Maximal Commuting Sectors in Boundary-Augmented Hybrid Fractional Calculus

Let \ (ₑ, ₒ: = (0, ) ʳˢ, \) \ (= (₁, , ᵣ) (0, 1) ʳ, \) \ ( (C_+) ˢ, \) and let \ (e₊, (x, y): = (₈=₁ʳxᵢ^kᵢᵢ (kᵢᵢ+1) ) e^, y\) \ ( (k₀ʳ, \) \) be the canonical hybrid basis. The preceding paper constructed weighted Banach completions X, ^p of the algebraic span of this basis and proved that the partial Caputo tuple acts there as a commuting family of weighted backward shifts. In the present paper we adjoin ordered boundary-trace sectors indexed by words in the one-sided coordinates and thereby construct a boundary-augmented Banach space\ (X, , ^p = X, ^p ₖ㶂Tw^p. \) On the canonical block the partial Caputo operators act exactly as in the commuting shift algebra, while on the trace blocks they lower residual grades and append new trace letters when a coordinate reaches grade zero. The spectral multipliers remain diagonal on every block. The resulting extended tuple is no longer commuting in general. Its commutator is explicit: for distinct free coordinates i and j and simultaneous vacuum in those coordinates, \ (Cᵢ, Cⱼtₖ, ₊, = tₖ₉₈, ₊^\{₈, ₉\, } - tₖ₈₉, ₊^\{₈, ₉\, }, \) while in all other cases the commutator vanishes. We then prove that the maximal closed graded invariant sector containing the canonical completion and carrying a commuting Caputo tuple is \ (K, , ^p = X, ^p ₐ (ₖ) ₁Tw^p, \) where q (w) is the number of free one-sided coordinates remaining after the ordered trace word w. Thus noncommutativity is localized entirely in defect layers with at least two free one-sided coordinates. The whole-space Weyl block remains diagonal and plays no role in the failure of commutativity.