Discrete Hybrid Shift-Spectral Algebra

Let \ (ₑ, ₒ^disc: =N₀ʳˢ, \) \ (= (₁, , ᵣ) (0, 1) ʳ, \) \ ( (E) ˢ, \) \ (E: =\z: |z|>1\. \) For k₀ʳ and, define \ (e₊, (n, m): = (₈=₁ʳ h₊㶁㶁 (nᵢ) ) ᵐ, \) \ (h_ (n): =n^{} (+1). \) We prove that the algebraic direct sum generated by these basis vectors provides a canonical discrete hybrid model joining one-sided discrete fractional calculus on Nʳ with the whole-space discrete Weyl calculus on Zˢ. More precisely, the left nabla fractional sums Jᵢ: =₀, ₈^-ᵢ and the Caputo nabla fractional differences Cᵢ: =^ C₀, ₈^ᵢ act as commuting forward and backward shifts on the grade index k, while generalized discrete Weyl operators and standard discrete Weyl fractional differences act diagonally on the spectral label. Hence the hybrid operator algebra is the tensor product of a unilateral shift algebra in the one-sided directions and a diagonal multiplier algebra in the whole-space lattice directions. We further show that higher-order defects are localized entirely in the one-sided block: \ (C^mJ^m=I, \) \ (J^mC^m=I-<₌\) \ ( (m₀ʳ), \) where <₌ is the projection onto the discrete boundary-layer sector. No spectral defect occurs in the Weyl variables. We also prove a uniqueness theorem: among rising-factorial--character lattices with one-dimensional homogeneous-spectral components, the canonical basis is forced, up to an independent nonzero scalar on each spectral fiber. Finally, we show that mixed constant-coefficient difference equations reduce to spectral multiplication in together with finite downward recursion in the one-sided grades.