Maximal Commuting Sectors in Boundary-Augmented Hybrid Fractional Calculus

Let \(Ωr,s:=(0,∞)^r^s,\) \({α}=(α_1,, α_r)∈(0,1)^r,\) \(Λ⊂(C_+)^s,\) and let \(ek,λ(x,y) := (∏ᵢ₌₁^r{x_ik_iα_i}{Γ(k_iα_i+1)})eλ,y\) \((k_0^r,\ λ∈Λ)\) be the canonical hybrid basis. The preceding paper constructed weighted Banach completions X_ρ,ηᵖ 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\(X_{{ρ},τ,η}ᵖ = X_{{ρ},η}ᵖ ⊕ w_rT_wᵖ.\) 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, \([C_i,C_j]tw,k,λ = t_{wji,k^{{\{i,j\}}},λ} - t_{wij,k^{{\{i,j\}}},λ},\) 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 \(K_{{ρ},τ,η}ᵖ = X_{{ρ},η}ᵖ ⊕ q(w)≤ 1T_wᵖ,\) 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.