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

Take \(m=128\) and \(\delta =0.05\) in Corollary [ftip-00A2]. After zero hits, the exact one-sided upper endpoint is

\[ u_{128,0.05} =1-0.05^{1/128} \approx 0.02313. \]

Under the iid Bernoulli model, probabilities above approximately 2.313 percent make a zero count less than five percent likely. The calculation does not turn a recorded \(0/128\) into proof that the true probability is zero, and it does not apply if the 128 evaluations have a different joint sampling law.

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

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]
✍️source

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

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

Proof.

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

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

Example. Equal averages and zero hits do not identify coverage [ftip-00IG]

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

Proof.