Taylor Tail Renormalization in Several Variables: Exact Lattice Shift Linearization, Rectangular Rigidity, and Anisotropic Fingerprints

We extend the renormalized-tail theory of {AbuGhuwaleh2026} from one complex variable to analytic power series in several variables. For \[ f(z)=∑α∈^d a_α z^α, a_α≠ 0, \] we attach to each multi-index α the normalized corner tail \[ T_α^f(w):=∑β∈^d{aα+β}{a_α}w^β. \] The orbit is governed by a commuting d-tuple of nonlinear renormalization operators \[ S_j(F)(w)={F(w)-F(w_1,,wⱼ₋₁,0,wⱼ₊₁,,w_d)}{w_j∂_jF(0)}. \] We prove the exact identities Tα+eⱼᶠ=Sⱼ(T_αᶠ) and show that the correct linearizing coordinates are no longer a one-dimensional ratio sequence but a multiplicative lattice cocycle \[ rα,j={cα+e_j}{c_α} \] satisfying the square compatibility relations rα,jrα+eⱼ,i=rα,irα+eᵢ,j. Our first main theorem is a bijective lattice shift linearization: the map F↦ (rα,j) is a conjugacy from the nonlinear ᵈ-action generated by (Sⱼ)ⱼ₌₁ᵈ to the coordinate translations on the space of admissible cocycles. This yields complete orbit realization. We then establish rigid orbit classifications. Common fixed points are exactly the product-geometric kernels ∏ⱼ₌₁ᵈ(1-ρⱼ wⱼ)⁻¹. Rectangular periodicity, \[ S_jm_jF=F (1≤ j≤ d), \] is equivalent to a finite-block rational form \[ F(w)={P(w)}{∏ⱼ₌₁^d(1-Λ_j w_jm_j)}, w_jP<m_j. \] In contrast, rank-one periodicity already carries infinite-dimensional strip moduli: in two variables, S₁S₂F=F if and only if \[ F(w_1,w_2)=U(w_1)+V(w_2)-1/1-Λ w_1w_2. \] Thus finite-index periodicity, rather than one-direction periodicity, is the genuinely rigid multivariate notion. On the asymptotic side we prove a universal product-geometric attractor theorem and the first anisotropic fingerprint along lattice rays. If the edge ratios near the ray nν admit first-order expansions with parameters (b₁,,bd), then \[ Tnν^f(w)=∏ⱼ₌₁^d1{1-ρ_j w_j} +1n[∏ⱼ₌₁^d1{1-ρ_j w_j}] ∑ⱼ₌₁^d b_jρ_j w_j/1-ρ_j w_j+o(n⁻¹), \] uniformly on compact polydiscs. We also show that separable multivariate models tensorize the full higher-order one-variable fingerprint hierarchy. The results show that Taylor tail renormalization is not a one-dimensional accident but a genuine lattice dynamical theory.