Forma-variance, Polynomial Gibbs Models, and Universality over Gaussian Mixtures

Classical variance is an expected quadratic energy and therefore privileges unimodal central tendency. We define forma-variance as the expected value of a nonnegative forma potential, and we introduce a companion moment calculus based on iterated antiderivatives that recovers classical central moments as a special case. We then construct a Laplace--parametric family of analytic densities of Gibbs type and derive a closed-form normalization via Laplace approximation about a dominant global mode. Finally, we prove a non-tautological universality theorem: restricting the forma function to the class of coercive polynomials (degree allowed to grow), the induced Gibbs densities are dense in the class of finite weighted Gaussian mixtures. The proof uses compact approximation of log-densities by polynomials plus tail control via high-degree coercive terms.