Finite-distance mixed-axis inequalities for real polynomials

Let p be a real polynomial. At a real point x and a common distance ρ>0, we compare the complexconjugate product |p(x+iρ)|² with the real symmetric product p(x+ρ)p(x−ρ). Their normalized difference Aρ and centered sum Eρ separate the odd and even sectors of the generalized Laguerre expansion. After recording the corresponding classical real-rootedness reformulation as background, we prove two sharp degree-five results. First, if a nonreal zero a+ib is locally compatible with Aρ,Eρ≥0, then deg p≥5; in the equality case the root configuration is unique up to a nonzero scalar factor and has the form C(z−a)((z−a)⁴−b⁴), yielding the complete global degree-five boundary family. Second, within a bounded nodal interval, L1[p](x)>0 can coexist with Aρ[p](x)<0 only from degree five onward, and an exact witness is exhibited. The two thresholds arise from distinct mechanisms: multiplicity saturation at an off-axis boundary and the first quadratic dependence of Aρ on ρ⁴.