Deformations and local arithmetic of Lucas polynomial observations

Lucas recurrence polynomials with a variable product admit several different observations whose factorization and root geometry need not agree. For the zero-companion polynomial \(W_n(X)=(V_n(2,1+X)-2)/X\), we determine the complete negative-slope Newton polygon at every odd prime. The terminal segment depends on a Frobenius-reduced index, demonstrating why a parameter derivative alone does not determine the finite polynomial's local arithmetic. We also give an exact odd-prime Eisenstein classification for this family. For the level polynomials \(U_n(2,1+X)-q\), we determine the exact real interval of levels \(q\) where positive zeros are simple and strictly interlace with those of the neighbouring level polynomial at index \(n+2\), including root-motion and endpoint-asymptotic formulas. Integral coefficient identities and finite-field Frobenius reduction permit transfer of specified small-degree factor profiles; known periodicity of reversed Dickson polynomials is used rather than claimed new. The paper distinguishes local Newton geometry and real interlacing from global rational irreducibility; no general factorization theorem for all Lucas levels is asserted.