SAT 3.1: Regulated Superfluid Framework for Stable Galactic Dynamics and Lensing without Dark Matter

ATTENZIONE: Questa versione è superata. Per la formulazione definitiva della teoria, consultare la SAT 4.0 al seguente link: https://zenodo.org/records/18812684 SAT 3.1: Regulated Superfluid Framework for Stable Galactic Dynamics and Lensing Author: Christian Pellegrino Version: 3.1 (Major Update - Regulated Edition) Date: February 27, 2026 Field: Theoretical Physics / Modified Gravity / Dark Matter Alternatives 1. Introduction and Objectives The SAT 3.1 (Silk Adaptive Tension) framework proposes a superfluid scalar field theory, termed "Silk," as a self-sufficient alternative to the Cold Dark Matter (CDM) paradigm. The primary objective of version 3.1 is to resolve mathematical instabilities found in previous formulations, ensuring a fluid and coherent model across all scales, from galactic cores to galaxy clusters. 2. The Regulated Kinetic Term (Solving the Singularity) The core innovation of version 3.1 is the regularization of the field's kinetic function. While previous iterations relied solely on a fractional power term (3/2) to generate MOND-like dynamics, SAT 3.1 introduces a quadratic regulator. Outer Regime: In galactic outskirts where gradients are weak, the fractional power component dominates, ensuring flat rotation curves and the correct radial acceleration. Central Regime (Bulge Stability): In the galactic core, the new quadratic term (Epsilon) prevents the superfluid’s speed of sound from dropping to zero. This eliminates the risk of field collapse and ensures that the superfluid remains stable and hydrodynamically robust even where baryonic density is at its peak. 3. Gravitational Lensing and Conformal Coupling To overcome the historical limitations of modified gravity theories, SAT 3.1 utilizes a conformal coupling between the scalar field and the matter metric. This ensures that the "Silk" field does not only influence visible matter but also curves the trajectory of photons. The result is a gravitational lensing profile consistent with observations from space missions (such as Euclid), without requiring invisible dark matter particles.