Hybrid Shift-Spectral Algebra on Mixed Regions

Let r,s, let Ωr,s:=(0,∞)ʳ× Rˢ, and fix α=(α₁,,αᵣ)∈(0,1)ʳ,Λ⊂(C₊)ˢ,₊:=: z>0\. For k₀ʳ and λ∈Λ, define ek,λ(x,y):= (∏ᵢ₌₁ʳ xᵢkᵢαᵢΓ(kᵢαᵢ+1))eλ,y, (x,y)∈Ωr,s. We prove that the algebraic direct sum H_α,Λᵃˡᵍ := (k,λ)₀ʳ×Λ Cek, λ provides a canonical hybrid model joining the multidimensional half-space calculus of Riemann--Liouville and Caputo type with the whole-space Weyl calculus. More precisely, the one-sided operators Jᵢ:=₀ Ixᵢαᵢ, Cᵢ:=₀CDxᵢαᵢ (1≤ i≤ r) act as commuting forward and backward coordinate shifts on the k-index, while generalized Weyl operators in the y-variables act diagonally on the spectral label λ. Thus the hybrid operator algebra is the tensor product of a commuting unilateral shift algebra in the one-sided directions and a diagonal multiplier algebra in the whole-space directions. We further show that higher-order Caputo defects are localized entirely in the one-sided block: for every m₀ʳ, CᵐJᵐ=I, JᵐCᵐ=I-Π<m, where Π<m is the projection onto the boundary-layer sector k≥ m,\ λ∈ΛCek,λ; no spectral defect occurs in the Weyl variables. We also prove a uniqueness theorem: among monomial--character lattices with one-dimensional homogeneous-spectral components, the canonical basis is forced, up to an independent nonzero scalar on each spectral fiber λ.