Structural Selection of the Einstein-Compatible Branch in Nonlinear Projection-Origin Geometry (Projection-Origin Gravity 10)

This work investigates the nonlinear dynamical behavior of projection-origin geometry in a discrete lattice setting. Previous studies established that the branch corresponding to β ≈ 0 satisfies discrete Bianchi compatibility and reproduces weak-field structures consistent with Einstein gravity.The present work examines whether this compatibility remains structurally preferred once nonlinear dynamical feedback is introduced. To address this question, we introduce a residual-driven constraint solver based on an energy-decreasing update rule.This framework allows the geometric potential to evolve together with perturbations while explicitly penalizing violations of thePoisson-type residual. Three independent diagnostics are evaluated:(i) solver acceptance statistics,(ii) the minimal achievable Bianchi residual,and (iii) the width of the admissible solver step-size window. Across these diagnostics the Einstein-compatible branch (β ≈ 0) is found to remain dynamically preferred.Deformed branches require significantly narrower parameter windows to maintain bounded evolution and produce substantially larger residual errors. These results suggest that Einstein compatibility may arise as a structurally stable branch within the nonlinear dynamical landscape of projection-origin geometry, rather than being imposed explicitly through the field equations. Note: Parts of the manuscript were linguistically and structurally refined with the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own. Note: This work represents Version 1.0 of an ongoing research program on the Order-Projection Principle (OPP). Minor typographical corrections and clarifications may appear in later versions. The core conceptual claims remain unchanged.