NSk--Weyl Spectral Counting Asymptotics and Realisation for the NSk/𝜓 Program

NSk--Weyl establishes the spectral counting asymptotics apparatus in the NSk/ψ programme. The PRE-PURE and PURE layers build a self-contained mathematical framework: counting functions, Weyl-type asymptotics in dimension three, the counting measure and the density of states in the distributional sense, and band-counting observables in the scales Λ, Γ, and u = ln Γ. The CORE layer realises this apparatus contractually for the base operator H = κ√A over the admissible geometry class Geo, in compatibility with the band formalism of NSk--Windows. In particular, the module provides the Weyl law in the Γ-scale, asymptotics of logarithmic band counts, uniformity with respect to band width, and a weighted version formulated through a Stieltjes counting functional. The EXEC layer provides the reference q³D realisation, estimators, and diagnostics of asymptotic consistency. The module exports the counting function, the leading asymptotic coefficient, the remainder term, the counting measure, band-counting observables, and the weighted apparatus to downstream modules of the programme, in particular to NSk--Entropy, NSk--GapRelRol, and other spectral modules. The document separates the formal core from explanatory background: the classical origin of the Weyl law through the heat kernel and the Tauberian passage is made explicit, while the formal core contains only those definitions, theorems, and proofs that are required as a stable spectral contract within the NSk/ψ architecture.