This preprint introduces the NSk–PLS–CORE module, a functional-inequalities block in the NSk/ψ program. We develop Poincaré and logarithmic Sobolev inequalities in a setting adapted to NSk: Dirichlet forms associated with a Riemannian metric and an NSk measure on manifolds of bounded geometry, together with a base NSk generator and a spectral window calculus. The construction has two layers. In the universal layer we formulate abstract functional inequalities (Poincaré, log-Sobolev, Sobolev) for Dirichlet forms on manifolds with bounded geometry. In the geometric layer we specialize to the quasi three-dimensional torus (q3D) used in NSk, and we combine the functional-analytic framework with spectral control by the NSk generator and its windows. For the q3D torus we prove existence and scaling of the Poincaré and log-Sobolev constants in NSk language, describe their dependence on the geometric scale parameter, and formulate per-band versions in the window calculus. On the semigroup side we show how these inequalities imply exponential decay of variance and entropy and hypercontractivity of the NSk semigroup. Conceptually, NSk–PLS–CORE provides a universal analytic layer of functional inequalities for later NSk/ψ modules in statics, global gap constructions, Yang–Mills and Wightman blocks, cosmology and numerical studies of correlation decay. All statements are expressed purely in terms of geometry, spectral data and Dirichlet-form theory, without data fitting or extra dynamical assumptions, in line with the zero-fit philosophy of the NSk/psi program. It is compatible with the global core interface formulated in NSk–MAIN–CORE–API (DOI: 10.5281/zenodo.17897274) and with the ontology of the ψ field presented in the NSk/ψ Manifest (DOI: 10.5281/zenodo.17782177).
Nowak et al. (Wed,) studied this question.