Exceptional points and Jordan-chain signatures in quantum first-passage statistics

Exceptional-point signatures in first-passage observables are governed by two independent survival mechanisms: nonlinear spectral defectivity must pass from the one-record or transfer description to the physical one-level first-passage ladder, and the resulting ladder Jordan mode must have nonzero overlap with the chosen preparation and terminal observation. For finite reset-form monitored quantum systems with upward-skip-free counting, we prove the exact threshold factorization $H_N(s)=R_s^N$, identify the physical Perron branch, and derive the transfer factorization $Q_s(r)=B_s(r)(R_s-rI)$. Invertibility of the cofactor gives local equality of Smith data, whereas a singular cofactor can contribute transfer multiplicity absent from the ladder. An order-two Keldysh formula yields an independent observation gate. We then prove a fixed-reset no-go theorem: even a defective full tilted generator cannot generate a cumulative-count Jordan polynomial when the post-count ladder is scalar. Finally, we construct a three-reset monitored-Lindblad witness, minimal within irreducible nonnegative ladders, with a genuine subleading $Nλ^N$ threshold term and a coherence-tuned model in which a binary terminal effect retains the exact polynomial signature at a transversal transform-domain exceptional point. The analysis is formulated at the level of exact transform-domain structure; time-domain asymptotics and perturbative robustness remain separate questions.