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Abstract

For homogeneous real-stable polynomials of degree n in n variables, with nonnegative coefficients and value one at the all-ones vector, we give explicit positive rational constants in every fixed degree for a global prescribed-gradient coefficient reserve. On an explicitly specified closed neighborhood of the balanced gradient, the squarefree coefficient deficit dominates both centered-root dispersion and the coefficient ℓ^1 distance from the corresponding power of a linear form. The proof replaces the compactness step in the earlier qualitative theorem by a radial extension to a degree-drop boundary and a positive contraction dichotomy. An explicit positive map then corrects the gradient while preserving the centered factors. The bounds include singular dispersion and stable coefficient boundaries. We also obtain an explicit Frobenius-distance reserve for Hermitian mixed discriminants. The constants are deliberately coarse.