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

This module formalises the band entropy apparatus within the NSk/𝜓 program at two logically independent levels.The PRE-PURE/PURE layers introduce abstract logarithmic frequency bands B_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 — the Half-Slope Law — states that if the band function satisfies S_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 averaging over a logarithmic grid and requires no spectral or geometric assumptions. The PURE layer also provides an A4-ready export form with explicit error bounds, and oracle/filtered variants including congruential filters compatible with the NSk--Algebra module.The CORE layer realises the same apparatus for the base operator H_NSk on a Riemannian manifold (M,g) satisfying Weyl's law. Under these spectral 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 and the geometry class Geo is established. The EXEC layer provides a slope estimator kappa_hat(U1, U2) and a data adapter interface for computational realizations.The module explicitly separates band entropy from Shannon entropy, documents all downstream contracts (NSk--A4, NSk--GapRelRol, NSk--Algebra), and contains no physical interpretation beyond the spectral realization in CORE.