Quantum Key Distribution (QKD) enjoys information-theoretic security, yet the most damaging attacks against deployed systems exploit the receiver, where the key bit is encoded in which one of a pair of never-identical detectors clicks. The minimal receiver, one rotatable polarizer and one threshold detector, removes that attack surface, and single-detector BB84 demonstrations already run sampled error estimation; the structure of its zero-probability error subensemble, however, has remained uncharacterized. We characterize exactly that structure, introducing a deterministic impossible-event certificate: a click behind a polarizer set orthogonal to the transmitted state has probability exactly zero on an ideal channel, so a single occurrence is a probability-one witness of disturbance; and, since loss deletes clicks and never creates them, the certificate is loss-robust. We prove it sound but incomplete over three polarization states, and show that the four BB84 states close the gap: a fixed-basis intercept-resend attack yields an ideal trip probability of $1/4$ per orthogonal round ($η/4$ observed at detection efficiency $η$), independent of the interception angle. An illustrative finite-size budget yields 256 retained bits from $\approx 62{,}000$ transmitted rounds at $η= 0.1$; under realistic detector noise ($q_0 = 10^{-6}$ per opened gate), each trip retains $\approx 12$ bits of evidence at a sub-percent honest false-abort probability per session. The core ideal trip-probability predictions are numerically verified on the Qiskit circuit simulator, via a released, seed-fixed implementation. Overall, by endowing the minimal-detector receiver of polarization QKD with a conclusive, loss-robust disturbance alarm, our solution lowers the hardware entry cost of security-monitored QKD, hence fostering its adoption at the cost-sensitive network edge.

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