Finite-Time Closure Theorem and Minimal-Energy Mesh Framework

This work presents a unified theoretical framework that interprets bound orbits as the three-dimensional projection of four-dimensional geodesic free-fall. By introducing minimal dissipation, we prove that all bound geodesics must coalesce within finite proper time — the Finite-Time Closure Theorem. This result redefines the concept of orbital motion: an “orbit” is not mechanical balance but continuous geodesic fall, with angular momentum simply bending the trajectory away from the radial plunge. Consequently, every gravitationally bound system — from planets to galaxies to the entire cosmic web — is in fall and must evolve toward closure or reorganization in finite proper time. Building on this formal result, we extend the framework to cosmology by proposing a Minimal-Energy Mesh with three or four macroscopic states: free thermal radiation (CMB), distributed free-energy from mass decay (dark energy), boundary-induced clustering (dark matter), and optionally a plasma-threshold cycle as the universal pressure limit. The paper includes testable hypotheses (H1–H3), falsifiable predictions, and a reproducible pipeline based on public datasets (gravitational waves, exoplanet timing, CMB/BAO spectra, X-ray accretion diagnostics). This preprint serves as a comprehensive statement of the Finite-Time Closure Theorem and its cosmological implications, providing a path for observational validation.