The work presents a minimal scalar-field mechanism in which the late-time evolution of a single k-essence field links the present-day cosmic expansion rate, H0H_0H0, to the characteristic acceleration scale, a0, observed in disk-galaxy rotation curves. Using a noncanonical kinetic term and a weak conformal coupling to matter, the scalar field admits a dynamical late-time attractor in which its background evolution tracks the Hubble expansion rate. In this regime, the field equations fix the resulting acceleration scale to a0 = c H0 / 6. This prediction is tested using independent low-redshift cosmological measurements, including cosmic chronometers and DESI baryon acoustic oscillation data, and is compared with the empirical acceleration scale inferred from galaxy rotation curves in the SPARC database. Agreement is quantified using the consistency parameter kappa = a0_emp / a0_pred which is found to be consistent with unity within observational uncertainties. Contents Manuscript PDF Reproducibility script (Python) used to generate all figures This deposit constitutes the first part of a multi-part series developing the broader Quantum Vibrational Relativity (QVR) framework and its implications for weak-field dynamics, cosmology, and quantum phenomena.