Finite-Orbit Rationality Tests for Renormalized Tail Dynamics

We continue the renormalized-tail program of {AbuGhuwaleh2026} in the direction suggested there by the rigidity of periodic and eventually periodic orbits. For an analytic power series \[ f(z)=∑n≥ 0 a_n z^n, a_n≠ 0, \] let \[ T_n^f(w):=∑k≥ 0{aₙ₊ₖ}{a_n}w^k \] be the normalized Taylor tails. The first paper proved that Tₙ₊ₘᶠ=Tₙᶠ for all large n if and only if f is a polynomial plus a rational tail of the form zN P(z)/(1-Λ zᵐ). The present paper asks the inverse finite-data question: what can be certified from a finite orbit segment or from finitely many Taylor coefficients? We prove four structural results. First, for every pair $(N,m)$ there is a canonical idempotent projector ΠN,m from normalized analytic germs onto the class of tails satisfying SN+mF=SNF; in coefficient form it is the unique polynomial-plus-rational-tail model matching the series through order $N+m$. Second, the gain in contact order of this model is measured exactly by the successive periodicity defects \[ Δ_n⁽ᵐ⁾:=a_n aₙ₊ₘ₊₁-aₙ₊₁aₙ₊ₘ. \] Third, no blind finite test can characterize exact eventual periodicity on the whole analytic class: for every finite jet length there exist eventually periodic and non-eventually-periodic germs with identical jets of that length. Fourth, once a tail TNᶠ has finite state dimension d (equivalently, is proper rational with denominator degree d), exact periodicity becomes finitely decidable: it is enough to compare the first d Taylor coefficients of TNᶠ and TN+mᶠ. We also prove Lipschitz and finite-horizon estimates showing that small observed defects force the orbit to stay close to its canonical periodic model on compact discs. These results convert the exact rigidity theorem of {AbuGhuwaleh2026} into effective finite-orbit rationality tests and constrained Pad\'e-type approximants adapted to the tail dynamics.