Geometric Framework for Gravitational Transitions

This work presents a conceptual and visual framework for interpreting gravitational behavior across scales using a normalized geometric language. The framework introduces Universal Radius Units (URU)—defined as normalized Hill-radius-based distance units—as a pedagogical and comparative tool for analyzing gravitational regimes from planetary to cosmic scales. The framework does not propose new forces, particles, or modifications to General Relativity or Newtonian gravity. Instead, it provides a geometric reinterpretation of weak-field behavior already predicted by General Relativity, focusing on how gravitational influence transitions smoothly rather than terminating abruptly. Key elements include: URU Scaling: A distance-normalization method allowing planetary, stellar, galactic, and cosmic systems to be compared using a common reference scale. Regime-Based Geometry: Visualization of motion and curvature across four regimes: Elliptical (strong-field confinement) Saddle-barrier (transition zone) Branch-like (weak-field trajectory divergence) Inertial (negligible curvature influence) Halo Interpretation: Halo regions are treated as weak-field geometric zones where gravitational effects persist due to spacetime curvature gradients, without requiring additional matter components in the framework itself. This approach remains agnostic regarding the particle nature of dark matter while offering an alternative geometric lens for understanding observed halo-scale phenomena. Resonance Descriptors: Labels such as Polarity Resonance Waves (PRW) and Thermo-Gravitational Resonance (TGRW2) are used strictly as descriptive visualization tools, not as claims of physical wave entities. Observational Alignment: The framework is consistent with known observational signatures including: flat rotation curves, gravitational lensing, Shapiro time delay, and gradual decay of curvature influence with distance. Scope and Intent This work is intended as: a teaching and visualization aid for advanced secondary, undergraduate, and interdisciplinary audiences, a conceptual bridge between classical orbital intuition and weak-field relativistic behavior, a discussion framework for exploring gravitational transitions without altering established physical laws. It is not presented as a unified field theory, alternative gravity theory, or empirical replacement for existing models. Contents The repository includes: Journal-ready vector figures illustrating URU scaling and gravitational regimes, Minimal-text visual aids suitable for lectures and preprints, A structured outline aligning figures with General Relativity concepts.