Probe with Participation Trophies: Random-Reward RL as a Probe of LLM Capability

Yu Mao; Lei Yu; Zining Zhu; Yusheng Zheng; Haohang Li; Freda Shi; Yutong Yin; Zhaoran Wang

cs.CLOct 1, 2026

Probe with Participation Trophies: Random-Reward RL as a Probe of LLM Capability

Authors: Yu Mao, Lei Yu, Zining Zhu, Yusheng Zheng, Haohang Li, Freda Shi, Yutong Yin, Zhaoran Wang, +1 more

Organizations: University of Toronto · Stevens Institute of Technology · University of California, Santa Cruz · University of Waterloo · Vector Institute · Northwestern University

Abstract

We connect the spurious-reward paradox to a model's reachability and propose random-reward reinforcement learning (RL) as a useful tool for the probing enterprise, addressing a decade-long debate over what probing performance actually reveals about a model. There are two prevailing explanations for the surprising finding that even random rewards can improve the performance of large language models (LLMs): one attributes the gains to particular mechanisms within RL training; the other to data contamination. Our results motivate a different view: spurious-reward RL can probe a model's reachability, or what further training can attain from its current state under specified constraints, beyond what is reflected in its current performance. Two OLMo checkpoints with the same accuracy on synthetic arithmetic (3.5%), for example, reach 8.5% and 55% in their best runs under the same correctness-rewarded RL. Examining OLMo checkpoints across pre-training and mid-training reveals three distinct regimes of training response: early on, RL produces little improvement even when correct answers are rewarded; later in pre-training, rewarding correct answers becomes effective while random rewards remain weak; and, upon entering mid-training, even random rewards can produce large gains. A similar ordering appears in a number-masked supervised fine-tuning (SFT) analysis of these checkpoints, suggesting that the pattern is not specific to a particular RL mechanism. Moreover, RL with random rewards offers a distinctive perspective on what training can make an LLM do, since its reward signal supplies no information about which answers are correct. By asking what training can attain without correctness feedback, it addresses the label-leakage side of a central problem in decodability-based probing: whether a successful probe reveals the model's capabilities or learns the task itself.

Figures & tables

Figure 1 : Development of LLM reachability across pre-training and mid-training (OLMo-2, synthetic GSM). Early checkpoints barely improve under either reward (dormant), later ones improve under GT rewards (receptive), and in mid-training some random-reward runs also reach high accuracy (autodidactic). Each line is a seed ending at its held-out accuracy after 500 RL steps, higher closer to the star; bars give the best across seeds, and $(\Delta_{\rm GT},\Delta_{\rm rand})$ marks a gain (+) or none (0). The landscape is schematic; overlapping endpoints (panel I; GT in panel III) are spread slightly.

Response	Learner	Initial	Terminal	Mean
source	Learner	score	max	gain
P1	P1	2.5	9.5	+0.6
P1	P2	13.5	80.0	+4.5
P2	P1	2.5	14.0	+2.9
P2	P2	13.5	57.0	+13.7

Table 1: Identical rollouts and random rewards yield different outcomes across OLMo-2 checkpoints. P1: end of pre-training; P2: +5B mid-training. Each row summarizes 16 runs, with terminal results at step 500. Scores are accuracies (%); gains are changes from initial accuracy (percentage points).

Figure 2 : With the same rollouts and rewards, P2 gains more on average and varies more. Panels split runs by rollout source, the x-axis by learner. Dots are the 16 runs per group and lines their means; changes are step-500 minus initial accuracy (percentage points).

Panel of Figure 3 — Figure 3 : Inputs used to separate task cues from correctness information.

Figure 4 : Random-reward training of OLMo-2 P2 +50B on sequences of randomly sampled tokens: (a) accompanied by the math instruction and (b) presented alone. Panel (c) shows terminal accuracy after 500 training steps. Thin lines and dots show individual runs (32 seeds per condition); highlighted lines show the highest accuracy across seeds at each step. The dashed line marks the initial accuracy of 20.5%. Further comparisons appear in Appendix 11.5 .

