Leading-Order QCD Equation of State in Strong Magnetic Fields at Nonzero Baryon Chemical Potential

We present continuum-estimated $(2+1)$-flavor lattice QCD results for the leading-order Taylor expansion coefficients of the equation of state in strong magnetic fields and at nonzero baryon chemical potential. Simulations employ the highly improved staggered quark (HISQ) action with physical pion masses on lattices of temporal extent $N_\tau = 8,\,12$, covering $145 \lesssim T \lesssim 165~\mathrm{MeV}$ and $eB \lesssim 0.8~\mathrm{GeV}^2$, imposing strangeness neutrality with baseline results at electric charge to baryon number ratio $r = 0.4$. We determine the $T$--$eB$ dependence of $q_1$ and $s_1$ (electric charge and strangeness chemical potential ratios), pressure coefficient $P_2$, baryon number density coefficient $N_1^{\rm B}$, and energy-like coefficients $\Theta_2$ (trace anomaly), $\epsilon_2$ (energy density), and $\sigma_2$ (entropy density). Magnetic fields induce temperature-band crossings for $q_1$ and $P_2$ and non-monotonic structures in the energy-like coefficients, with $\Theta_2$ at strong fields possibly vanishing or turning negative at higher $T$, indicating dominance of the pressure term over the energy contribution. We also examine the $r$-dependence, finding that $r=0$ (charge-neutral matter) shows the most muted magnetic-field enhancement of $P_2$ despite larger $|q_1|$, providing a useful reference for neutron-star-like conditions. Comparisons with the hadron resonance gas (HRG) model show qualitative agreement at low $T$ and weak $eB$, with clear deviations near the crossover and at strong fields. These results provide useful input for constraining models and effective theories of QCD matter in strong magnetic fields at finite baryon density.