The Multiplicative Equilibrium of Galactic Kinematics: Resolving the Rotation Curve Anomaly via Conjugate Scaling Topologies

The rotational velocity of stars in galactic disks presents a fundamental challenge to classical Newtonian dynamics. Historically, the discrepancy between predicted Keplerian decay and empirically observed flat rotation curves has been rectified through the postulation of non-baryonic dark matter. This paper investigates the possibility that the anomaly does not stem from "missing mass, " but originates from a structural limitation in applying linear, 1D additive vector superposition to macroscopic systems governed by 3D volumetric scaling. This research proposes a novel geometric framework grounded in the Unitary Symmetry Series (USS). It shifts the equilibrium baseline from a net-zero additive state to a multiplicative identity (Unity Baseline, 1. 0). Within this reflexive topology, volumetric spatial expansion is intrinsically coupled with a conjugate metric contraction to conserve the total "Information Mass" of the system. Key Achievements of this Framework: - Resolution of Rotation Curves: Derives a dynamically stabilized kinematic equation where a logarithmic scaling index perfectly offsets classical Newtonian decay, yielding flat rotation curves without extrinsic free parameters. - Gauge Independence & Domain Restrictions: Mathematically proves that the arbitrary scaling operator () self-corrects algebraically, completely eliminating the "free parameter" problem. It strictly defines the physical boundaries of expansion (> 1), resolving mathematical anomalies associated with static limits or singularity collapses. - Localized Newtonian Collapse: Demonstrates a seamless mathematical collapse back to pure Newtonian dynamics (n=0) for "relaxed vacuums" (e. g. , the Solar System) that lack a central singularity anchor, preserving extreme local precision. - Resolving the Infinity Paradox: Utilizes proof by contradiction within the conjugate balance equation (r S = 1. 0) to prove that infinite central singularities are mathematically impossible, bounding core states to finite metrics dictated by macroscopic limits. - Empirical Alignment: Naturally resolves gravitational lensing cross-sections, the collision dynamics of the Bullet Cluster, and derives the Baryonic Tully-Fisher Relation (Vf⁴ M) directly from first principles. This scale-invariant model provides a rigorous, mathematically complete alternative to dark matter, aligning perfectly with modern, highly-resolved observational data from the SPARC database and recent high-redshift discoveries by the James Webb Space Telescope (JWST).