Figure 5 : Three representative OLMo-2 checkpoints illustrate the dormant, receptive, and autodidactic stages of training response on synthetic GSM. Top: GT, four seeds per checkpoint; bottom: Random, eight. Thin lines show sampled rollout accuracy (trailing 20-step mean); black lines show pointwise maxima across seeds; dashed lines show frozen greedy held-out accuracy.

Panel of Figure 6 — Figure 6 : Both RL and SFT show the same three-stage developmental pattern across OLMo-2 checkpoints (synthetic GSM). (a) Frozen and maximum terminal accuracy after 500 RL steps; faint points show individual seeds. (b) Initial and best reported SFT accuracy. Stage bands in both panels follow the RL results in (a). See Appendix 11.4 for the SFT comparison.

Figure 7 : OLMo-2 results on code (a) and lookup (b). The overall trend is similar, although the timing of the transitions is idiosyncratic to each task. Lines show frozen accuracy and maximum accuracy across seeds after 500 RL steps; faint points show individual runs. On code, GT and Random gains first appear together in P2. The dashed line marks chance accuracy on lookup (25%).

	Synthetic GSM	Code	Invented-fact lookup
P1 / P2 checkpoints	25 / 12	8 / 4	8 / 4
Training / evaluation problems	800 / 200	764 / 200	800 / 200
GT seeds, P1 / P2	4 / 4	1 / 4	2 / 2
Random seeds, P1 / P2	8 / 8	2 / 8	8 / 8
Response cap (tokens)	96	256	96
Training steps	500	500	500

Table 2: Checkpoint-scan settings. Seed counts are per checkpoint. GT and Random use reference coefficients 0 and 0.01, respectively, in addition to different rewards. Appendix 10.8.5 defines the reference term.

Arm	Training content	Math template	Seeds
B	Original GSM question	Yes	32 (8 reused)
C	Word-shuffled GSM question	Yes	16
A	Unrelated MMLU question stem	Yes	16
D	Random vocabulary tokens	Yes	32
E	Random-token complete prompt	No	32

Table 3: The five training-input conditions. All use random rewards and are evaluated on the same GSM questions. The math template contains the instruction and the Question: and Answer: markers.

ID	Training position	Training step
SmolLM2-1.7B
S1	0.26T	step-125000
S2	5.77T	step-2750000
S3	9.96T	step-4750000
S4	10.75T	step-5125000
OLMo-3-7B

Table 4: SmolLM2 and OLMo-3 checkpoints used in the developmental comparison. P2 token counts are additional to P1.

configuration	reward	reference coefficient	$G$
r_g4 / r_g8	independent random	0.01	4 / 8
gt_g4	ground truth	0	4
gt_K50	ground truth for 50 steps, then random	0.01	8
anti_K50	inverted ground truth for 50 steps, then random	0.01	8

Table 5: Group-size and steering conditions in the earlier 55-run study. All use learning rate

3\times 10^{-5}

, 500 updates, and evaluation every 100 steps. Steering supplies correctness-dependent rewards for the first 50 steps.

Batch	Runs	Flagged steps / all steps	Runs with a flagged step
Completed checkpoint scan	636	519 / 318,000	155
Expanded input controls	152	492 / 76,000	112
SmolLM2 / OLMo-3 checkpoints	80	46 / 40,000	21
Earlier control study	55	0 / 27,500	0

Table 6: Clipping diagnostics across experiments. A flagged step contains at least one selected token with a probability ratio outside [0.8, 1.2]. The count does not measure the resulting gradient change.

Checkpoint	Seed	Lenient	Strict	Format rate
P2 +5B	31003	67.5	0.0	100.0
P2 +5B	31004	59.0	0.0	0.0
P2 +5B	31006	77.5	0.0	0.0
P2 +5B	31007	77.0	0.0	50.5
P2 +9B	31004	78.0	43.0	54.0
P2 +9B	31008	55.0	4.5	9.5

Table 7: Answer-format diagnostics for all GSM Random runs ending at or above 50% lenient accuracy. Strict accuracy requires a standalone answer line; format rate records the presence of an answer marker. Values are percentages.

