Passive spectral-admittance bounds and exact continuum certificates for multiresonator quantum-memory interfaces

Broadband quantum-memory interfaces are often assessed by center-frequency impedance matching or by a sampled efficiency curve. Neither supplies an operational continuous-band certificate, and absorption is not automatically reversible storage. We model a passive one-port multiresonator interface by a positive-real spectral admittance with explicitly identified controlled output channels. If their one-photon subspace is mapped isometrically into long-lived registers, the write probability for a normalized spectrum $f$ supported in a band $\mathcal{B}$ is $1-\int_{\mathcal{B}} |r(i\omega)|^2 |f(\omega)|^2\,d\omega$, and the worst-case write efficiency is $1-\|r\|_{L^\infty(\mathcal{B})}^2$. We prove that a finite passive rational interface cannot have zero reflection on a nonzero interval and derive the Bode--Fano floor $\|r\|_\infty \geq \exp[-\pi\kappa/(2B)]$ for a band of half-width $B$. At fixed pole locations, minimax synthesis is a quasiconvex semi-infinite problem in the oscillator strengths. We then give an exact computer-assisted certificate: after a decimal design is converted into an explicit rational system, the continuum reflection bound becomes positivity of one univariate polynomial and is proved by Sturm root counting; exact Routh--Hurwitz determinants certify stability and minimum phase. In units $\kappa=2$ and $B=1$, an 11-mode design obeys $0.064112405 \leq \|r\|_\infty < 0.0641125$, implying a conditional uniform write guarantee above $0.995889587$. This is a reproducible certificate for a specified interface, not a claim of global movable-pole optimality or of an experimentally complete memory.