M18b The Halbzeug: Physics of the Operational Standard Model, the Half-Rank Ladder, Gauge Structure, Fermion Masses, Star Sector, Gravity, and Cosmology from the Symmetric Core

This monograph develops the physical interpretation of the Halbzeug, extending the Operational Standard Model to half‑integer rank physics and identifying the Halbzeug as a central organising principle for mass, chirality, renormalisation flow, gravity, and cosmology. At the algebraic level, the Halbzeug appears as the half‑rank midpoint between the identity and absorber of Apow. Physically, this midpoint governs chiral decomposition. The Apow Hermit unit τ admits two square roots under Apow, forming a cascade Weyl pair whose product is the vacuum absorber. These two roots define left‑ and right‑chiral components, with chiral conjugation implemented by the functional‑equation involution. Chirality is thus not imposed but arises algebraically from half‑rank structure. The monograph derives mean‑field critical behaviour directly from half‑rank operations. The Halbzeug corresponds to the universal critical exponent 1/2, identifying half‑rank as the mean‑field universality class. This provides an intrinsic explanation for the appearance of square‑root scaling in critical phenomena, independent of microscopic modelling. A central result is the formulation of a renormalisation‑group flow generated by cascade dynamics. The half‑rank ladder produces a linear RG beta function with a Gaussian infrared fixed point at the absorber. The imaginary part of the Halbzeug coordinate functions as an anomalous dimension, giving a precise algebraic meaning to scaling violations. The monograph introduces a cascade dispersion relation as a mass‑shell condition derived from the NC quadratic. The Hermit and Companion units appear as the two on‑shell modes of this relation. Interaction vertices are computed explicitly, yielding vertex amplitudes governed by a Z/4 selection rule, which determines which interaction channels collapse to vacuum. Gravity and cosmology are analysed in the star sector, where half‑rank structure controls scaling, flow, and asymptotic behaviour. In this setting, gravitational attraction is interpreted as drift toward the absorber in native coordinates, and cosmological scaling laws follow from cascade geometry rather than from phenomenological assumptions. M18b thus identifies the Halbzeug as the physical control point of the operational hierarchy: the locus where chirality, criticality, renormalisation, mass shells, and gravitational scaling intersect. The monograph establishes these correspondences structurally, without invoking perturbative quantum field theory or model‑dependent fitting, and prepares the ground for later phenomenological refinement.