We show that the Schur complement of the nodal admittance matrix, which reduces a multiport electromagnetic environment to the driving-point admittance $Y_{\mathrm{in}}(s)$ at the Josephson junction, naturally leads to an eigenvalue-dependent boundary condition determining the dressed mode spectrum. This identification provides a four-step quantization procedure: (i) compute or measure $Y_{\mathrm{in}}(s)$, (ii) solve the boundary condition $sY_{\mathrm{in}}(s) + 1/L_J = 0$ for dressed frequencies, (iii) synthesize an equivalent passive network, (iv) quantize with the full cosine nonlinearity retained. Within passive lumped-element circuit theory, we prove that junction participation decays as $O(ω_n^{-1})$ at high frequencies when the junction port has finite shunt capacitance, ensuring ultraviolet convergence of perturbative sums without imposed cutoffs. The standard circuit QED parameters, coupling strength $g$, anharmonicity $α$, and dispersive shift $χ$, emerge as controlled limits with explicit validity conditions.