Spacetime Memory and Emergent Galactic Dynamics(SMG): A Field-Theoretic Origin of the Observed Acceleration Law

The missing mass problem in galaxies remains unresolved within the standard cosmological model. This paper presents Spacetime Memory Gravity (SMG), a covariant scalar-tensor modification of general relativity in which an auxiliary field encodes the accumulated curvature history of spacetime. By localising a non-local curvature integral, the theory yields a well-defined action within the Horndeski class, preserving a strictly luminal gravitational wave speed. In the weak-field limit, the modified field equations produce a non-linear Poisson equation that integrates to an implicit acceleration law, a + αa3/2 = aN, predicting a logarithmic slope of 2/3 in the low-acceleration regime. A global fit to the full SPARC sample (175 galaxies, 2,847 data points) determines a universal coupling α = (1.20 ± 0.06) × 10−10 m−1/2s, yielding a mean reduced χ2ν = 1.12 ± 0.14. The same field dynamics naturally generate late-time cosmic acceleration with w ≈ −1 and dynamically link the galactic acceleration scale to the Hubble constant, a0 ≈ 0.1 cH0. Solar System constraints are satisfied via Vainshtein screening. The framework provides a unified, parameter-efficient description of galactic and cosmological dynamics, with clear falsifiable predictions for next-generation rotation curve, weak-lensing, and gravitational-wave observations.