Corollary. Exact one-sided bound after zero hits [ftip-00A2]
✍️source

    Assume Theorem [ftip-00A1] with \(m\in \mathbb N_{\geq 1}\) and fix \(0<\delta <1\). If \(K_m=0\), inversion of the exact zero-count likelihood gives the one-sided upper endpoint

    \[ u_{m,\delta }=1-\delta ^{1/m}. \]

    Indeed, \(\Pr _{u_{m,\delta }}(K_m=0)=\delta \), and for every \(p>u_{m,\delta }\) one has \(\Pr _p(K_m=0)<\delta \). Thus the observed zero count excludes probabilities above \(u_{m,\delta }\) at exact level \(\delta \) under the declared iid model.

    Proof. 

      By Theorem [ftip-00A1], the zero-count likelihood is \((1-p)^m\), which is strictly decreasing in \(p\). Solving \((1-p)^m=\delta \) gives the endpoint and the stated strict inequality.

      This is a likelihood inversion derived from the displayed finite setup, not a source theorem and not a support certificate.