Hybrid Shift-Spectral Algebra on Mixed Regions

Let r, s, let ₑ, ₒ: = (0, ) ʳ Rˢ, and fix = (₁, , ᵣ) (0, 1) ʳ, (C_+) ˢ, _+: =\z: z>0\. For k₀ʳ and, define e₊, (x, y): = (₈=₁ʳ xᵢ^kᵢᵢ (kᵢᵢ+1) ) e^, y, (x, y) ₑ, ₒ. We prove that the algebraic direct sum H, ^alg: = (₊, ) 䃐⋒ Ce₊, 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 Iₗ㶁^ᵢ, Cᵢ: =₀^CDₗ㶁^ᵢ (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. 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, C^mJ^m=I, J^mC^m=I-<₌, where <₌ is the projection onto the boundary-layer sector ₊ ₌, \ Ce₊, ; 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.