Galactic Rotation Curves: A First Principle Derivation

We present the exterior theory of the Bipolar Helicoidal Core (BHC) framework, a geometric approach to galactic dynamics rooted entirely in general relativity. Starting from a finite helicoidal interior and smooth matching at the core boundary, we derive a correction term ε(r) that modifies the Schwarzschild exterior without introducing additional matter. This correction is fixed by a second‑order vacuum equation together with boundary conditions, asymptotic flatness, monotonicity, and curvature regularity. The resulting velocity law contains no free parameters and links orbital dynamics, gravitational lensing, and outer‑disk structure through a single geometric function. We show that the universal asymptotic behavior ε′(r) ~ 1/r produces flat rotation curves, a characteristic acceleration scale, and the major empirical scaling relations (BTFR, MDAR, RAR, mass–size, and size–velocity). The same geometric tail explains outer‑disk stability, truncation, flaring, and counterrotation, while drift and preferred‑direction fields generated by the interior helicoidal structure account for warps, lopsidedness, bar thresholds, spiral persistence, satellite planes, and jet alignment. All predictions arise from the same geometric function ε(r), enabling stringent cross‑consistency tests across independent observational domains. Paper 1 establishes the exterior theory. Paper 2 will derive ε(r) from first principles by solving the interior helicoidal geometry, fixing the amplitude of the geometric tail and completing the theoretical foundation.