The Madelung cascade: generating series of the tetrahedral counting volume

A previous note showed that the Madelung–Klechkowski filling rule is the second finite difference of a tetrahedral counting volume C(m) = Te(2m). Here the whole chain is shown to be a single object: the generating series of the volume has the closed form G(x) = 4x(1+x)/(1−x)^4. Each multiplication by (1−x) yields one level of the chain; the Madelung rule is multiplication by (1−x)^2. The cascade terminates at the germ 4x(1+x), whose coefficients (4, 4) sum to 8 = Δ³C. The Ehrhart polynomial of the tetrahedron contains Euler’s characteristic (L(0) = 1 = χ) and, through Ehrhart reciprocity, the nesting of tetrahedra. The volume decomposes exactly into constructed family volumes V_k(m) = 4(2k−1)·Te(m−k+1). All statements are verified in exact integers for m = 1…7, with no adjustable parameter. Combinatorial result only: no dynamical derivation of energies is claimed. The value D(8) = 120 remains beyond the 118 known elements. Une note précédente a établi que la règle de Madelung–Klechkowski est la seconde différence d’un volume tétraédrique C(m) = Te(2m). Nous montrons que toute la chaîne est un seul objet : G(x) = 4x(1+x)/(1−x)^4. La règle de Madelung est la multiplication par (1−x)^2. Résultat combinatoire, sans dérivation énergétique.