proposition. A necessary-event bound for an evolving lineage [ftip-00MP]
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    Fix \(B\in \mathbb N\) charged elementary transitions and a \([0,1]\)-valued realized acquisition score \(Z\), including its evaluation randomness. Let \(G_t\) mean a specified necessary structural event has occurred by transition \(t\). Assume \(G_0\) is false, \(G_t\subseteq G_{t+1}\), and \(Z\leq \mathbf 1_{G_B}\) almost surely. For deterministic \(\epsilon _t\in [0,1]\), suppose that every admitted campaign and every permitted positive-probability history before \(G\) satisfy

    \[ \Pr (G_t\mid \text {that history at }t-1)\leq \epsilon _t, \qquad t=1,\ldots ,B. \]

    For general history spaces use the corresponding conditional-kernel bound almost everywhere for each admitted campaign. Include histories after training and controller replacement. Stopped runs are padded by absorbing transitions. Then every admitted \(P\) satisfies

    \[ Q_{\mathrm {acq}}(P)=\mathbb E Z \leq 1-\prod _{t=1}^B(1-\epsilon _t) \leq \min \{1,\sum _{t=1}^B\epsilon _t\}. \]

    Proof. 

      Conditional on not having reached \(G\) by \(t-1\), survival at the next transition has probability at least \(1-\epsilon _t\). Conditional expectation gives

      \[ \Pr (G_t^c)\geq (1-\epsilon _t)\Pr (G_{t-1}^c). \]

      Induction from \(\Pr (G_0^c)=1\) gives the product bound. The union bound on first-entry events gives the sum bound, while probabilities are at most one. Taking expectations of \(Z\leq \mathbf 1_{G_B}\) proves the score bound. For \(B=0\), the empty product is one and the bound is zero.

      If the premise holds with \(\epsilon _t=\epsilon (n)>0\) at every work budget, then \(B_{\mathrm {closed}}(n,\tau )\geq \frac {\tau }{\epsilon (n)}\) for integer work budgets. A bound measured for one fixed checkpoint does not supply this premise. If other routes can achieve positive score without \(G\), the domination assumption fails. Calling \(G\) simply “success” does not explain a discovery mechanism without an independent probability bound. Recognition and learning obstructions may require other arguments.