The Al-Mizan Al-Mutakamil Relational Framework: Discrete Scale Operators, Galactic Kinematics, and Cosmological Expansion

I present a unified, rigorous theoretical formulation of the Al-Mizan Al-Mutakamil relational framework. I replace continuous particle dark matter halos and a fine-tuned cosmological constant with a discrete relational scale-step operator acting on baryonic angular-momentum flux tensors. I derive the Mizan Master Equation governing state evolution across discrete scale steps k ∈ ℕ, obtain the continuum limit of the non-local scale-step kernel R_k(r, R_c, k), and calibrate the framework globally against 175 rotationally supported galaxies from the Spitzer Photometry and Accurate Rotation Curves (SPARC) database using solely the observed baryonic mass distribution M_bar(r). With strictly three global scalar constants—an amplitude normalization A_0 = 5.234951 (log10 A_0 = 0.7189), a mass-scaling exponent γ = -0.138325, and a relational scale-step ratio exponent K_Global = 0.978294—I achieve a median galaxy-by-galaxy RMS residual of 0.1157 dex and a global mean residual bias of -0.0079 dex across approximately 3400 radial data points, matching the empirical intrinsic scatter floor of the Radial Acceleration Relation. The negative mass-scaling power naturally accounts for the elevated mass discrepancies of low-surface-brightness dwarfs without artificial dark-matter cores. I further derive the dual-trace scale factor of the Al-Musi'a expansion sector, rectify the algebraic derivation of the cosmic acceleration transition time t_acc (ä = 0), obtain the exact physical root x_acc = (√33 - 5)/4 · Ω_m/Ω_v, and demonstrate that the transition occurs at z_acc ≈ 2.15. Numerical evaluation of the AlMizan Cosmology Pipeline against DESI BAO and RSD data yields enhanced comoving distances and a suppressed peak fσ_8(z) ≈ 0.295. I construct the covariant relational stress-energy tensor from the off-diagonal coherence operator, prove its conservation, and supply a thermal decoherence mechanism that accounts for the observed mass-gas offset in systems such as the Bullet Cluster.