NSk--Entropy Band Entropy for the NSk/𝜓 Program

This module formalises the band entropy apparatus within the NSk/𝜓 programme at two logically independent levels. The PRE-PURE/PURE layers introduce abstract logarithmic frequency bandsB_i = [exp(u_i), exp(u_i + Delta_u)), band entropy S_band(u_i) = (1/d) ln(Delta_N_i),and mean band entropy S_mean(U). The central abstract result — theHalf-Slope Law — states that if the band function satisfiesS_band(u_i) = A + u_i + eta_i with eta_i -> 0, then S_mean(U) = C_0 + (1/2) U + o(1) as U -> infinity. This universal slope 1/2 is a pure consequence of arithmetic averagingover a logarithmic grid and requires no spectral or geometric assumptions.The PURE layer also provides an A4-ready export form with explicit errorbounds, and oracle/filtered variants including congruential filterscompatible with the NSk--Algebra module. The CORE layer realises the same apparatus for the base operator H_NSkon a Riemannian manifold (M,g) satisfying Weyl's law. Under thesespectral conditions the per-band asymptotics Delta_N_i ~ C_W Vol(M,g) exp(d u_i)reduce the CORE half-slope theorem to a direct corollary of the PURE engine.Stability of the slope with respect to the logarithmic step Delta_u andthe geometry class Geo is established. The EXEC layer provides a slopeestimator kappa_hat(U1, U2) and a data adapter interface for computationalrealizations. The module explicitly separates band entropy from Shannon entropy, documentsall downstream contracts (NSk--A4, NSk--GapRelRol, NSk--Algebra), andcontains no physical interpretation beyond the spectral realization in CORE.