Fractal Spatio-Temporal Gravity (Fsg)

FRACTAL SPATIO-TEMPORAL GRAVITY (FSG) Abstract We propose a modified theory of gravity based on a Fractal and Non-Local Spacetime Geometry (FSG), obtained from an effective action containing the fractional non-local operator X=◻−1R, motivated by an infrared flow of the spectral dimension towards dS≈2. In the static weak-field limit, the fractional structure f(X)∼X−1/2 reduces to a nonlinear modified Poisson equation of Milgrom (AQUAL) type, whose exact deep-IR solution is a logarithmic potential. This leads to: Flat galactic rotation curves without dark matter. An exact Baryonic Tully-Fisher relation (V4=GMa0), derived analytically. An acceleration scale derived from the cosmic memory fields: a0=cH0,|V0|3/2/|U0|≈1.14×10−10 m/s2, within 5% of observation with no free parameter. A perturbative treatment of the non-local sector shows that Solar System constraints are satisfied without any screening mechanism (|γPPN−1|∼10−16–10⁻²⁰), as in the related RR non-local model. Regarding cluster dynamics, a 1D wave simulation demonstrates that the non-local field possesses the intrinsic capacity for transient ballistic overshoot — a qualitative mechanism relevant to the Bullet Cluster; the quantitative confrontation (3D, realistic velocities, lensing) is explicitly listed as open. Cosmologically, the monotonic growth of the memory fields structurally implies a phantom equation of state (w0<−1), with a computed first-order bracket w0∈[−1.7,−1.1] and |ΔH/H|∈[1 for 0.5<z<2 — testable by Euclid across the whole range. The geometry of the CMB acoustic peaks is compatible with FSG (sound horizon unchanged; the induced shift is absorbed by the (w,H0) degeneracy); the peak amplitudes require a dedicated non-local Boltzmann code and remain the decisive open test.