We extend the inverse side of renormalized Taylor tail dynamics in four directions. First, for dominant coefficient packets \ aₙ=ⁿ n^-1Qₙ (1+ (n^-1-) ), Qₙ==₁L c__ⁿ, \ we prove that the normalized tails no longer approach a single geometric kernel. On every stable subsequence they admit the universal interference expansion \ Tₙᶠ (w) =ₙ (w) +-1n\, xₓₙ (x) |ₗ= ₖ+o (n^-1), \ where \ ₙ (x) =1Qₙ=₁L c__ⁿ1-_ 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 \ u, (n) =n^- (n) ^, \ covering \ (n^-j (n) ^-r\), \ (n^-j-\), and \ ( (n) q n^-j\). For every stable finite family of such scales, a ratio expansion \ a₍+₁aₙ = (1+_ c_ u_ (n) +o (u_* (n) ) ) \ propagates into a universal tail expansion with rational fingerprints \ (P_\), and the hierarchy is triangular: \ P_' (0) =c_. \ Third, for the multivariate product-type class \ f (z) =₉=₁ᵈ (1-ⱼ zⱼ) ^-ⱼ, \ the corner-tail orbit along a ray \ (n\) 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.
Mohammad Abu-Ghuwaleh (Sat,) studied this question.