Checkpoint	Base	1	2	3	4	Max	$n$
P1 5B	0.0	0.0	0.0	0.0	0.5	0.5	4
P1 34B	0.5	2.0	0.5	0.0	1.0	2.0	4
P1 462B	3.5	4.5	3.5	4.5	8.5	8.5	4
P1 839B	3.5	55.0	33.5	16.5	19.5	55.0	4
P1 1,259B	2.0	66.5	72.5	1.5	57.0	72.5	4
P1 1,469B	2.5	18.5	18.0	12.0	19.5	19.5	4

Table 8: Synthetic GSM GT terminal accuracy at step 500 (%). Seed headings omit 3100; Max is the row maximum. Dashes mark unrun configurations. Base follows the frozen-score convention in Appendix 10.7.1 .

Checkpoint	1	2	3	4	5	6	7	8	Max
P1 5B	0.0	0.0	0.0	0.0	0.5	0.5	0.5	0.5	0.5
P1 34B	0.5	2.0	1.0	1.5	1.0	0.0	0.5	1.5	2.0
P1 462B	4.0	2.5	4.0	3.5	4.5	0.0	2.5	3.0	4.5
P1 839B	1.5	2.5	1.5	3.0	4.0	4.0	4.0	3.5	4.0
P1 1,259B	3.0	2.5	2.5	1.0	2.0	2.5	2.5	3.5	3.5
P1 1,469B	2.0	1.0	1.0	1.5	2.0	2.5	1.0	0.0	2.5

Table 9: Synthetic GSM Random terminal accuracy at step 500 (%). Seed headings omit 3100; Max is the row maximum. Dashes mark unrun configurations.

Panel of Figure 8 — Figure 8 : Frozen and maximum terminal accuracy after 500 updates on code and lookup. Faint points show individual seeds. Lookup pools 104 one-hop and 96 two-hop items, with uniform choice among four cities shown at 25%. Panel scales differ. Checkpoints are equally spaced in training order; P2 tokens are additional to P1. Table 2 gives seed counts.

Checkpoint	Base	1	2	3	4	Max	$n$
P1 5B	0.0	0.0	—	—	—	0.0	1
P1 462B	0.5	0.5	—	—	—	0.5	1
P1 839B	0.0	0.0	—	—	—	0.0	1
P1 1,259B	0.5	0.5	—	—	—	0.5	1
P1 2,098B	1.0	11.0	—	—	—	11.0	1
P1 2,937B	0.0	0.5	—	—	—	0.5	1

Table 10: Code GT terminal accuracy at step 500 (%). Seed headings omit 3100; Max is the row maximum. Dashes mark unrun configurations. Base follows the frozen-score convention in Appendix 10.7.1 .

Checkpoint	1	2	3	4	5	6	7	8	Max
P1 5B	0.0	0.0	—	—	—	—	—	—	0.0
P1 462B	0.0	1.0	—	—	—	—	—	—	1.0
P1 839B	0.0	0.0	—	—	—	—	—	—	0.0
P1 1,259B	0.0	0.0	—	—	—	—	—	—	0.0
P1 2,098B	1.5	0.0	—	—	—	—	—	—	1.5
P1 2,937B	0.5	0.5	—	—	—	—	—	—	0.5

Table 11: Code Random terminal accuracy at step 500 (%). Seed headings omit 3100; Max is the row maximum. Dashes mark unrun configurations.

Checkpoint	Base	1	2	3	4	Max	$n$
P1 5B	0.0	0.0	0.0	—	—	0.0	2
P1 462B	3.0	97.5	97.0	—	—	97.5	2
P1 839B	22.5	98.5	99.5	—	—	99.5	2
P1 1,259B	27.0	99.0	98.0	—	—	99.0	2
P1 2,098B	20.0	99.5	99.0	—	—	99.5	2
P1 2,937B	5.0	99.0	98.5	—	—	99.0	2

Table 12: Compositional lookup GT terminal accuracy at step 500 (%). Seed headings omit 3100; Max is the row maximum. Dashes mark unrun configurations. Base follows the frozen-score convention in Appendix 10.7.1 .

