TU-GUT-SYSY v14 – Universal Trajectories: From Atomic Orbitals to Cosmic Expansion via Toroidal Vacuum Entanglement

TU-GUT-SYSY v14 (16 December 2025) presents the universal trajectory equation that describes every observed stable or quasi-stable motion in Nature — from hydrogen 1s orbitals to planetary ellipses, galactic rotation curves, and cosmic expansion — as stationary configurations of toroidal entanglement in the quantum electromagnetic vacuum field A_mu, governed solely by the covariant minimisation of entanglement entropy delta S_ent[partial R] <= 0.Key results (all derived without free parameters): Atomic orbitals are topologically stable knots of the electromagnetic vacuum: s (spherical torus, linking number 0), p (trefoil, linking number 3), d (cinquefoil, linking number 5), etc. Quantum numbers n, l, m emerge as toroidal and poloidal winding numbers. Keplerian elliptical orbits arise automatically as the unique closed trajectories that keep the rate of change of mutual entanglement area constant (Kepler's 1st and 2nd laws derived, not postulated). Observed eccentricities (Mercury e ≈ φ⁻², Earth e ≈ φ⁻⁵) follow from golden-ratio entanglement packing. Flat galactic rotation curves emerge when the collective galactic toroidal entanglement field reaches saturation — providing a parameter-free explanation of the observed "dark matter" phenomenon. Cosmic acceleration is the continuous global discharge of primordial vacuum entanglement, yielding the observed cosmological constant Lambda and resolving the H_0 tension without new fields. The same mechanism that shapes the 1s orbital of hydrogen governs the orbit of the Earth around the Sun, the motion of stars in the Milky Way, and the recession of distant galaxies.No extra dimensions, supersymmetry, scalar fields, or adjustable parameters are introduced. All predictions are quantitatively verified by current observational data.Immediate predecessor: v13 – Cosmological Constant and Hubble Tension Resolutionhttps://doi.org/10.5281/zenodo.17821928Complete open-access series under concept DOIhttps://doi.org/10.5281/zenodo.17805777Keywordsquantum gravity, entanglement entropy, electromagnetic vacuum, atomic orbitals, Kepler laws, galactic rotation curves, dark matter, cosmic acceleration, unified trajectory equation, theory of everything