The $\mathrm{U}$-Derivative Associated with Linear Recurrence Sequences

We associate a canonical operator to every linear recurrence sequence admitting a distinct-root Binet representation ${\mathrm{U}}_n={\sum }_{j=1}^k{A}_j{\lambda }_j^n\mathrm{.}$ The construction is based on the spectral Binet operator ${\mathcal{B}}_{\mathrm{U}}={\sum }_{j=1}^k{A}_j{T}_{{\lambda }_j},$ where $T_{{\lambda }_j}f(x)=f({\lambda }_{j}x)$ is the dilation operator associated with the characteristic root ${\lambda }_j$. The corresponding recurrence derivative is defined by ${\mathrm{D}}_{\mathrm{U}}=x^{-1}{\mathcal{B}}_{\mathrm{U}}\mathrm{.}$ Its fundamental property is the monomial identity ${\mathrm{D}}_{\mathrm{U}}(x^n)={\mathrm{U}}_n{x}^{n-1},$ which establishes a direct correspondence between recurrence sequences and degree-lowering operators. We develop the operator theory of ${\mathrm{D}}_{\mathrm{U}}$, including iteration, a spectral product rule, the associated Leibniz defect, and a quotient rule. The construction reveals a natural bifurcation determined by the value ${\mathrm{U}}_0$, producing the Ward and non-Ward regimes as intrinsic operatorial phenomena. The product structure is shown to be governed by $k-2$ residual spectral channels, leading to notions of spectral cancellation and spectral balance. As an application, we obtain a complete qualitative classification of second-order recurrence calculi. Using Horadam sequences as a universal second-order model, we prove that every such calculus is determined by the invariant triple $({\mathrm{W}}_0,\mathrm{\Delta },Z_{\mathrm{W}}),$ where ${\mathrm{W}}_0$ identifies the Ward/non-Ward type, $\mathrm{\Delta }$ determines the spectral geometry, and $Z_{\mathrm{W}}$ determines regularity or singularity.