Maximal Commuting Sectors in Boundary-Augmented Discrete Hybrid Fractional Calculus

Let \(Ωr,sᵈⁱˢᶜ:=N_0^r^s,\) \({α}=(α_1,,α_r)∈(0,1)^r,\) \(Λ⊂(E)^s,\) \(E:=\{z:|z|>1\},\) and let \(ek,λ(n,m) := (∏ᵢ₌₁^r hk_iα_i(n_i))λ^m\) \((k_0^r,\ λ∈Λ)\) be the canonical discrete hybrid basis. The preceding paper constructed weighted Banach completions X_ρ,ηp,disc of the algebraic span of this basis and proved that the discrete 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 \(X_{{ρ},τ,η}p,disc = X_{{ρ},η}p,disc ⊕ w_rT_wp,disc.\) On the canonical block the partial discrete 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 discrete Caputo tuple is \(K_{{ρ},τ,η}p,disc = X_{{ρ},η}p,disc ⊕ q(w)≤ 1T_wp,disc,\) 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 discrete Weyl block remains diagonal and plays no role in the failure of commutativity.