composition graph, and the obstruction set. A product of two adjacent projected blocks vanishes exactly when the image produced by the first block lies entirely in the kernel of the second. The two-step composition graph is therefore contained in the square of the support graph, with equality requiring an additional image-kernel nondegeneracy condition. When the group action and all sector projectors preserve the same physical blocks, projected composition cannot connect sectors with disjoint block support at any finite length. In the canonical 228-dimensional Rubik realization, the registered sectorization contains nine sectors and its direct support graph has ten edges. Five distinguished two-step support paths satisfy the physical-block obstruction. For each path, all 324 ordered pairs of the eighteen face-turn generators were evaluated. The adjacent factor norms are of order one, while the resulting product norms lie between 1.10 times ten to the power of minus sixteen and 3.02 times ten to the power of minus fifteen. The numerical values are computational certificates for the declared complex128 realization. Exact vanishing relies on the stated block-preserving hypotheses. The five paths are finite witnesses that Boolean support-graph reachability can overapproximate projected-composition reachability; they do not imply that every support path is obstructed. Image-kernel compatibility is the missing local promotion datum. Source repository: https://github.com/dooven-prime/rime-lite Paper III of the Representation-Induced Mechanics and Evolution (RIME) program. Independent researcher; no institutional affiliation.