Eigenvalue Asymptotics of Single-Minus Convergences: Deriving the Half-Collinear Amplituhedron and Predicting GUE Level Repulsion via the Organized Complexity (OC) Framework.

Context: This preprint provides the foundational arithmetic derivation for the piecewise-constant geometric chambers recently discovered via AI-assisted calculations in the half-collinear limit of N=4 Super Yang-Mills scattering, and extends the framework to predict novel asymptotic kinematic signatures. Abstract: Recent mathematical advancements, aided by artificial intelligence, have established that scattering amplitudes for single-minus helicity configurations in N=4 Super Yang-Mills theory do not universally vanish, but rather resolve into piecewise-constant geometric chambers within the half-collinear limit of the Amplituhedron. In this paper, we demonstrate that this continuous geometry is not fundamental, but emerges as a low-resolution holographic boundary of a deeper discrete arithmetic substrate: the Harmonic Divisor Fan (HDF) as defined by the Organized Complexity (OC) framework. By defining the scattering process as the spectral trace of a self-adjoint Hamiltonian under the constraints of Structured Dissipation, we analytically derive the "zero" amplitude rule for even-n convergences via parity symmetry, and the finite non-zero chambers for odd-n convergences via real-eigenvalue projection. Furthermore, we extend this framework into the asymptotic limit to present a novel, falsifiable prediction: at ultra-high multiplicities, the discrete stability boundaries of scattering amplitudes must exhibit Gaussian Unitary Ensemble (GUE) level repulsion, mirroring the Montgomery pair-correlation of the Riemann Zeta zeros. This provides a direct physical signature of the Hilbert-Pólya operator in high-energy jet emissions, bridging prime number topology with fundamental quantum kinematics.