M19c The Summit Monograph: A Framework for the Riemann Hypothesis

This monograph is the summit of the Garycki Manifold Programme’s engagement with the Riemann Hypothesis, synthesising the results developed across the Symmetric Core, Cascade, and Halbzeug series and identifying precisely what remains open. The work integrates several previously independent strands: NC quadratics, the LP gap, the carry cocycle, and the merge‑theory spectral ladder. Classical zeta, the cascade zeta families, and Halbzeug‑based constructions are unified into a single framework in which all known symmetries of the zero set are made explicit. A central theme is the identification of the remaining gap: after imposing functional‑equation symmetry, conjugation symmetry, cascade duality, and Halbzeug gap symmetry, the only obstruction to a full proof is a single, sharply stated structural condition. The monograph does not claim closure of this gap; it isolates it. The carry cocycle is introduced as the residual obstruction that cannot be eliminated by coordinate change, symmetry, or spectral transport alone. It measures the failure of naive merge rules to respect additive carry structure across scales. This cocycle is shown to be the only place where zero migration can escape the LP locus. The merge‑theory spectral ladder organises zeta‑type objects by successive merge operations, unifying the classical Euler product, the Kneser Euler product, and the Halb‑zeta into one ladder. Within this ladder, the LP gap appears as a rigidity condition rather than a numerical accident. M19c thus serves two roles simultaneously:(1) a synthesis of all structural reductions achieved in M13–M18, and(2) a precise frontier statement, isolating the last unsolved component of the Riemann Hypothesis within the Garycki Manifold framework. No claim of a completed proof is made. The contribution is the exact identification of the remaining mathematical gap.