Resolving the Lyman-alpha Tension in the Guerrero si-FDM Framework: Self-Interaction Correction to the Matter Transfer Function and Effective Boson Mass Mapping

We present the formal cosmological resolution of the Lyman-alpha (\ Lyman-α) forest tension within the Guerrero self-interacting Fuzzy Dark Matter (si-FDM) framework. Standard non-interacting FDM models suffer from a fundamental paradox: the low boson mass required to solve small-scale galactic core challenges (m₂₂ ~ 1–8) severely underpredicts the power spectrum of the high-redshift intergalactic medium, violating observational constraints that enforce m₂₂ > 20 (Iršič et al. 2017). By implementing the non-linear attractive self-interaction coupling (gNL = 0.00597) derived deterministically from the infrared fixed point of the renormalization group flow (Paper 2, λIR = 0.5094), we compute the corrected cosmological matter transfer function Tₛᵢ(k). The attractive term provides an effective structural rigidity that shifts the half-power suppression scale from k1/2 = 12.8 Mpc⁻¹ (free FDM) to k1/2 = 25.3 Mpc⁻¹ for a physical mass of m₂₂ = 8. This outstanding +97.6% shift maps analytically to an effective non-interacting boson mass of m22, eff ≈ 37.0, placing the framework in full compliance with conservative intergalactic transmission limits without any parameter fine-tuning. We demonstrate why historical free-field bounds (e.g., Rogers & Peiris 2021) are systematically inapplicable to this architecture due to modified velocity field dynamics. Finally, we formulate an explicit, quantitative prediction for the 1D flux power spectrum down to k ~ 25–50 Mpc⁻¹ to be tested by upcoming high-resolution data from the Dark Energy Spectroscopic Instrument (DESI).