We continue the renormalized-tail program initiated in AbuGhuwaleh2026. For an analytic power series\ f (z) =₍ ₀ aₙ zⁿ, aₙ 0, normalized Taylor tailsₙᶠ (w): =₊ ₀a₍+₊aₙwᵏ the renormalization orbit studied in AbuGhuwaleh2026. That paper established the universal geometric limit profile and the first two asymptotic fingerprints of the orbit. Here we prove the full higher-order hierarchy. Assuming₍+₁aₙ= (1+₁n+₂n²++ₘnᵐ+ (n^-m-) ), ^, \ >0, show that for every fixed s (0, 1), ₙᶠ (w) =11- w+₉=₁ᵐ Pⱼ (w;₁, , ⱼ) nʲ+o (n^-m) on \| w| s\. Each Pⱼ is a universal rational function depending only on (₁, , ⱼ), and we give a constructive recursive algorithm for all of them via logarithmic cumulants, Faulhaber sums, and Newton interpolation. The hierarchy is triangular: the coefficient ⱼ is recovered directly from the linear term of Pⱼ. In particular, the first two fingerprints from AbuGhuwaleh2026 are recovered as the cases m=1, 2, while the higher-order terms are new. We also compute the third and fourth fingerprints explicitly and derive the resulting hierarchy for algebraic singularities.
Mohammad Abu-Ghuwaleh (Sat,) studied this question.