Rollouts from the base model allenai/Olmo-3-1025-7B on all 500 MATH-500 problems, with each of 20 short openings ("token cues") prefilled at the start of the response, plus a no-cue baseline. The 20 cues are the depth-2 beam-search nominees at the end of the RL-Zero prompt.

21 conditions x 500 problems x 4 rollouts = 42,000 responses
Sampling: temperature 0.6, top-p 0.95, up to 31,744 new tokens, stop at end-of-text or a new Problem: block (vLLM)
Prompt ({q} is the problem):

Solve the following math problem step by step.
The last line of your response should be the answer to the problem in form Answer: $Answer (without quotes) where $Answer is the answer to the problem.
{q}
Remember to put your answer on its own line after "Answer:"

The model's input is the prompt string followed by the cue, tokenized together. completion is the cue followed by the generated text, so every cued response starts with its cue; the generated text alone is completion[len(cue):].

Conditions

File tag	Cue	pass@1	Cap rate	Median tokens
`none`	(no cue)	0.415	0.12	740
`r01_p1`	`.\n`	0.262	0.09	622
`r02_p2`	`.\n\n`	0.766	0.15	1,436
`r03_spand`	`and`	0.432	0.13	698
`r04_p2Okay`	`.\n\nOkay`	0.769	0.16	1,488
`r05_p1Problem`	`.\nProblem`	0.063	0.09	722
`r06_p1Answer`	`.\nAnswer`	0.130	0.03	4
`r07_spin`	`in`	0.468	0.14	848
`r08_spparen`	`(`	0.456	0.10	667
`r09_spandspto`	`and to`	0.419	0.14	738
`r10_spinspthe`	`in the`	0.484	0.14	837
`r11_p1The`	`.\nThe`	0.378	0.08	224
`r12_spandspbefore`	`and before`	0.451	0.13	691
`r13_spparenwithout`	`(without`	0.441	0.12	679
`r14_p3`	`.\n\n\n`	0.735	0.15	1,438
`r15_spwithout`	`without`	0.451	0.11	707
`r16_p2The`	`.\n\nThe`	0.628	0.09	2,004
`r17_spbut`	`but`	0.441	0.14	711
`r18_p3Okay`	`.\n\n\nOkay`	0.774	0.15	1,504
`r19_p2Answer`	`.\n\nAnswer`	0.172	0.01	5
`r20_p2To`	`.\n\nTo`	0.567	0.06	368

Files

rollouts/rollouts_base_<tag>_p<start>.jsonl: one JSON object per line, 50 problems x 4 rollouts per file.

Field	Meaning
`problem_key`	MATH-500 problem id
`question`	the problem text
`answer`	the ground-truth answer
`rollout_index`	0-3
`arm`	condition tag (table above)
`cue`	the prefilled opening for this arm; `""` for `none`
`prompt`	the prompt with the problem filled in, without the cue
`input_token_ids`	Olmo-3 tokenizer ids of `prompt + cue`, tokenized together, as the model saw it (the tokenizer adds no BOS)
`completion`	the cue followed by the generated text
`generated_token_ids`	Olmo-3 tokenizer ids of the generated text, `completion[len(cue):]`, tokenized on its own
`correct`	graded with the MATH-500 grader (last `Answer:` line or \boxed{})
`committed`	whether a final answer was found
`hit_token_cap`	reached the 31,744-token limit
`output_tokens`	generated length reported by vLLM

The sampled token ids were not stored, so generated_token_ids is a re-tokenization of the generated text. It is tokenized on its own, not joined to the input, because generation continues from the input ids and cannot merge a token across that boundary; the two agree except where the sampler happened to produce a token split the tokenizer would not choose. output_tokens is the sampler's own count and can differ from len(generated_token_ids) by a token or two.

Tokenizer: allenai/Olmo-3-1025-7B. Read JSONL by splitting on \n only: some completions contain U+2028/U+2029, which str.splitlines() treats as line breaks.

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