Nonlinear Connection Geometry and Predictive Finite Transport in a Finite-Energy Electron Core

This work is a companion study to the previously published orientational-transport analysis of a finite-energy electron core, which revealed anisotropic finite-angle transport, nonlinear memory effects, curvature-driven path dependence, hysteresis, and low-rank collective transport organization. The present study reconstructs the effective nonlinear transport structure underlying that finite-angle geometry. The reconstruction begins with Hamiltonian-compatible factorization attempts and progressively develops toward an emergent nonlinear transport-connection interpretation. The analysis identifies an approximately invariant quadratic metric, discovers a rank-2 nonlinear connection collapse, validates coherent principal-angle evolution of the reduced nonlinear manifold, reconstructs smooth nonlinear coefficient laws, predicts finite transport maps, reproduces holonomy and path-ordering structure, and establishes an amplitude-bridge diagnostic linking nonlinear connection coefficients to observable transport-deformation amplitudes. The principal result is that nonlinear finite transport appears primarily as a low-rank renormalization of the transport connection rather than as a nonlinear potential-energy correction. The surviving structure is highly organized and remains simultaneously constrained across invariant reconstruction, collective low-rank organization, predictive finite transport, principal-angle evolution, amplitude-correlated coefficient dynamics, and holonomy recovery. A major outcome of the reconstruction is the emergence of a mathematically coherent effective geometric formulation capable of organizing the observed finite-angle transport phenomena within a unified nonlinear transport-connection framework. Key results include: • Approximately invariant quadratic geometry • Rank-2 nonlinear connection collapse • Coherent principal-angle evolution of the reduced nonlinear manifold • Predictive finite-map reconstruction with strict out-of-sample validation • Stable holonomy and path-ordering recovery • Amplitude-correlated nonlinear connection coefficients • Emergence of an effective nonlinear transport-connection geometry The reconstruction determines operational finite-angle connections (K_a(\theta)). The construction of an autonomous state-dependent connection (K(a)) and its associated nonlinear transport dynamics remains an open direction for future work. Companion framework paper:DOI: 10.5281/zenodo.20401538