We extend the renormalized-tail theory of AbuGhuwaleh2026 from one complex variable to analytic power series in several variables. For \ f (z) =㵧 a_ z^, a_ 0, \ we attach to each multi-index the normalized corner tail \ T_ᶠ (w): =㵧a+a_w^. \ The orbit is governed by a commuting d-tuple of nonlinear renormalization operators \ Sⱼ (F) (w) =F (w) -F (w₁, , w₉-₁, ₀, ₖ_₉+₁, , wd) wⱼⱼF (0). \ We prove the exact identities T+₄䲛ᶠ=Sⱼ (T_ᶠ) and show that the correct linearizing coordinates are no longer a one-dimensional ratio sequence but a multiplicative lattice cocycle \ r, ₉=c+₄䲛c_ \ satisfying the square compatibility relations r, ₉r+₄䲛, ₈=r, ₈r+₄㶁, ₉. Our first main theorem is a bijective lattice shift linearization: the map F (r, ₉) is a conjugacy from the nonlinear ᵈ-action generated by (Sⱼ) ₉=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 ₉=₁ᵈ (1-ⱼ wⱼ) ^-1. Rectangular periodicity, \ Sⱼ^mⱼF=F (1 j d), \ is equivalent to a finite-block rational form \ F (w) =P (w) ₉=₁ᵈ (1-ⱼ wⱼ^{mⱼ) }, ₖ䲛P<mⱼ. \ In contrast, rank-one periodicity already carries infinite-dimensional strip moduli: in two variables, S₁S₂F=F if and only if \ F (w₁, w₂) =U (w₁) +V (w₂) -11- w₁w₂. \ 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 n admit first-order expansions with parameters (b₁, , bd), then \ T₍ᶠ (w) =₉=₁ᵈ11-ⱼ wⱼ +1n[₉=₁ᵈ11-ⱼ wⱼ ₉=₁ᵈ bⱼⱼ wⱼ1-ⱼ wⱼ+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.
Mohammad Abu-Ghuwaleh (Sat,) studied this question.