Checkpoint	1	2	3	4	5	6	7	8	Max
P1 5B	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
P1 462B	26.0	2.0	25.0	1.0	3.0	5.5	5.5	4.0	26.0
P1 839B	20.5	20.5	22.0	0.0	27.0	23.5	22.0	1.5	27.0
P1 1,259B	17.0	24.5	3.5	33.0	17.5	33.0	24.0	0.0	33.0
P1 2,098B	12.0	11.5	16.5	19.0	15.0	15.5	10.0	12.5	19.0
P1 2,937B	0.0	2.0	3.0	3.0	23.0	16.0	7.0	0.0	23.0

Table 13: Compositional lookup Random terminal accuracy at step 500 (%). Seed headings omit 3100; Max is the row maximum. Dashes mark unrun configurations.

Panel of Figure 9 — Figure 9 : Checkpoint development in SmolLM2 and OLMo-3. Curves show frozen greedy accuracy and maximum terminal accuracy after 500 updates; faint points show the two GT and eight Random seeds at each checkpoint. All use the same 200 GSM questions. Horizontal spacing follows checkpoint order; shading marks OLMo-3 mid-training.

Checkpoint	Raw	GT max	Random max	Random mean	Gain	Random peak
S1	0.5	1.0	1.0	0.56	0/8	1.0
S2	0.0	1.0	4.0	1.12	0/8	4.0
S3	1.0	3.0	8.5	2.44	0/8	8.5
S4	3.0	78.0	7.5	4.69	0/8	7.5
O1	1.0	0.5	1.0	0.63	0/8	1.0
O2	15.5	93.5	15.0	8.31	0/8	31.5

Table 14: Additional-model checkpoint results (%). Maxima use terminal scores, except Random peak, which also includes intermediate evaluations. Gain counts terminal improvements greater than ten points.

	GT		Random
ID	1	2	1	2	3	4	5	6	7	8
S1	1.0	0.5	0.5	0.5	0.5	0.5	0.5	1.0	0.5	0.5
S2	0.5	1.0	0.5	0.5	2.0	1.5	0.0	0.0	0.5	4.0
S3	3.0	1.0	8.5	2.0	1.0	4.0	1.0	1.5	0.5	1.0
S4	54.0	78.0	7.0	3.5	2.5	5.0	7.5	5.0	5.0	2.0
O1	0.5	0.5	0.5	0.5	1.0	1.0	1.0	0.0	0.5	0.5

Table 15: All 80 terminal scores in the additional checkpoint comparison (%). GT columns use seeds 31001–31002 and Random columns use seeds 31001–31008.

Checkpoint	S1	S2	S3	S4	O1	O2	O3	O4
Full output	0.5	0.0	1.0	3.0	1.0	15.5	18.0	30.0
First block	3.0	4.0	14.0	27.5	1.0	53.0	35.0	34.0

Table 16: Frozen accuracy with the primary full-output extractor and the first-block diagnostic (%). The developmental figures retain the primary extractor before and after training.

Figure 10 : Terminal GSM accuracy under the five OLMo-2 input conditions. Dots show every seed; diamonds mark maxima. Original GSM and the two random-token conditions have 32 seeds each, and shuffled GSM and MMLU have 16. The dashed line is the frozen score of 20.5%.

Training input	$n$	Max	Mean	Median	Gain	Collapse
B: GSM	32	78.0	13.98	6.75	3/32	15/32
C: shuffled GSM	16	55.5	15.56	14.25	2/16	6/16
A: MMLU	16	86.5	23.28	18.50	2/16	1/16
D: tokens + math template	32	68.5	17.50	13.50	4/32	8/32
E: raw tokens	32	23.0	20.36	20.50	0/32	0/32

Table 17: OLMo-2 input-control outcomes at step 500. Scores are percentages. Gain means improvement greater than ten points from a run's initial score; collapse means terminal accuracy at or below 5%.

Seed	B	C	A	D	E
31001	23.5	8.5	23.0	16.5	19.5
31002	7.5	0.5	15.5	22.0	19.5
31003	15.0	21.0	22.5	8.5	23.0
31004	6.0	35.0	50.5	68.5	21.0
31005	0.0	20.5	19.0	23.0	20.5
31006	21.5	55.5	29.5	22.5	21.5

Table 18: All OLMo-2 input-control terminal scores (%). B uses original GSM, C shuffled GSM, A MMLU stems, D tokens within the math template, and E tokens without the template. Dashes indicate seeds not run.

