Inverse Singularity Detection from Renormalized Tail Orbits

We continue the renormalized-tail program initiated in AbuGhuwaleh2026 by addressing an inverse problem: what dominant singular data can be recovered from the orbit of normalized Taylor tails \ Tₙᶠ (w): =₊ ₀a₍+₊aₙwᵏ \ of an analytic power series f (z) =₍ ₀ aₙ zⁿ with no vanishing coefficients? We prove a universal inverse transport principle. If \ a₍+₁aₙ= (1+c uₙ+o (uₙ) ) \ on an admissible logarithmically flat scale uₙ, then uniformly on each closed disk \ | w| s\ with 0<s<1, \ Tₙᶠ (w) =11- w+c uₙ w (1- w) ²+o (uₙ), \ and conversely the same tail expansion forces the same ratio expansion. Thus the first deviation of the orbit from the geometric attractor has a rigid universal shape, and its scalar amplitude is exactly the ratio perturbation. We then combine this transport law with standard transfer theorems of singularity analysis to build an inverse dictionary for three dominant model classes: algebraic singularities, algebraic-logarithmic singularities, and confluent algebraic singularities. In particular, the orbit determines the dominant singularity location =^-1, the algebraic exponent, the logarithmic power, and for confluent terms b (1-z/) ^ with 0<<1, both the exponent and amplitude b. We also prove that first-order orbit data alone cannot distinguish algebraic, algebraic-logarithmic, and confluent models sharing the same principal exponent; secondary scales are genuinely necessary. The results provide a concrete inverse diagnostic procedure for renormalized tail dynamics.