Filling and Interlayer Superexchange Control Superconductivity in La$_3$Ni$_2$O$_7$
A central puzzle in bilayer nickelate superconductors is why pressure, epitaxial strain, oxygen stoichiometry, and chemical substitution produce systematic but apparently different changes in the superconducting T$_\text{c}$. Here we show that these trends can be organized by a two-parameter control principle based on the $d_{x^2-y^2}$-orbital filling $n_x$ and the effective interlayer antiferromagnetic superexchange $J_\perp$. Starting from a physical picture in which the nearly half-filled $d_{z^2}$ orbital supplies localized spin correlations while the nearly quarter-filled $d_{x^2-y^2}$ orbital carries superconductivity, we study an effective bilayer $t-J_\parallel-J_\perp$ model with parameters constrained by first-principles calculations. Combined slave-boson mean-field and density-matrix renormalization group calculations place realistic La$_3$Ni$_2$O$_7$ in an overdoped-cuprate-like regime where T$_\text{c}$ is governed mainly by the pairing scale. In this regime, hole doping reduces $n_x$ and suppresses T$_\text{c}$, whereas tuning routes that enhance $J_\perp$ raise T$_\text{c}$. This framework accounts for the suppression by over-oxidation and Ca/Sr substitution, the half-dome oxygen-stoichiometry response in the film, the enhancement by Nd/Sm substitution in the pressurized bulk, the right-triangle-like pressure dependence in the bulk, and the enhancement by compressive strain in the film. It also separates clean carrier doping from oxygen-vacancy tuning: clean electron doping mainly increases $n_x$, whereas oxygen vacancies weaken the apical-oxygen-mediated exchange path and introduce disorder. This leads to a falsifiable materials-design prediction: clean electron doping should enhance the pairing scale if introduced without oxygen vacancies or strong pair-breaking disorder.