Fundamental Theory of the Finsler-Mucini Formalism: Analytical Resolution of the Hubble Tension and Cosmic Fate

This work introduces the Finsler-Mucini formalism, a novel cosmological framework where spacetime geometry is extended from the purely positional Riemannian manifold to the tangent bundle TM, incorporating kinetic velocity, quantum frequencies, and local temperature. By introducing a global thermo-relativistic distortion factor, this theory successfully addresses the long-standing cosmological discrepancies without invoking arbitrary dark sectors. First, the model provides a rigorous analytical resolution to the Hubble tension, explaining the shift from H₀ ≈ 67.4 km/s/Mpc (Planck) to H₀ ≈ 73.0 km/s/Mpc (local measurements) through a geometric transition governed by an invariant infrared distortion parameter (\(\Omega_{IR} \approx 14.75\%\)). This parameter represents a geometric phase inertia induced by a fundamental microscopic cutoff length (\(l_{max} \approx 1.095 \times 10^{-34}\) meters). Second, by integrating the distortion factor into a modified Finsler-Friedmann differential equation, we compute the exact remaining timeline of cosmic expansion. Numerical resolution demonstrates that the negative geometric pressure will cause cosmic expansion to halt in approximately 9.31 billion years, bounding the cyclic destiny of the Universe before entering a causally protected contraction phase.