Polynomial-Exponential Counterexamples to a Derivative-Zero Half-Plane Classification

Let P be the compact-open closure of the polynomials whose zeros lie in the closed lower half-plane. The entire function exp(iz)+z+2i has every zero of every derivative in that half-plane, but lies neither in P nor in the exceptional families C exp(az) and C(exp(ibz)-a), with complex C,a and real b. This gives a negative answer to the precise alternative in Eremenko's version of a problem of B. Ya. Levin and in the frozen UnsolvedMath record AMR-036-0017. For every positive integer d, an explicit affine holomorphic d-parameter family gives the stronger property that every derivative is zero-free in the closed upper half-plane, while remaining outside P and the stated exceptional families. These exceptions cannot be covered by countably many holomorphic families of uniformly bounded finite dimension. The proof uses polynomial dominance and a logarithmic-derivative condition. Buhovski's earlier exponential example is credited. This is not a classification of all admissible entire functions. The preprint is AI-assisted, self-audited and unrefereed, with no independent review, formal verification or absolute priority claim.