"Prime Geography Atlas: Finite-Scale Organization of Normalized Prime Gaps as a Weakly Coupled Dual-Core Landscape"

AbstractWe present the Prime Geography Atlas, a systematic data-driven framework for mapping the finite-scale organization of normalized prime gaps \( g_n = (p_{n+1} - p_n)/\log p_n \). Through multifractal analysis, ridge detection, mechanism-separation indices, and information-flow diagnostics across logarithmic scales, we uncover a robust three-landmark skeleton consisting of an Organizer Core near \( s \approx 8.22 \), a Persistence Core near \( s \approx 10.0 \), and an auxiliary L2 Transition Layer (\( s \approx 9.3-9.7 \)) that connects them. Extensive robustness checks confirm that the normalized prime gap landscape is best described as a weakly coupled dual-core system, where information propagates multi-path across scales. These findings reveal a previously unrecognized organized structure in prime distributions at finite scales, providing a phenomenological foundation for deeper theoretical investigations.