Forma Potentials, Forma Means, and Forma-Variance: Coercive Polynomial Energy-Based Models, Laplace Regimes, and Universality over Finite Gaussian Mixtures

Classical variance is the expectation of a quadratic energy and therefore privileges unimodal central tendency. We introduce forma-variance as the expectation of a nonnegative forma potential, which yields dispersion functionals compatible with multimodal geometry. We then study a structured one-dimensional family of Gibbs/energy-based densities\[q_{\phi,\Sigma}(\tau)\; \propto\; \exp\!\left(-\frac{\phi(\tau)}{2\Sigma}\right),\]where $\phi$ is an admissible (in particular, coercive) potential and $\Sigma>0$ is a temperature/scale. Our first contribution is an explicit Laplace approximation for the partition function $Z_{\phi,\Sigma}$ in a dominant-well regime, together with a multimodal mixture-Laplace extension; both are proved with quantitative remainder control. Our second contribution is a universality theorem: restricting $\phi$ to the class of coercive polynomials (degree allowed to grow), the induced Gibbs densities are dense in the class of finite Gaussian mixtures in $L^1(R)$, equivalently in total variation. We then connect the framework to energy-based learning: the coercive polynomial class provides a tractable EBM hypothesis space where partition-function approximations (Laplace regimes) can be used to accelerate likelihood-based training and to sharpen the theory of EBM normalization.