The Madelung cascade: generating series of the tetrahedral counting volume — mother equation, Madelung-MK (2026) and the MK Cone

The Madelung–Klechkowski filling rule (here Madelung-MK, 2026) is the second finite difference of a tetrahedral counting volume C(m) = Te(2m). The whole chain is one object: the generating series G(x) = 4x(1+x)/(1−x)^4. Each multiplication by (1−x) yields one level. Madelung-1 is (1−x)^2 G, i.e. Δ²C = 8m−4. Families s, p, d, f have volumes V_k(m) = 4(2k−1)·Te(m−k+1) with Δ²V_k = 4(2k−1)(m−k+1). The Ehrhart polynomial carries Euler’s characteristic (L(0) = χ = 1) and, by reciprocity, the nested tetrahedra (interior of the n-paving = (n−4)-paving). Exact integers for m = 1…7, no free parameter. The cascade for m ≥ 8 (Z ≥ 119) is written before measurement. Combinatorial structure of the rule, not a dynamical derivation of energies. Version 1.1: Ehrhart interior n−4 (not n−2); pyramidal identity for family shares; explicit Madelung-1,2,3,4; MK Cone DOIs.