Curvature Integrated: From Atomic Probability to Relativistic Orbits — A Cross-Scale Framework Unifying Quantum Mechanics, Orbital Dynamics, and General Relativity - Weber

Quantum Mechanics: The hydrogen 1s radial probability density reveals parallel stability structures at atomic scales, centered around the Bohr radius. General Relativity: Mercury’s observed perihelion precession is matched precisely by Einstein’s correction, completing the progression from Newtonian to relativistic curvature. Unification Theme: These examples are not isolated calculations — they show a continuous curvature-based framework that transcends scale, suggesting that the stability of matter, orbits, and even relativistic corrections may all be understood through a shared lens of radial effective potentials. This paper is designed as both a technical integration and an eye-opening visualization tool for physicists, researchers, and educators. Each figure has been carefully constructed to highlight the deep analogies between orbital mechanics and quantum probability, culminating in a demonstration that General Relativity’s curvature corrections fit seamlessly into the same framework. Broader Implications: Reinforces the unity of physical law across scales, from atoms to planetary systems. Provides a didactic framework for teaching fundamental physics by showing common mathematical structures. Opens pathways for future theoretical work on curvature-driven stability and dimensional unification, including applications to dark matter, quantum gravity, and 5D models. This work is part of the continuing Quantum Big Bang / Curvature Integration Project, aiming to build bridges between quantum, classical, and relativistic physics in a unified presentation. - RJW