M21g The Half-Étage Tower: Protocol Means, Conserved Quantities, and Arithmetic Applications from the AGM to the Inter-Shelf Half-Étage

M21g develops the Half‑Étage Tower: the canonical family of half‑integer Hyper Core (HC) ranks obtained by alternating adjacent HC means in an (n,n+1)-protocol. These half‑étages fill the structural gaps between integer ranks of the HC elevator and form a systematic tower indexed by R=n+1/2, each associated with a protocol mean, a conserved quantity, a governing special function, and a precise arithmetic role. The monograph begins by resolving a persistent taxonomy confusion between SC and HC half‑rank constructions. Although both define (n,n+1)-protocols, they are genuinely different because the underlying means differ (e.g. HC Cpow vs. SC Apow). Three structural types are identified: intra‑K‑shelf half‑étages (R=1.5, 2.5), the unique inter‑shelf half‑étage (ISHE, R=3.5), and intra‑slog‑shelf half‑étages (R=4.5, 5.5, …). At HC R=1.5, the half‑étage is exactly the AGM, with full proofs of convergence, the Landen transformation as protocol symmetry, and BSD / Gross–Zagier formulas rewritten cleanly in HC notation. This anchors the half‑étage tower in classical arithmetic and elliptic‑curve theory. At HC R=2.5, alternating the geometric mean and the Cpow mean produces the Heun half‑étage. The limit defines a new mean HC2.5, whose conserved quantity K2.5 is currently unknown. The central conjecture is that the limit satisfies a confluent Heun equation, placing half‑étages as intermediate rungs in the special‑function ladder between elliptic integrals and Lambert‑W. A major arithmetic conjecture identifies the Bost–Mestre genus‑2 period algorithm with this HC (2,3)-protocol, yielding a conjectural genus‑2 BSD formula in HC notation. The structural centerpiece is the Inter‑Shelf Half‑Étage (ISHE, R=3.5), the only half‑rank that crosses a shelf boundary by alternating Cpow (K‑shelf) and Ttet (slog‑shelf). The ISHE has a compact domain, a conjectured conserved quantity K3.5, and introduces a new special function: the mixed logarithm LB(x)=sqrt(ln⁡(x)*slogB(x)). The ISHE Hermit unit is characterized by simultaneous K‑shelf and slog‑shelf conditions and satisfies the universal NC quadratic in a mixed coordinate. Crucially, the ISHE generates an ONS with a finite prime set, a phenomenon unique in the programme. M21g then formulates the Genus–Rank Correspondence, conjecturing that HC rank R=1+g/2 governs period computations for genus‑g curves. In this view, the ISHE at R=3.5 corresponds to genus 3 and leads to the ISHE Schottky Conjecture, proposing an operational characterization of Jacobians among principally polarized abelian threefolds. Finally, the monograph shows that the safe‑haven structures of integer étages—NC quadratic, companion algebra, and clone algebras—extend to half‑étages in modified (mixed‑coordinate) form. The paper closes with a sharply delimited list of open problems, ranging from explicit conserved quantities and Heun equations to BSD generalizations, Schottky theory, and complex HC rank.