GUFT - Geometric Unified Framework Theory: An Approach to Galactic Dynamics with Curvature Regulated by Memory

Abstract The lack of direct detection of particles related to dark matter is the cause for the search of purely geometric alternatives to explain galactic dynamics. In this paper, we present the Geometric Unified Framework Theory (GUFT), a modified gravity model that introduces the mechanism of spacetime "memory", encoded in a scalar field. This memory function must be, and is, regulated by a local Kretschmann invariant (K). The intention of this document is to demonstrate that by setting a universal transition scale of K₀ ≈ 10⁻⁶⁰ m⁻⁴, the theory naturally provides a robust environmental screening mechanism. Thanks to this, in high-curvature regimes, such as the Solar System or in compact binary systems, the memory effect is parametrically turned off. In this way, the framework is parametrically compatible with post-Newtonian (PPN) constraints, maintaining the suppression of anomalous dipole radiation by a factor of 50 orders of magnitude. However, when moving to areas of extremely weak fields, such as in low surface brightness (LSB) galaxies, the screening of the memory function vanishes. Being completely active, the memory function induces an effective geometric density profile (₆₄₎ r^-2) that sustains asymptotically flat rotation curves (v → const) without having to resort to any kind of exotic dark matter. This framework offers a self-consistent local formulation, thanks to that Kretschmann scalar, which allows it to be coherent with the precision tests of the Solar System and, at the same time, with galactic infrared phenomenology. Notes on Version 2: Major Manuscript Expansion: Transition from the preliminary 8-page conceptual sketch (Version 1) to the fully formalized 54-page comprehensive manuscript. Mathematical Formalization: Inclusion of complete mathematical appendices formally deriving the isothermal geometric density profile (₆₄₎ r^-2) and demonstrating the emergence of asymptotically flat rotation curves (v → const). Phenomenological Validation: Expanded sections detailing the strict compatibility with the Baryonic Tully-Fisher Relation (BTFR) and Solar System PPN constraints. Ongoing Development: This manuscript establishes the fundamental action and core asymptotic limits. Several advanced phenomenological applications currently outlined in the text will be fully expanded in future updates.