M24 Hodge in ONS: The Hodge Conjecture via the Operational Number System and the HyperCore half-Etage tower: complete synthesis

This monograph presents a complete synthesis of a three‑paper programme (Hodge I–III) reformulating the Hodge conjecture within the framework of the Operational Number System (ONS) and the Hyper Core (HC) half‑etage tower. The central observation is that codimension‑p Hodge theory is governed by a natural HC period machine at rank R = p + 1/2. At p = 1 this machine is the classical AGM (Gauss, 1799) ; at p = 2 it is conjecturally governed by the Heun equation; at p = 3 by the ISHE mixed logarithm. The transcendental complexity of the governing special functions increases in exact parallel with the difficulty of the Hodge problem. The main synthesis result is the Operational Hodge Theorem: the Hodge conjecture in codimension p is equivalent to the conjunction of two independent conditions: (H‑Conv) The HC half‑etage (p, p+1) protocol converges at rank R = p + 1/2, equivalently the Koenigs multiplier satisfies |lambda₏+₁/₂| = 2. Its resolution would simultaneously prove the Hodge conjecture in all codimensions and resolve the analogous commutator gap in the Riemann programme, highlighting a deep structural unity between Hodge theory and RH within the ONS/HC framework.