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

• 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

• Source → formalized: Appendix Theorem 2, Condition 1 selects a start after a realized first report within the observation window. The current Lemma 2 model instead requires an unconditional exponential first arrival to lie before a fixed finite horizon $H$ for every outcome.
• Why this needs repair: for reporting rate $\lambda>0$, the exponential law gives $P(T_1>H)=e^{-\lambda H}>0$, contradicting that bound. No process satisfies both premises, so the Lemma 2 and Appendix D.8.2 proofs do not establish the source claims.
• Scope: the main likelihood-factorization proof uses a separate causal observation model. This finding concerns the formalization’s conditioning, not a counterexample to the paper’s waiting-time result.