Inverse Singularities from Renormalized Tail Orbits: Multi-Point Interference, Mixed-Scale Fingerprints, and Anisotropic Recovery

We extend the inverse side of renormalized Taylor tail dynamics in four directions. First, for dominant coefficient packets \[ a_n=ρ^n nα-1Q_n(1+(n-1-η)), Q_n=∑₌₁^L c_ω_^n, \] we prove that the normalized tails no longer approach a single geometric kernel. On every stable subsequence they admit the universal interference expansion \[ T_n^f(w)=_n(ρ w)+α-1/n\,x∂_x_n(x)|x=ρ w+o(n⁻¹), \] where \[ _n(x)=1/Q_n∑₌₁^L c_ω_^n/1-ω_ x. \] Thus multi-point dominant singularities produce periodic or quasi-periodic orbit profiles whose poles recover the dominant locations. Second, we build a finite mixed-scale algebra based on \[ uσ,μ(n)=n-σ(log n)^μ, \] covering \(n⁻ʲ(log n)⁻ʳ\), \(n-j-ϑ\), and \((log n)^q n⁻ʲ\). For every stable finite family of such scales, a ratio expansion \[ {aₙ₊₁}{a_n} = ρ(1+∑λ∈ c_λ u_λ(n)+o(u_*(n))) \] propagates into a universal tail expansion with rational fingerprints \(P_λ\), and the hierarchy is triangular: \[ P_λ'(0)=c_λ. \] Third, for the multivariate product-type class \[ f(z)=∏ⱼ₌₁^d (1-ρ_j z_j)-θ_j, \] the corner-tail orbit along a ray \(nν\) recovers the edge point, the anisotropic exponents, and, after one calibration ray is known, any second ray direction. Finally, we give a finite detector scheme: an \(L\)-point interference packet is reconstructed from \(2L\) probes by a Hankel-Prony system, each new mixed-scale coefficient from one additional residual, and the multivariate anisotropic data from first-derivative residuals on two selected rays.