Corollary. Sparse-reward discovery within a rollout budget [ftip-00A5]
✍️source

Suppose the events \(\{r_j^{\mathrm {sp}}>0\}\) are independent and have a common probability \(q\). For every positive rollout budget \(B\),

\[ \Pr (J_+\leq B)=1-(1-q)^B. \]

Proof.

Apply the independent-attempt theorem Theorem [ftip-0078] with \(E_j=\{r_j^{\mathrm {sp}}>0\}\). Its discovery event is exactly \(\{J_+\leq B\}\).

This is a specialization proved from the displayed hypotheses. It does not supply independence, stationarity, or the value of \(q\) for a training run, and it does not equate positive proxy reward with task utility.

Corollary. Sparse-reward discovery within a rollout budget [ftip-00A5]
✍️source

Proof.

Finite initial-coverage measurement [ftip-009X]

Definition. Initial evaluation hit count [ftip-009Y]

Definition. Empirical initial success rate [ftip-009Z]

Remark. Zero observed hits are not zero probability [ftip-00A0]

Theorem. Zero-hit likelihood under iid evaluation [ftip-00A1]

Proof.

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

Proof.

Example. The 128-sample zero-hit bound [ftip-00A3]

Definition. First positive sparse-reward time [ftip-00A4]

Corollary. Sparse-reward discovery within a rollout budget [ftip-00A5]
✍️source

Proof.

Example. The AIME 128-sample measurement record [ftip-00A6]

Remark. An observed plateau is not a universal probability threshold [ftip-00A7]

Theorem. Discovery under independent attempts [ftip-0078]

Proof.