We rigorously demonstrate that the time-dependent Schrödinger equation emerges as the long-wavelength limit of the Al-Ani Fabric Theory. Starting from a discrete three-dimensional Planck-scale cubic lattice governed by the single Mother Equation, with only two calibrated cosmological parameters (Γ = 2.4 × 10⁻¹⁸ s⁻¹, λ = 200 kpc), we perform the continuum limit followed by the complete Madelung transformation. The Planck constant ħ, the probability interpretation, the mass spectrum, and the uncertainty principle all arise directly without additional postulates. All derivations are explicit, step-by-step, and numerically verified.