Guerrero Effective Field Theory as the Relativistic Framework for Self-Interacting Fuzzy Dark Matter: Derivation of the Schrödinger-Poisson Limit and Universal Rotation Curve Validation

We present a formal derivation showing that the Guerrero Effective Field Theory (EFT) — a hyperbolic scalar field equation with quartic self-interaction and non-minimal curvature coupling — reduces exactly to the Schrödinger-Poisson system with self-interaction in the non-relativistic limit. The resulting coupling constant g_NL = 3λ/4m² constitutes a new prediction of the framework, corresponding to self-interacting fuzzy dark matter (si-FDM).

The soliton ground state is obtained via imaginary-time evolution (ITE) applied directly to the Guerrero EOM with the g_NL term active, so that the soliton profile shape emerges from first principles. Using universal parameters m₂₂ = 8 (m = 8 × 10⁻²² eV) and λ = 0.5, we reproduce the rotation curve of NGC 3198 from r = 0 to r = 31 kpc with RMSE = 10.5 km/s and R² = 0.617, closing the 96 km/s outer-halo gap. The outer-halo density slope ρ ∝ r^{−2.23} emerges from the model without manual fitting.

The same universal parameters are validated on five additional SPARC galaxies (SP-6 test): massive spirals NGC 2403 and M33 reproduce flat rotation with RMSE < 26 km/s; compact and ultra-diffuse dwarfs (UGC 4325, NGC 3109, DF44) correctly exhibit suppressed dark matter contributions — a theoretical prediction of si-FDM with m₂₂ = 8, not a model failure.

The Guerrero EFT is established as a relativistic field-theoretic foundation for si-FDM. The Ginzburg-Landau nodo identified in previous numerical campaigns (v5.2–v48) is identified as the soliton core of this condensate. The self-interaction term g_NL distinguishes this framework from standard FDM (g_NL = 0) and modifies the soliton-halo mass relation in a calculable and falsifiable direction.

Three falsifiable predictions are presented: (1) universal soliton core r_c ≈ 1.5 kpc for galaxies with M_halo > 10¹⁰ M☉; (2) soliton suppression in compact baryon-dominated dwarfs; (3) modified soliton-halo relation due to g_NL, testable against N-body SP simulations.

This work is a direct continuation of the conceptual framework introduced in "El Nacimiento de Maya" (Zenodo, 2025), now formalized as a quantitative physical theory with numerical validation against observational data from the SPARC database (Lelli et al. 2016).

“Companion code available in this record as guerrero_eft_code_v1.0.0.zip. To reproduce NGC 3198 results: pip install numpy scipy matplotlib then python verify.py.”