Atomic Orbitals via Coherence Geometry

This paper presents a coherence-geometric model for the emergence of atomic orbital structures from curvature-driven field dynamics. Rather than imposing Schrodinger eigenfunction bases, complex-valued wavefunctions, or predefined orbital boundary conditions, the paper investigates whether familiar orbital morphologies can arise as metastable attractors in a real-valued amplitude field shaped by angular tension gradients, phase relations, curvature flow, and topological constraints. The paper develops the Coherence Geometry (CG) orbital model through analytic structure, Euler-Lagrange equations, numerical relaxation, topological stability arguments, and simulated orbital morphologies. Reported forms include s-, p-, d-, f-, and g-like orbital structures, including px, pz, dz2, fz3, and gz4-type geometries. Starting from isotropic or minimally structured initial conditions, the CG relaxation process produces recognizable nodal orbital geometries without inserting standard quantum-mechanical orbital equations directly. The resulting forms reproduce key topological features associated with conventional orbital classes, including nodal number, orientation, and surface type. The paper focuses on orbital identity and nodal topology as emergent coherence-geometric structures. Broader chemistry-scale modeling, spectral calibration, and predictive electronic-structure applications are left to future work. Its purpose is to show that orbital-like nodal topology can emerge from coherence-geometric relaxation and curvature constraints. This paper is released in its original September 2025 form. Some references reflect the framework references available at the time of writing. Later canon and Foundations materials provide the current public reference layer for the underlying CG formalism. Internal reference: CGI-RSR-000024.