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Abstract

For a family of nonnegative Hermitian quartics on complexifications of real subspaces, we identify the exact failure set of eventual holomorphic squared-norm certification: the subspaces containing a sign vector. At fixed dimensions and positive perturbation, the least norm-multiplier degree grows uniformly as the inverse square of the distance to this set. This geometric law gives exact moment thresholds for random certificate degree. On Haar-distributed real d-planes in ℝ^n, the degree has tail order M^(-(n-d)/2); it is finite almost surely but has infinite mean in codimensions one and two. Soft-constraint matrices with independent uniform entries on [-1,1] give the corresponding threshold in their number of rows. The probability estimates remain valid under changes of density bounded above and away from zero. The same classification yields coNP-hardness for already nonnegative integer quartics with exactly one negative coefficient eigenvalue and cubically bounded coefficients. A sharp uniform inertia threshold strengthens that coefficient restriction. Explicit hierarchy bounds supply global degree upper estimates, while logarithmic curvature gives the matching lower scale. In a weighted two-dimensional restriction we determine both an exact fixed-perturbation leading constant and the joint vanishing-perturbation crossover, including finite degree plateaus. The random and joint-limit results quantify how eventual certification can coexist with large or nonintegrable degree costs.