Source clarifications

Source: official arXiv version, main model and Appendices B.1–B.2, C.2.

Lemma 1 and Proposition 1: calendar-time first reports

• Birth-window eventual reports → first reports inside the calendar-time window. An incident born earlier can count if first reported inside the window. The proof must therefore displace retained incident births by their first-report delays, rather than only thin births. The stationary marked displacement theorem gives the observed homogeneous Poisson process.
• Proposition 1 nonidentifiability construction → positive observed rate. This makes its two compensating latent incident rates positive. The zero-detection process is covered separately by Lemma 1.

Appendix B.2: likelihood-factorization algebra

• Condition 1: point probability → conditional density. In continuous time, read the displayed $P(S=t)$ as a density $g(s\mid T_1)$ for the selected start $S$, given the first report time $T_1$. The density integrates to one, does not depend on the reporting rate, and selects a start between $T_1$ and the observation horizon $T$ when $T_1\le T$. The likelihood uses its value at the observed start. This preserves the paper’s selection rule and independence from subsequent reports.
• Equation (30) first post-start gap → $t_{m+1}-s$, where $s$ is the selected start after report $m$ and $t_{m+1}$ the next report time.
• Equation (31) residual normalization → multiply by $M!/(e-s)^M$, where $M$ is the post-start count and $e$ the endpoint. This converts $\lambda^Me^{-\lambda(e-s)}$ into the Poisson count mass with mean $\lambda(e-s)$. It repairs the algebra without changing the likelihood factorization. For a history-responsive endpoint, that factor alone is not a Poisson law conditional on the endpoints: the residual records selection.

Equation (3): rate-estimation convention

• Count-over-exposure estimator → nonnegative rates and strictly positive total exposure. A zero count then has the attained maximum-likelihood estimate zero, rather than an unattained positive-rate limit.

Lemma 2: conditioning on an observed first report

• Observed first report: condition on the actual first report, then select a start between that report and the observation horizon. The subsequent interarrival times are independent exponential draws. This proves the waiting-time conclusion in Lemma 2 and the shifted-process result in Appendix D.8.2.
• The first report need not occur before the horizon on every possible path; the claim concerns paths on which it has been observed. This is a conditioning convention, not an additional behavioral assumption.