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 ($\ Lyman\text{-}\alpha$) 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 ($m_{22} \sim 1\text{–}8$) severely underpredicts the power spectrum of the high-redshift intergalactic medium, violating observational constraints that enforce $m_{22} > 20$ (Iršič et al. 2017). By implementing the non-linear attractive self-interaction coupling ($g_{NL} = 0.00597$) derived deterministically from the infrared fixed point of the renormalization group flow (Paper 2, $\lambda_{IR} = 0.5094$), we compute the corrected cosmological matter transfer function $T_{\rm si}(k)$. The attractive term provides an effective structural rigidity that shifts the half-power suppression scale from $k_{1/2} = 12.8\text{ Mpc}^{-1}$ (free FDM) to $k_{1/2} = 25.3\text{ Mpc}^{-1}$ for a physical mass of $m_{22} = 8$. This outstanding +97.6% shift maps analytically to an effective non-interacting boson mass of $m_{22,\rm eff} \approx 37.0$, 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 $k \sim 25\text{–}50\text{ Mpc}^{-1}$ to be tested by upcoming high-resolution data from the Dark Energy Spectroscopic Instrument (DESI).