Saturated Grid Theorem M²¹: A Geometric Alternative to Dark Matter

This work presents the Saturated Grid Theorem, a geometric alternative to dark matter within the M²¹ theory framework (21-dimensional Universe). We derive a universal equation for the gravitational amplification factor g(Σ,z) that depends solely on baryonic surface density Σ and redshift z, with zero adjustable parameters. Key constants emerge from the 21D structure: g_min = 6 (matter partition 6/21), g_max = ξ₀^(21/4) = 97.37 (grid saturation), and Σ_c = 6.49×10⁵ M☉/kpc² (critical transition density). Cosmological evolution is governed by a coupling function f(z) centered on z_eq ≈ 3, corresponding to the peak of cosmic star formation rate. Validation on SPARC data (spirals), dwarf galaxies and clusters yields an average error of 3.3%. The model naturally explains: (1) flat rotation curves of spirals (g ≈ 6), (2) excess "dark matter" in dwarfs (g → 97), (3) the apparent absence of dark matter at high redshift observed by JWST, and (4) the geometric origin of MOND's critical acceleration a₀. The theory predicts that no dark matter particle will ever be detected, and imposes a saturation limit g_max = 97.37 that MOND does not predict.