M12b The Hyper Core Analytic Extensions and the Tetraalgebra

This monograph extends the Hyper Core framework developed in M12a by introducing the analytic and algebraic machinery required to operate beyond the foundational layer. Central results include the K‑Shelf theorem, identifying the first genuine mutation of the conserved quantity KR at rank 4 via the superlogarithm, and the formulation of law mutation, whereby classical analytic results are lifted uniformly to rank‑R analogues. This yields the Hyper‑Fourier transform (with unitarity and Plancherel identity proved), the Hyper‑Gamma function, operational sine products, and Hyper‑Stirling asymptotics. The work derives the Operational Theta function and establishes universal modular symmetry across rank, as well as the Hypereuler spectrum, generalising Euler‑type identities to fractional and intermediate ranks. Connections between the Hyper Core, Tetraalgebra, and the operator structures of M11 are made explicit. The purpose of this deposit is to document the analytic extensions of the Hyper Core established in M12a, together with their structural consequences, while remaining independent of number‑theoretic interpretation.