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) =₍ ₀ aₙ zⁿ, aₙ 0, \ let \ Tₙᶠ (w): =₊ ₀a₍+₊aₙwᵏ \ be the normalized Taylor tails. The first paper proved that T₍+₌ᶠ=Tₙᶠ for all large n if and only if f is a polynomial plus a rational tail of the form zN P (z) / (1- 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 ₍, ₌ from normalized analytic germs onto the class of tails satisfying S^N+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 \ ₙ^ (m): =aₙ a₍+₌+₁-a₍+₁a₍+₌. \ 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 T₍+₌ᶠ. 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.
Mohammad Abu-Ghuwaleh (Sat,) studied this question.