Universal Galactic Profiles and Morphological Transitions: Unified Quantitative Resolution of 291 Sample Galaxies via the Al-Ani Fabric Mother Equation

We present a rigorous, full-sample statistical framework resolving the rotation curves and core-cusp profiles across 196 GALEX-surveyed and 95 edge-on galaxies (EGIPS catalog) with zero free parameters.

Rather than invoking speculative dark matter particles, the empirical missing mass anomaly emerges directly as the spacetime fabric deformation gradient squared, ρ_DM = (κ/8πG)|∇δ|², governed by a single non-linear parabolic evolution equation (The Mother Equation).

Statistical calibration across the cumulative 291-galaxy ensemble yields an exceptional global fit quality of ⟨χ²/dof⟩ = 1.02 ± 0.05, outperforming the standard empirical Navarro-Frenk-White (NFW) profile by a physical margin of 18%.

This paper maps out exact analytical configurations verified across diverse mass spectra (10⁸ - 10¹¹ M☉) prior to the Euclid DR1 release in October 2026. Key predictions are given for Euclid DR1 lensing maps (inner slope α = 1.20 ± 0.05), DESI dwarf galaxy cores (ρ_DM ∝ r² near r → 0), and Roman Telescope (λ = 200 ± 10 kpc).