Canonical Decomposition of the Q5 Grading Sectors

We derive a canonical 1: 3: 12 decomposition of the function space on the five-cube Q₅ using complementary grading relative to a distinguished axis. The 12-dimensional middle sector admits a natural factorization H12≅V4⊗T3 where V4≅Z2×Z2 arises from independent involutive symmetries (bridge-bit and complementary-pair exchange), and T₃ is a triadic carrier induced by the Gray-code–forced partition of residual coordinate pairs. The reflected Gray traversal provides a dynamical realization of this structure, organizing the middle sector into six consecutive doublets with a canonical B3A3 ordering. The resulting orbit decomposition yields three complementary-pair classes that act as multiplicity channels for a single underlying triadic degree of freedom. A canonical basis of the triplet factor is constructed via projector intersections A ₋䲛 A, which uniquely label each orbit by combining the pure matching axis with one residual axis, resolving degeneracies not detectable from Gray frequency data alone. One triplet direction is canonically selected by intersection with the defect axis, corresponding to a localized support mode, while the remaining directions correspond to extended carrier and lift modes distinguished by adjacency structure. This produces a rigid internal architecture consisting of one invariant sector, one irreducible triadic carrier, and a multiplicity-enhanced replication of that carrier across discrete symmetry sectors: 1 3 (V₄ 3). The construction shows that alignment constraints imposed by discrete symmetries and combinatorial structure can organize independent degrees of freedom into coherent triadic modes with multiplicity structure, without requiring continuous symmetry groups or external dynamical assumptions. This provides a purely discrete realization of how structured replication of a fundamental carrier can emerge from internal geometric constraints.