This preprint defines the geometric backbone of the NSk/ψ program in the form of a self-contained module NSk–Geo–CORE. We introduce a class of Riemannian manifolds with bounded geometry and discrete Laplace–Beltrami spectrum, denoted by Geo, and construct an explicit adapter (M, g) inGeo→DataOut (M, g) (M, g) in Geo → DataOut (M, g) (M, g) inGeo→DataOut (M, g) which exports all geometric, spectral, functional and topological data needed by higher NSk modules. On the geometric side, Geo includes compact manifolds (with or without boundary) and special subclasses such as quasi three-dimensional tori (q3D), channel-type geometries and selected product manifolds. On the analytic side, we develop the Dirichlet-form framework for the Laplace–Beltrami operator −Δg, prove regularity and Markov properties, and collect the functional inequalities that enter later modules: Poincaré, Sobolev, log-Sobolev, Korn and Cheeger inequalities, together with their associated constants. We also state a uniform Weyl law on a suitable subclass of Geo and stability results for spectra and resolvents under metric deformations and potential-type perturbations (Kato–Rellich, KLMN, Mosco convergence). Conceptually, NSk–Geo–CORE is designed as a “zero-fit” layer: all parameters in DataOut (M, g) are defined purely geometrically and analytically, without any reference to experimental data. The module provides a single, consistent geometric language for all higher-level NSk/ψ constructions and for the analytic CORE blocks built on top of it. 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.