FRACTAL SPATIO-TEMPORAL GRAVITY (FSG)

FRACTAL SPATIO-TEMPORAL GRAVITY (FSG) — Version 6.1: The Pillar, Cluster Memory Modulation, and the Route to a Covariant Completion Abstract FSG replaces dark matter by an infrared modification of the effective dimensionality of spacetime. The founding principle — the pillar — is that the gravitational flux propagates through an effective number of spatial dimensions d_eff that flows from 3 in the high-acceleration regime to 2 in the deep infrared, with the transition set by the acceleration scale a₀. Flux conservation yields, for a point mass, the exact deep-MOND law g = √(g_N·a₀), the logarithmic potential, flat rotation curves, and the exact baryonic Tully–Fisher relation V⁴ = GM·a₀ (numerically < 0.01%). The observed radial acceleration relation is shown to be a direct measurement of d_eff(g); because the reduction is geometric (Φ = Ψ), gravitational lensing follows the dynamics by construction. The acceleration scale is fixed with no free parameter by the cosmic memory fields U = □⁻¹R and V = □⁻²R: a₀ = c·H₀·|V₀|^{3/2}/|U₀| ≈ 1.14×10⁻¹⁰ m/s² (within 5% of observation); the same sector drives a phantom-like dark energy (w₀ < −1). The MOND mass deficit in galaxy-cluster cores is resolved by a memory modulation of a₀, closing the full 12-cluster X-COP sample with a single parameter and cross-validated to ~10% by CLASH lensing. Solar-System tests are satisfied without any screening mechanism, and in the strong field Schwarzschild/Kerr are recovered up to O(e^{−g/a₀}). New in Version 6.1. The principal advance since v6.0 is the identification of a route to the theory's central missing piece — a covariant action for the pillar. The pillar's single-potential structure places FSG in the Aether–Scalar–Tensor (AeST) class (Skordis & Złośnik 2021), the relativistic MOND framework already established to reproduce gravitational lensing and the CMB acoustic peaks. FSG's flux dimension fixes the AeST free function F(X), reproducing its required deep-MOND (⅔·X^{3/2}) and Newtonian (X) limits, while the cosmological dark sector is carried by the same memory field U = □⁻¹R, whose energy density scales dark-matter-like (∝ a⁻³) through the CMB epoch and dark-energy-like (→ const) today — driven by the same U₀ = −16.1 that fixes a₀. This explains why two earlier covariant-action attempts failed and converts the central open problem from "no known route" into a mapping onto a known, viable class. Epistemic status (stated honestly). FSG is presented as an effective law with a clearly identified path to a fundamental formulation, not as a validated fundamental theory. The following remain open: (i) completion of the covariant action — the correspondence above fixes the free function and the dark-sector scaling but not the amplitudes, which require a perturbed non-local (Einstein–Boltzmann) computation; (ii) the Bullet Cluster lensing offset; and (iii) the CMB acoustic-peak amplitudes. These are the decisive tests of the framework. Mots-clés (champ Keywords) modified gravity · MOND · dark matter · dark energy · non-local gravity · Aether-Scalar-Tensor · fractal spacetime · spectral dimension · galaxy rotation curves · galaxy clusters · cosmology