Taylor Tail Renormalization Theory Exact Shift Linearization, Orbit Rigidity, and Asymptotic Fingerprints of Analytic Power Series

Let f (z) = ∑n≥0 anzn be analytic at the origin, and assume that no Taylor coefficient vanishes. We study the normalized Taylor tails Tfn (w) := ∑k≥0 an+k/anwk, not as isolated remainders but as a discrete renormalization orbit on the space of normalized analytic germs. The governing map is the nonlinear operator S(F)(w) := F(w)−1 wF′ (0) , which acts as a coefficient shift followed by canonical normalization. The exact identity Tfn+1 = S(Tfn) turns the Taylor coefficients of f into a dynamical system. We develop a self-contained theory of this dynamics. First, we prove that the nonlinearity of S is exactly linearized in ratio coordinates: the map F→ (Sn F)′(0) n≥0 conjugates S to the one-sided shift on an explicit space of admissible ratio sequences. This yields complete reconstruction of the orbit and a realization theorem for all admissible orbits. Second, we classify the rigid orbit types: fixed points are exactly geometric series, periodic points are exactly rational functions with denominator 1 − Λwm, and eventual periodicity is equivalent to a polynomial plus a rational tail. Third, we exhibit genuinely rich internal dynamics by constructing compact invariant subsystems on which S is conjugate to full shifts on finite alphabets. Fourth, on the asymptotic side, ratio limits force universal geometric profiles, while first- and second-order corrections to the coefficient ratios produce universal corrections to the tail orbit. In particular, dominant algebraic singularities leave a precise first asymptotic fingerprint on the renormalized tails. We also prove exact transport laws under differentiation and Hadamard products. The basic normalized tail object overlaps, up to an index shift, with the normalized remainders recently studied in the special-functions literature. The present contribution is different in focus: it isolates the renormalization operator itself, proves exact shift linearization and orbit realization, identifies symbolic invariant subsystems, and develops rigidity and asymptotic classification results for the resulting dynamical flow.