M21b The TetraCore: Further Work Olympus Programme, Paper II: K4 Invariance via Koenigs, the Universal NC Quadratic Theorem, Étage-ONS Induction and Universal Properties, No-Singular-Birth as a Universal ONS Property, and the Path to Generalised ONS

M21b gives a precise, structural account of the TetraCore (TC) and situates it within a unified theory of Operational Number Systems (ONS). Building on what was established in M21a, the monograph inventories the results that distinguish the TC as a genuine new étage: it is neither a deformation of the Symmetric Core (SC) nor a special case of the Hyper Core (HC), but a slog‑native operational system with its own invariants, units, and convergence properties. A central result is the proof of K4 invariance for the TetraCore via the Koenigs isomorphism. In Koenigs coordinates, the TC operation reduces to a quadratic normal form, allowing conserved quantities to be identified and showing that the Tet iteration is well defined and commutative. This establishes the TC as an ONS with a verified invariant, in contrast to SC and HC behavior at comparable étages. . The monograph then proves the Universal NC Quadratic Theorem, valid for all n≥2n. It shows that the Hermit condition z²−z=i*pi/2 is invariant across all étages and that each n‑Core realizes the same underlying Hermit structure in different Koenigs coordinates. This unifies SC, TC, PentaCore (PC), and higher n‑Cores at the level of NC geometry, while keeping their operational behaviors distinct M21b introduces the PentaCore as the n=4 étage, defines its operations, and proves that its Bell tower converges and its prime warp is contracting. A key structural contrast is established: the SC is the unique divergent étage, while TC, PC, and all higher n‑Cores form bounded, contracting ONS with convergent Bell towers. The work develops étage‑wise induction, showing which properties are universal (spectral equation, functional equation, NC quadratic, Bell convergence) and which are étage‑specific (prime warp, domain, Hermit unit values). An inductive transfer theorem formalizes how properties propagate across étages via Koenigs conjugation. Finally, the TC zeta function is defined, its Euler product is shown to converge rapidly, and the TC Canonical‑Abel Equivalence (CAE) is established. Structurally, CAE is equivalent to the non‑existence of TC ghost primes. The monograph closes by outlining the path toward generalized ONS, including non‑integer ranks and non‑commutative extensions, and lists open problems to be addressed in later M21 series.