Lower Bounds for Positive-Power Exponents in a Hermitian Polynomial Family

For a fixed integer k >= 2, let N_k(t) be the least positive integer n for which ((1+x)^(2k)-t*x^k)^n has nonnegative coefficients. Known positivity theorems imply that N_k(t) is finite for 0 <= t < T_k = 2^(2k-1), whereas no exponent works at t = T_k. We give an explicit necessary lower bound as t approaches this boundary. Writing epsilon = T_k-t, L = log(1/epsilon), and beta_k = T_k(k-1)/(4k), we prove liminf[epsilon*N_k(T_k-epsilon)-beta_k*L-beta_k*log(L)] >= C_k, with an explicit Gamma-function constant C_k. The proof compares a Gaussian contribution to a selected coefficient with a negative contribution flat to order 2k. The estimates are uniform in t near the boundary. Equivalently, they bound the exponent needed to express a power of a known bihomogeneous Hermitian form as a holomorphic squared norm. No matching upper bound or general criterion for powers of Hermitian polynomials is asserted. This AI-assisted preprint is self-audited and unrefereed; it does not resolve the general AIM problem, and absolute priority is not certified.