M21a The TetraCore: Olympus Programme, Paper I (Revised): The Hyper Core as Universal Lift, Étages as Shelves, Three Constructions of the TC, the Shift Theorem, the Half-TC and the Slog-AGM, and the Ttet K4 Identifcation

M21a establishes the TetraCore (TC) as the first genuine étage other than the Symmetric Core (SC). The central structural correction is that the Hyper Core (HC) is not an étage but the universal elevator: it traverses all ranks without plateau, while étages are shelves—horizontal regions where a single tamer governs a range of ranks. Within this framework, the TetraCore is identified as the slog‑shelf, occupying HC rank 4, with superlogarithm (slog) as its native Abel function. The monograph develops the TetraCore in three a priori distinct but ultimately equivalent constructions: Additive (slog) construction, Cpow‑native construction, and Koenigs construction. The Koenigs Unification Theorem proves that all three are conjugate descriptions of the same algebraic structure. Several foundational results follow. The Shift Theorem shows that the TetraCore Hermit cascade is formally identical to the Symmetric Core cascade when expressed in log‑Koenigs coordinates, establishing the universality of the NC quadratic across étages. The Half‑TC is identified with the slog‑AGM, the arithmetic–geometric mean transported into slog‑space, placing it naturally between HC ranks 4 and 5. The Ttet operation is shown to coincide with the HC rank‑4 symmetric operation and to carry the K₄ invariant, aligning the TetraCore’s ground floor with the HC elevator. The TetraCore is further shown to form a fully valid Operational Number System (ONS) with Abel function slog, sharing the universal spectral data (theta function, Euler product, functional equation) of all étages. Unlike the Symmetric Core, the TetraCore exhibits contracting prime warping and a convergent Bell Tower, marking the transition from unbounded to bounded operational domains. Together, these results establish the TetraCore as the first non‑trivial étage above the Symmetric Core and provide the template for higher étages (PentaCore, etc.) in the Olympus hierarchy.