We present i-ICST v12, a mathematically rigorous derivation of the H(z)-tracking Symmetron mechanism. Starting from the Hinterbichler–Khoury Symmetron Lagrangian, we show that the assumption of a constant nonminimal coupling ξ to the Ricci scalar R cannot produce the required µ²(z) ∝ H²(z) behaviour, because R(z)/H²(z) is not constant in a flat ΛCDM background (it varies by a factor ~2.5 from z=0 to z=2). The correct derivation proceeds via the Friedmann equation: µ²(z) = µ 0 ² H²(z)/H0 ² is the unique dimensionally consistent, physically motivated extension of the Symmetron mass that (a) reduces to µ0 ² at z=0, (b) grows monotonically as required by MUSE-DARK III, (c) preserves cosmological Symmetron screening, and (d) is radiatively stable under the near-conformal symmetry of the massless limit. This yields β(z) = β 0 H(z)/H0 in galaxy halos and a0,eff(z) = √(1+2β 0 ² H²(z)/H0 ²) × cH(z)/(2π). Fitting to SPARC (z=0) and MUSE-DARK III (0.33 < z < 1.44) data gives β 0 = 0.520 ± 0.101, p = 0.487 ± 0.389 (consistent with p = 0.5 at 0.03σ), χ²/N = 0.097. The cosmological background remains fully screened (Geff,cosmo = 1.000) at all redshifts. Two critical open issues (CMB l a gap and MCMC) remain.
Fransisko Fransisko alfredo ikson saputra (Sun,) studied this question.