Right $q$-Vector Differentiation and Localized Fischer Decompositions on Radial Algebras

We construct a right $q$-vector derivative on radial algebras that is intrinsic to the radial variables. A coordinatewise replacement of ordinary partial derivatives by Jackson derivatives does not preserve even the one-vector radial subalgebra, so the deformation must act on the mixed scalar invariants associated with a distinguished vector variable. After extending scalars to $\text{B}=\mathbb{R}[q,q^{-1},Q,(1-q{)}^{-1}]$, where $Q$ records the formal dimension through the specialization $Q=q^M$, we define for every finite set $Y\subset S\setminus \{x\}$ a right $q$-vector derivative ${\mathrm{\partial }}_{x,q}^{Y,\mathrm{R}}$ on $R_{\text{B}}(\{x\}\cup Y)$. These operators are equivariant under relabelling and compatible with inclusions of finite sets; hence they induce a direct-limit operator on every radial algebra. Two Fischer-type results are obtained. The anticommutator with exterior creation is triangular with explicit diagonal resonance factors, and its localized inverse yields a Green decomposition by complementary projections. For full left multiplication $L_x$, the finite homogeneous operators ${\mathrm{\partial }}_{x,q}^{Y,\mathrm{R}}{L}_x$ have nonzero generic determinants. Localizing by these determinants gives a right-monogenic Fischer decomposition on arbitrary radial algebras. The determinant factors are computed completely for one and two vector variables, and a support filtration reduces the general denominator problem to exact-support determinants. A degree-zero factor in even support rank produces genuine resonances for $0<q<1$, showing that a universal unlocalized theorem requires explicit exclusion of resonance values.