Panel of Figure 11 — Figure 11 : Training trajectories with unrelated MMLU questions and shuffled GSM questions. Both retain the math template. Thin lines show all 16 seeds; thick lines show the pointwise maximum, which can switch seeds. The dashed line marks frozen accuracy.

Panel of Figure 12 — Figure 12 : Random-token input controls on Qwen2.5 and Llama. Dots show the eight terminal scores in each condition; diamonds mark maxima. Dashed lines show mean pre-update scores, 44.3% and 25.5%. Input construction differs in both template and token filtering (Appendix 10.3 ).

Model	Input	Max	Mean	SD	Gain	Collapse
Qwen2.5-7B	D	91.5	50.75	29.78	4/8	1/8
	E	51.0	46.38	2.45	0/8	0/8
Llama-3.1-8B	D	64.0	16.63	19.30	1/8	3/8
	E	31.5	19.56	6.94	0/8	0/8

Table 19: Qwen2.5 and Llama input-control results (%), with eight seeds per condition. D retains the math template; E omits it. Gain means an improvement above ten points; collapse means terminal accuracy at or below 5%.

	Qwen2.5-7B		Llama-3.1-8B
Seed	D	E	D	E
31001	57.0	51.0	1.0	31.5
31002	34.0	43.0	3.0	14.5
31003	0.0	45.0	1.0	9.0
31004	25.5	48.0	19.0	16.0
31005	91.5	44.0	13.5	18.0

Table 20: All Qwen2.5 and Llama input-control terminal scores (%). D uses tokens within the math template and E uses tokens alone.

Source	Paired quantity (P2 minus P1)	Counts	$p$	Holm $p$
P1	Gain	6/9/1	0.6072	0.8405
P1	Improved	3/0	0.2500	0.8405
P1	Absolute gain	13/2/1	0.0074	0.0443
P2	Gain	11/5/0	0.2101	0.8405
P2	Improved	5/1	0.2188	0.8405
P2	Absolute gain	12/4/0	0.0768	0.3841

Table 21: The six planned shared-rollout tests, with 16 paired blocks per source. Sign-test counts are positive/negative/tied differences. Improved-run counts are pairs where only P2 improves / only P1 improves. Holm correction covers all six tests.

Source	Learner	Primary score	Prefix score	Prefix gain	Prefix gain
		before $\to$ after	before $\to$ after	$>10$ pp	$<-10$ pp
P1	P1	$2.5\to 3.1$	$41.5\to 28.8$	2/16	10/16
P1	P2	$13.5\to 18.0$	$14.5\to 38.4$	8/16	0/16
P2	P1	$2.5\to 5.4$	$41.5\to 21.2$	1/16	11/16
P2	P2	$13.5\to 27.2$	$14.5\to 39.3$	8/16	0/16

Table 22: Shared-rollout answer-extraction diagnostic. Scores are percentages; the terminal score is the mean of 16 runs. Prefix scoring stops at the first generated newline followed by Question: . These descriptive results use the same rule before and after training.

Replicate	P1 $\to$ P1	P1 $\to$ P2	P2 $\to$ P1	P2 $\to$ P2
51001	3.0	4.0	1.5	48.5
51002	4.0	44.0	7.0	14.5
51003	4.0	23.0	1.0	13.5
51004	8.0	2.0	4.0	10.5
51005	1.0	43.5	3.0	47.5
51006	0.5	11.5	7.0	19.0

Table 23: All shared-rollout terminal scores (%) on the 200 held-out GSM questions. Arrows give response source

\to

learner. Each row is one paired block of four runs.

seed	P1 (3,896B)	P2 +5B	P2 +50B
r_g4 ( $G$ = 4)
777	3.0 / 7.5 (+4.5)	13.0 / 26.5 (+13.5) ⋆	21.0 / 1.5 ( $-$ 19.5)
888	1.0 / 27.0 (+26.0) ⋆	9.0 / 2.5 ( $-$ 6.5)	17.5 / 0.5 ( $-$ 17.0)
999	4.5 / 9.0 (+4.5)	20.0 / 17.5 ( $-$ 2.5)	27.0 / 76.5 (+49.5) ⋆
1001	4.5 / 0.0 ( $-$ 4.5)	14.0 / 19.5 (+5.5)	19.5 / 6.0 ( $-$ 13.5)
1002	1.0 / 8.0 (+7.0)	14.5 / 49.0 (+34.5) ⋆	16.0 / 85.5 (+69.5) ⋆

Table 24: All 30 random-reward runs in the earlier group-size comparison. Entries give initial / terminal accuracy in percent, with changes in parentheses. Stars mark gains above 10 points. Every run uses learning rate

3\times 10^{-5}

, reference coefficient 0.01, and 500 updates.

