Sharp Positivity Tests for Polynomial Hermitian Kernels

Let V be an r-dimensional space of complex polynomials and let P_k denote positivity on every k-point evaluation matrix. We prove that a Hermitian kernel supported on V is a holomorphic squared norm whenever it belongs to P_k and r is at most 2k. For every one-variable polynomial space the exact stability index is ceil(r/2), independently of its degrees. An indefinite polynomial kernel in P_k has at least 2k positive eigenvalues, sharply. An explicit Hankel moment construction has signature (2k,1), belongs to P_k but not P_(k+1), and has a squared-norm square for k at least 2. Combined with the established Cauchy-Schwarz necessity for squared-norm roots, this gives a complete criterion at coefficient rank at most four: a nonzero real-valued polynomial has a squared-norm power if and only if its coefficient matrix is positive or negative semidefinite. The least successful exponent is respectively one or two. The rank cutoff and the corresponding univariate bidegree cutoff (3,3) are sharp. The proof uses classical polynomial-secant geometry, constructibility and exact moment certificates. This is a complete scoped result, not a resolution of the unrestricted AIM self-power question. The source package includes 22,703 supplementary exact finite checks, replayable in normal and optimized Python. This unrefereed, AI-assisted preprint is self-audited, not independently reviewed or formally verified; absolute priority is not certified.