Unifying Gallium and Reactor Neutrino Anomalies in a 3+2 Sterile Neutrino Framework: eV-Scale for Anomalies and keV-Scale as Dark Matter Candidates

This theoretical framework rigorously unifies the gallium anomaly ($\sim$20-24\% deficit at $>5\sigma$ \citep{barinov2022results,barinov2022search,semenov2023revised}) and reactor antineutrino anomaly (originally $\sim$5\% deficit, reduced to 0.8-1.4$\sigma$ in recent flux models \citep{giunti2022reactor,zhang2024reactor,giunti2022gallium}) in a 3+2 sterile neutrino model, with an eV-scale sterile for anomalies and keV-scale as dark matter. Oscillation parameters are $\Delta m^2_{41} = 1.3^{+0.7}_{-0.4} \, \mathrm{eV}^2$, $\sin^2(2\theta_{14}) = 0.26^{+0.09}_{-0.09}$ from global fits \citep{banks2024broad,giunti2022gallium}. A broad sterile with $\Gamma_s = 10^{-8}$ eV (corrected for unit consistency) yields approximately 22\% gallium and 5\% reactor deficits ($\chi^2/\mathrm{d.o.f.} = 1.02$ \citep{banks2024broad}). Matter effects are negligible in labs due to low densities ($A \ll \Delta m^2 / E$). For the keV sterile, QKE-based relic density is $\Omega_s h^2 \approx 0.119 \pm 0.002$ ($m_s = 7$ keV, $\sin^2(2\theta) = 10^{-10}$ \citep{abazajian2017sterile,dhuria2023synergy,krivonos2024strong}), compatible with updated Lyman-$\alpha$ bounds ($m_{\mathrm{th}} \approx 4.1$ keV \citep{irsic2023lyman}) and recent X-ray limits from NuSTAR \citep{krivonos2024strong}). Sensitivity analyses quantify degeneracies. Tensions with KATRIN, solar data, reactor spectra, STEREO \citep{almazan2023stereo,giunti2022gallium}, and 2025 experiments (e.g., KATRIN extended data excluding sterile neutrinos at 0.45 eV upper limit \citep{katrin2025sterile}, MicroBooNE final results showing no evidence \citep{microboone2025}, Fermilab SBN/MicroBooNE two-beam search excluding sterile neutrinos at 95\% CL \citep{fermilab2025sterile}) are addressed within $2-3\sigma$, though recent results strongly challenge the sterile interpretation. Detailed derivations include scalar self-interactions (with full mathematical derivation of $V_s$) and QKE transitions, with corrected and reproducible codes. Sterile effects in supernovae (flavor evolution, SN1987A bounds, nucleosynthesis) and neutron stars (heating, mass-radius via modified TOV equations, cooling via NSCool) are derived in detail, including EOS modifications. Predictions for DUNE ($3\sigma$ deviation) and XRISM ($E_\gamma \sim m_s/2$) provide tests, though updated constraints are considered. Updated cross-sections for $^{71}$Ga neutrino capture are incorporated \citep{gallium2025cross}. This advances resolution of tensions via broad sterile, with insights into neutron stars and dark matter, while acknowledging alternative explanations like refined cross-sections.