Remark. An observed plateau is not a universal probability threshold [ftip-00A7]
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[clay2026demystifying, Section 5.2 and Figure 2] report that the Qwen3-1.7B Base movie-quote cell, whose initial empirical match rate is about 0.5 percent, does not optimize under the source's sparse-reward run, whereas its dense-reward run rises toward fifty percent. [clay2026demystifying, Appendix Figure 8] gives final match rates 10.0 percent for sparse reward and 48.8 percent for dense reward. Thus ``failed'' here means failure of the reported sparse run to optimize as intended, not zero final matches.

A plateau is indexed by the model, target, reward implementation, optimizer, sampling process, and budget in its experiment cell. One observed transition therefore does not identify a universal initial-probability threshold for RL learning, support, or capability acquisition.

Remark. An observed plateau is not a universal probability threshold [ftip-00A7]
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Reference. Demystifying Reinforcement Learning Post-Training of Language Models [clay2026demystifying]

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]

Proof.

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

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