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 |lambdap+1/2| < 1 on the compatibility domain. (H‑Tate)Every Abel‑rational stasis class with finite Koenigs orbit is generated by algebraic cycles. A systematic gap analysis shows that all operational axioms (faithfulness, phase splitting, cycle generation, analyticity, commutator vanishing) reduce to these two statements. Three new theorems are proved: Abel Frame Commutator Vanishing:The rank‑deformation generator commutes with the Abel‑frame Fourier transform because Abel values of warped primes are rank‑invariant. Hodge–Fourier Equivalence:The Hodge conjecture is equivalent to the equality of the canonical half‑etage Fourier transform and the Abel‑frame Fourier transform (absence of Hodge ghost cycles). Tate–Koenigs Correspondence:Over finite fields, Frobenius eigenvalue q^p is equivalent to Koenigs phase zero, identifying (H‑Tate) with the Tate conjecture in characteristic p. Two independent operational routes to the Hodge conjecture are identified:(i) the EDT route, proving Hodge from (H‑Conv) alone via equivariant degeneracy, and(ii) the A3 route, combining (H‑Conv) with (H‑Tate). The monograph isolates the irreducible remaining gap as the convergence of the HC half‑etage protocol for p >= 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.