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 bandsBᵢ = [exp (uᵢ), exp (uᵢ + Deltaᵤ) ), band entropy Sband (uᵢ) = (1/d) ln (DeltaNᵢ), and mean band entropy Sₘean (U). The central abstract result — theHalf-Slope Law — states that if the band function satisfiesSband (uᵢ) = A + uᵢ + etaᵢ with etaᵢ -> 0, then Sₘean (U) = C₀ + (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 HNSkon a Riemannian manifold (M, g) satisfying Weyl's law. Under thesespectral conditions the per-band asymptotics DeltaNᵢ ~ CW Vol (M, g) exp (d uᵢ) reduce the CORE half-slope theorem to a direct corollary of the PURE engine. Stability of the slope with respect to the logarithmic step Deltaᵤ andthe geometry class Geo is established. The EXEC layer provides a slopeestimator kappaₕat (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.