M18a The Halbzeug: Algebra and Fractional-Rank Structure

This monograph introduces the Halbzeug, the algebraic structure arising at half‑integer ranks of the Symmetric Core, and develops its full algebraic, lattice, and group‑theoretic framework. The central result is the Halbzeug identity, which gives an exact interpolation between identity and absorber across fractional rank: Coreₖ₊₁₊ₜ(u, a_k) = u^(1−t) · a_k^t At t = 1/2, this yields the Halbzeug Corek+3/2(u, a_k) = sqrt(u · a_k), the geometric mean between an element and its cascade absorber. M18a develops the half‑rank operations (AddMult, MultPow, PowTow), the ghost and Halbzeug lattices, and the full p‑lattice hierarchy. The ±Balance involution is introduced as a structural symmetry, and its consequences for unit structure and fixed points are analysed. The monograph develops the group and algebra theory of the Halbzeug, including cycle polynomials, Atow₃ spectral theory, and a systematic classification of units (“the Book of Units”). The Halbzeug is shown to be the correct algebraic object governing fractional rank, not a perturbation of integer‑rank operations.