Cluster Mass from the Non Linear Superposition of Galactic Scalar Fields

Galaxy clusters exhibit mass discrepancies far larger than those observed in galaxies. In the scalar-gradient framework introduced in [1], an internal geometric scalar field T(x) generates an effective density ρ_eff = Σa |∇T_a|² + Σ{a<b} ∇T_a · ∇T_b, a quadratic expression whose global amplitude is fixed by the galactic relation A = v_flat². The second term represents non-linear cross-interactions between overlapping scalar gradients and constitutes the central novelty of the mechanism. Because gravitational-lensing observables are intrinsically projected, we reconstruct the scalar field as a two-dimensional function T(X,Y) in the plane of the sky. The resulting effective density naturally separates into diagonal and cross-term contributions, the latter arising from the geometric overlap of scalar gradients associated with galaxies, intracluster gas, and subclusters. Using the observed baryonic distribution of the Coma cluster, we find that the projected effective-density map reproduces the main qualitative features of weak-lensing reconstructions. A key numerical result is the dominance of cross-terms, ρ_cross ≃ 0.9 ρ_eff, demonstrating that the projected mass distribution arises primarily from the geometric superposition of scalar gradients. The typical projected effective density Σeff ≃ 10⁹ M⊙ kpc⁻² is compatible with the critical surface density Σ_crit of Coma. A direct pixel-level comparison with the Subaru weak-lensing convergence map κ(X,Y) yields a strong spatial correlation (r ≃ 0.79), providing the first quantitative validation of the scalar-gradient mechanism at the cluster scale. These results show that the scalar-gradient mechanism provides a parameter-free geometric explanation for the projected mass distribution of the Coma cluster, offering a falsifiable alternative to collisionless dark matter at the cluster scale.