Deformations and local arithmetic of Lucas polynomial observations

A common second-order recurrence does not force common factorization behaviour. We study this distinction for the Lucas source with variable product and for its level and fixed-product observations. The parameter derivative at an integer index is controlled by two explicit logarithmic series, which explain candidate valuation patterns. Using direct integral coefficient identities and Frobenius reduction (rather than a formal parameter-jet transfer), we determine the entire negative-slope Newton polygon of \(W_n=(V_n(2,1+x)-2)/x\) at every odd prime. The last side depends on a Frobenius-reduced index and cannot in general be read from the first derivative alone. For the level family \(U_n(2,1+x)-q\) we determine the exact interval of strict positive interlacing with the neighbouring index \(n+2\), including its endpoint asymptotics. Finally, known periodicity identities for reversed Dickson polynomials, combined with Möbius inversion, transfer distinct small-degree factor counts on the nonsingular locus; the periodicity itself is not claimed as new. Explicit cyclotomic and quadratic observations delineate what these geometric and local conclusions do not imply about rational irreducibility. The final section gives an exact odd-prime Eisenstein classification for the zero-companion \(W_n\) and stable interlacing chains, which distinguish arithmetic factorization from real root motion.