Ordinary and Dark Matter as Phase Regimes of the ψ Field in the NSk Energy–Spectral Anchor Formalism

We develop a mathematical definition of ordinary matter and dark matter in the language of an existence density field psi and the NSk energy-spectral anchor, on a class of manifolds with bounded geometry. Building on the NSk anchor operator H = hbar c sqrt(-Delta_g), spectral windows satisfying assumptions (A1)-(A4), and the Anchor-Strong Theorem for the anchoring function F(Gamma_eff), we define ordinary matter as regions of full stabilisation (P = |psi|^2 = 1) that are locally compatible with the anchor equation, and dark matter as semi-stable regions (0 < P < 1) where no local anchoring scale exists, while the gradient grad P generates an effective gravitational field in the NSk model. We formulate a local anchoring postulate, prove stabilisation lemmas and a classification theorem for regions of the psi field into fully stabilised, semi-stable and diffuse regimes. In a quasi-3D torus model we show, on simple psi-profiles, that the halo region 0 < P < 1 naturally yields a dark-to-baryonic mass ratio of order 5, consistent with cosmological data, without introducing new fields or parameters. We interpret ordinary matter, dark matter and dark energy as three phase regimes of the same existence field psi and relate the construction to other NSk modules (psi-field postulate, NSk anchor, dark energy from Catalan's constant). This preprint contains the original formulation of the NSk-based definition of ordinary and dark matter as phase regimes of the ψ field.