An exact Jordan signature in cumulative-count first-passage statistics
Abstract
For cumulative-count first-passage problems, the Laplace transform of the time 𝒯N of the Nth count is naturally expressed through powers of a one-count kernel. A Jordan defect of that kernel does not, however, automatically survive the complete sum over terminal phases. We give an exact three-state Markov-renewal example in which it does. At s=1, the one-count kernel has spectrum {1/2,1/8,1/8}, with a one-dimensional eigenspace at 1/8, and for initial phase $1$, 𝔼1[e−𝒯N] =5/6 2−N+(N+1)/6 8−N. Thus the ordinary, unconditioned cumulative-count transform contains the explicit Jordan contribution N/(6 8N) at every threshold N≥1. For the same fixed stochastic process, the two eigenvalues meeting at 1/8 unfold as λ±(s)=1/8±1/96√(s−1) −131/2304(s−1)+O((s−1)3/2), and the characteristic discriminant has a simple zero at s=1. We place the example beside two structures that suppress such a marginal signature: fixed post-count reset, which gives scalar renewal, and a common total holding rate, for which terminal summation scalarizes the matrix power. The model also admits a monitored-Lindblad realization, but the mechanism is entirely classical. The result provides a small exact benchmark for how generalized spectral modes can remain visible in finite-threshold first-passage statistics.
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BibTeX
@misc{bridges2026exact,
title = {{An exact Jordan signature in cumulative-count first-passage statistics}},
author = {Lachlan Bridges},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20148},
eprint = {2609.20148},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20148},
}
RIS
TY - UNPB
TI - An exact Jordan signature in cumulative-count first-passage statistics
AU - Lachlan Bridges
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.20148
UR - https://arxiv.org/abs/2609.20148
AB - For cumulative-count first-passage problems, the Laplace transform of the time 𝒯_N of the Nth count is naturally expressed through powers of a one-count kernel. A Jordan defect of that kernel does not, however, automatically survive the complete sum over terminal phases. We give an exact three-state Markov-renewal example in which it does. At s=1, the one-count kernel has spectrum {1/2,1/8,1/8}, with a one-dimensional eigenspace at 1/8, and for initial phase $1$, 𝔼_1[e^−𝒯_N] =5/6 2^−N+(N+1)/6 8^−N. Thus the ordinary, unconditioned cumulative-count transform contains the explicit Jordan contribution N/(6 8^N) at every threshold N≥1. For the same fixed stochastic process, the two eigenvalues meeting at 1/8 unfold as λ_±(s)=1/8±1/96√(s−1) −131/2304(s−1)+O((s−1)^3/2), and the characteristic discriminant has a simple zero at s=1. We place the example beside two structures that suppress such a marginal signature: fixed post-count reset, which gives scalar renewal, and a common total holding rate, for which terminal summation scalarizes the matrix power. The model also admits a monitored-Lindblad realization, but the mechanism is entirely classical. The result provides a small exact benchmark for how generalized spectral modes can remain visible in finite-threshold first-passage statistics.
ER -