Exact Multimode Quantization of Superconducting Circuits via Boundary Admittance

We show that the Schur complement of the nodal admittance matrix, which reduces a multiport electromagnetic environment to the driving-point admittance Yin(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 Yin(s), (ii) solve the boundary condition sYin(s)+1/LJ=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, we prove that junction participation decays as O(ω−1n) 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.