The Fractal Gravity Postulate: A Field-Theoretic Treatment of Variable Dimensionality as a Solution to Galactic Scale Anomalies

The ΛCDM paradigm currently requires a non-baryonic matter component to account for the observed dynamicsof spiral galaxies and the deflection of light in cluster environments. We propose a fundamental departure fromparticle-based solutions, introducing the Fractal Gravity Postulate (FGP). In this framework, the effective spatialdimensionality D is a dynamical scalar field χ(x) that flows from D = 3 to D ≈ 2 as a function of the localacceleration scale η = a/a0. We demonstrate that this transition effectively modifies the gravitational flux, recoveringthe Baryonic Tully-Fisher Relation (BTFR) and providing a geometric explanation for the phenomenology usuallyattributed to Dark Matter. This paper provides the rigorous field-theoretic derivation, screening mechanisms, andobservational predictions of the FGP.