Beyond Product-Type Smooth Critical-Point Fingerprints for Renormalized Tail Orbits: Directional Edge Maps, Same-Scale Shape Separation, and Finite-Ray Recovery

The product-type multivariate theory of renormalized Taylor tails has constant edge ratios and a separable first fingerprint. Genuine smooth critical-point models are richer: the effective edge vector depends on the ray, and mixed pole patterns can already appear at the first nontrivial scale. We isolate this phenomenon at the orbit level. For a fixed ray ν∈ᵈ we assume a smooth critical-ray transport law of the form \[ {anν+β}{anν} = ρ(ν)^β ( 1+{P1,ν(β)}{n}+{P2,ν(β)}{n^2} ) +n,β(ν), \] where ρ(ν)∈(^×)ᵈ is a directional edge map and P1,ν,P2,ν are transport polynomials. We introduce the fingerprint calculus \[ ρ[P](w):=∑β∈^dP(β)ρ^β w^β = P(D)∏ⱼ₌₁^d1/1-ρ_j w_j, D_j=x_j∂x_j, \] and prove the universal expansion \[ Tnν^f(w)=Gρ(ν)(w)+1/nρ(ν)[P1,ν](w)+1/n^2ρ(ν)[P2,ν](w)+o(n⁻²). \] The map (ρ,P₁,P₂)↦ (G_ρ,_ρ[P₁],_ρ[P₂]) is injective. Hence the first two orbit fingerprints recover the directional edge vector and the first two transport polynomials exactly. In particular, distinct polynomial shapes at the same asymptotic scale give distinct fingerprints; the novelty is therefore not merely hierarchical in the scale variable. We then treat the genuinely coupled model \[ f_σ(z)=(1-∑ⱼ₌₁^d σ_j z_j)-θ. \] Its edge map is \[ ρ_j(ν)=σ_j|ν|_1/ν_j, \] and its first transport polynomial is quadratic: \[ P1,ν(β)= ∑ⱼ₌₁^d(θ/|ν|_1-1{ν_j})β_j +12∑ⱼ₌₁^d(1{|ν|_1}-1{ν_j})β_j(β_j-1) +1{|ν|_1}∑i<jβ_iβ_j. \] Thus mixed first-order kernels appear with a universal nonzero coefficient. This gives a clean orbit-theoretic distinction between exact product-type transport and genuine smooth coupling. Finally, for degree-two first fingerprints we give explicit finite detectors from finitely many coefficient ratios or, equivalently, finitely many derivatives of the normalized tail at the origin. Richardson extrapolation recovers the edge map with deterministic (N⁻²) bias, one-step residuals recover the linear coefficients with (N⁻¹) bias, and cross-ratios recover the mixed quadratic coefficients with (N⁻¹) bias. On a finite stencil of nearby rays this yields a first-order reconstruction of the local edge-map jet, hence of the local smooth critical transport.