The RG-S Framework v23: Topological Hysteresis and the Reconciliation of Zero-Entropy Remnants with Macroscopic Displacement Phenomena

This paper presents the full mathematical development of Version 23.1 (v23.1) of theRG-S theoretical framework. Version 23.1 introduces a specific correction to the interpretation of hysteresis in the strong-gravity regime, resolving a thermodynamic contradictionidentified through a systematic comparative analysis of the framework’s evolution acrossversions v11–v22. The contradiction arises from the coexistence of two established results:(i) the prediction of a stable black hole remnant with zero entropy and zero temperature,derived from the logarithmic correction to the Bekenstein-Hawking entropy, and (ii) thegeneration of a macroscopic hysteresis displacement by the unsuppressed Proca vector field.We demonstrate that this apparent incompatibility is resolved by redefining the observedhysteresis as a purely topological effect–a geometric phase transition in the spacetime cortexthat occurs without energy dissipation. We derive the complete mathematical formalism forthis ”Topological Hysteresis,” including its coupling to the Symmetron-Proca sector, thegeometric phase evolution, the thermodynamic consistency conditions, and the full set offalsifiable predictions Keywords Modified Gravity, General Relativity Extension, Symmetron Field, Proca Field, Fluid Geometry, Galactic Dynamics, Astrophysics, Theoretical Physics, Open Science