Paper VI: Observational Predictions of Geometric Memory: Hubble Tension and Dark Matter Abundance in the REM Framework

It closes the five-paper REM series by converting its theoretical structure into quantitative observational predictions. The paper derives from first principles the dynamic vacuum energy sourced by the running Weyl memory function, integrates it against a realistic ΛCDM background using the Hopkins et al. SMBH mass function, and obtains a Hubble tension correction consistent with the observed discrepancy ΔH₀ ≈ 5.8 km/s/Mpc. It simultaneously closes the dark matter chain by treating the hidden-sector intermediate scale M∗ as a prediction rather than an input: requiring the correct dilaton relic abundance with natural initial conditions uniquely fixes M∗ ≈ 5.6 × 10¹³ GeV and a hidden-sector confinement scale Λ_NP ≈ 7.5 TeV. A cross-consistency test confirms that the same memory profile responsible for the Hubble correction saturates within the BBN bound and reproduces the correct dark matter abundance, closing the programme without additional free parameters.