From Quaternionic Spacetime to MOND: A Single ε Anchored to <em>H</em>₀ Links Galaxies and Cosmology

The multimessenger constraint on the speed of gravitational waves, |c_T/c - 1| ≤ 10^-15, provides a powerful discriminator among modified-gravity scenarios. In this work, we perform a fully phenomenological and data-driven test of a symmetry-motivated weak-field limit that yields MOND-like dynamics governed by a single, global, dimensionless parameter ε (epsilon). This framework predicts three linked consequences: (1) the MOND acceleration scale emerges as a_0 = εcH_0; (2) the observed diversity of galaxy rotation curves is controlled by a geometry-only efficiency factor κ ≈ &lt;h/r&gt;; and (3) the galactic parameter a_0 and the cosmological parameter Ω_Λ share a common origin through ε.We test these predictions using a full-likelihood analysis of the SPARC rotation-curve sample. The model is decisively preferred over standard MOND, with ΔBIC ≈ 38. A novel galaxy-thickness test confirms the geometric nature of κ, revealing a strong correlation (R^2 ≈ 0.818) between the dynamically inferred efficiency, disk thickness, and surface density. Finally, a cross-scale consistency check shows that κ naturally reconciles the galactic and cosmological determinations of ε.These results demonstrate that this single-parameter weak-field limit provides a predictive and observationally consistent description of several long-standing empirical regularities, while remaining compatible with the gravitational-wave speed constraint.