Nowak-Stachowiak Energy–Spectral Anchor on Manifolds of Bounded Geometry

We introduce and prove the Nowak-Stachowiak energy–spectral anchor for the class Geo of Riemannian manifolds with bounded geometry. We start from a finite–volume manifold (M,g) endowed with the Laplace–Beltrami operator –Δ_g and the anchoring operator H = ħ c √{–Δ_g}. On such a manifold we construct local shell profiles w_r concentrated at scale r around a point x₀ and a class of NSk frequency windows W_NSk that modulate the contribution of spectral modes of H. Our first result is local. The shell energy E(r) = ⟨w_r, H w_r⟩ satisfies the universal scaling law E(r) = (ħ c / r) C_w(gx₀) + O(ħ c r / ℓ_geom(x₀)) as r → 0, with a positive shape constant C_w(gx₀) and a local geometric scale ℓ_geom(x₀) > 0, in line with classical semiclassical analysis of the Laplace–Beltrami operator. The second result is global and concerns the spectral energy function F(Γ_eff) = (1 / Vol(M)) ∫₀Γ_max γ W(γ / Γ_eff) dN(γ), where N is the counting function of the spectrum of H, W belongs to the NSk window class, and Γ_max is an energy cut–off. Under assumptions (A1)–(A3) the limit F_max = limΓ_eff→∞ F(Γ_eff) exists, is finite and positive, and the image of F contains the whole interval (0, F_max). Adding the non–degeneracy condition (A4) makes F strictly increasing, so that for each energy density 0 < ρ < F_max there exists a unique anchoring scale l₀ = c / Γ_eff. The NSk–Anchor assumption package (A1–A4) is designed as a purely analytic–geometric interface for other NSk modules (VacuumSelection, GlobalGap, q³D torus models, etc.). The paper can be regarded as the current, mathematically refined version of the original Energy Anchor Hypothesis (V1) and the Energy–Spectral Anchor Hypothesis (V2).