Axiomatic Second-Variation Geometry: Symmetric and Skew-Symmetric Decompositions of Real Functionals

An axiomatic framework for real functionals on separable Fréchet spaces decomposes the second Fréchet variation into symmetric (coercive Hilbert geometry) and compact skew-symmetric sectors under five structural axioms, yielding the Universal Convexity–Gap Theorem that bounds negative curvature along polynomial families, illustrated by embeddings in integer theory, Dirichlet series, and Boolean functions.