Local Microstructure in the NSk/ψ Programme: Excitation States, Local Microstructural Classes and Topological Completeness in q³D

Local Microstructure in the NSk/ψ Programme: Excitation States, Local Microstructural Classes and Topological Completeness in q³D This module establishes the local microstructural apparatus of the NSk/ψ programme across four layers: PRE-PURE, PURE, CORE, and EXEC. It occupies the position downstream of NSk–Physicality and upstream of NSk–Randomness, with its exports further consumed by NSk–Gravity. PRE-PURE layer. The partition lattice, refinement structure, and transition operator T₂₁ are imported upstream from NSk–MathFoundation and localised to the discrete carrier of the module's microstructure. On this apparatus the module defines the local coarse-graining loss Δ₂₁S and an entropic certificate for finite refinement chains. PURE layer. A local state alphabet −1, 0, +1ᵈ is introduced together with the activation rank a (σ) and three local microstructural classes B, K, D. The layer uses the imported averaging and block-projection bundle (T₂₁, L₁₂, P₂), P₂: = L₁₂T₂₁, without redefining these operators. Minimal cost classes are determined by the minimal critical pair (i*, j*) ; which of the two minimal classes is locally preferred constitutes a downstream contract passed to NSk–Randomness. CORE layer. The orbit-configuration apparatus Γ, the local orbit count kₗoc (Γ), and the topological completeness condition kᵢnd (Γ) = 3 are introduced. The entropic certificate and the completeness condition are exported downstream as two independent components; the module contains no theorem asserting their equivalence. EXEC layer. The reference realisation is the torus q³D (d = 3, A = diag (1, 4, 2) ), treated as a canonical but non-unique realisation of the framework. The module does not define the physicality gate ΦPhys (NSk–Physicality), the branch-selection mechanism (NSk–Randomness), the gravitational field (NSk–Gravity), or the operator package (T₂₁, L₁₂, P₂) — the last remaining upstream in NSk–MathFoundation.