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: gₘin = 6 (matter partition 6/21), gₘax = ξ₀^ (21/4) = 97. 37 (grid saturation), and Σc = 6. 49×10⁵ M☉/kpc² (critical transition density). Cosmological evolution is governed by a coupling function f (z) centered on zₑq ≈ 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 gₘax = 97. 37 that MOND does not predict.