Relational Geometric Mechanics X: Cosmology — Dark Matter, Dark Energy, the Hubble Tension, and the Big Bang as Foam Decompression

This is the tenth paper in the Relational Geometric Mechanics (RGM) series. We apply the framework to the four largest open problems in observational cosmology. Dark matter is identified as the gravitational contribution of the attention matrix cross-terms that General Relativity discards when it approximates matter as a smooth fluid — no new particles are required. The cosmological constant problem — a 122-order-of-magnitude discrepancy between quantum field theory and observation — is dissolved by holographic mode counting at the cosmological horizon, giving the dark energy density ρ_Λ = ħc/ (4π·LQF²·RH²) with 4. 6% agreement against the measured value. Inverting this formula yields a first-principles prediction for the Hubble constant: H₀ = 68. 99 km/s/Mpc, sitting between the Planck CMB measurement (67. 4) and the SH0ES local ladder measurement (73. 0), offering a potential resolution of the Hubble tension. The Big Bang is derived — not assumed — as the inevitable decompression of a maximally compressed foam. The dark energy equation of state w (z) ≠ −1 is identified as a falsifiable prediction testable by EUCLID and the Nancy Grace Roman Space Telescope.