Checkpoint	$G$ = 4 successes	$G$ = 4 mean $\Delta$	$G$ = 8 successes	$G$ = 8 mean $\Delta$
P1 3,896B	1/5	+7.5	0/5	+0.8
P2 +5B	2/5	+8.9	0/5	$-$ 4.0
P2 +50B	2/5	+13.8	0/5	$-$ 6.1

Table 25: Group-size outcomes at each checkpoint, with five seeds per condition. Success means a terminal gain above 10 points; mean changes are in percentage points.

arm	after accuracy (777 / 888 / 999 / 1001 / 1002)	mean $\Delta$
gt_g4 P1	90.5 / 92.5 / 81.5 / 81.0 / 84.0	+83.1
gt_g4 P2 +5B	99.0 / 97.0 / 98.0 / 99.5 / 100.0	+84.6
gt_g4 P2 +50B	100.0 / 98.0 / 100.0 / 98.0 / 21.5	+63.3
gt_K50	90.5 / 1.0 / 62.0 / 3.0 / 54.0	+28.0
anti_K50	1.5 / 1.5 / 16.5 / 3.5 / 9.0	$-$ 7.7

Table 26: GT and early-steering outcomes in the earlier study. Terminal accuracies follow seed order 777, 888, 999, 1001, 1002 and are reported in percent; mean changes are in percentage points. Steering is applied at P2 +5B.

Quantity	D	E	Difference	$p$	$p_{H}$	$p_{\rm all}$
OLMo-2
Mean signed gain (pp)	$-$ 3.00	$-$ 0.14	$-$ 2.86	0.3337	1	1
Mean absolute change (pp)	12.59	1.11	+11.48	$4.66\!\times\!10^{-10}$	$6.98\!\times\!10^{-9}$	$1.35\!\times\!10^{-7}$
Gain $>10$ pp	4/32	0/32	+12.50	0.125	1	1
Absolute change $>10$ pp	16/32	0/32	+50.00	$3.05\!\times\!10^{-5}$	$4.27\!\times\!10^{-4}$	0.0088
Terminal score at most 5%	8/32	0/32	+25.00	0.0078	0.1016	1

Table 27: Paired D/E comparisons on OLMo-2 (32 seeds), Qwen2.5 (eight), and Llama (eight). Effects are D minus E in percentage points; binary rows also report event counts.

p_{H}

corrects the 15 tests in this table, and

p_{\rm all}

corrects all 290 exploratory tests.

Model	Input	Gain: count [95% CI]	Change: count [95% CI]
OLMo-2	B	3/32 [2.0, 25.0]	21/32 [46.8, 81.4]
OLMo-2	C	2/16 [1.6, 38.3]	10/16 [35.4, 84.8]
OLMo-2	A	2/16 [1.6, 38.3]	5/16 [11.0, 58.7]
OLMo-2	D	4/32 [3.5, 29.0]	16/32 [31.9, 68.1]
OLMo-2	E	0/32 [0.0, 10.9]	0/32 [0.0, 10.9]
Qwen2.5	D	4/8 [15.7, 84.3]	7/8 [47.3, 99.7]

Table 28: Input-control event counts and marginal 95% Clopper–Pearson intervals (%). Gain means

\Delta>10

points; change means

|\Delta|>10

points. Each interval describes one model, input condition, and protocol.

Pair	$n$	Mean diff.	$p$	$p_{H}$	Rate diff.	$p$	$p_{H}$
B $-$ C	16	$-$ 2.59	0.5857	1	$-$ 6.25	1	1
B $-$ A	16	$-$ 10.31	0.0023	0.0417	$-$ 6.25	1	1
B $-$ D	32	$-$ 3.64	0.4467	1	$-$ 3.12	1	1
B $-$ E	32	$-$ 6.50	0.0672	1	+9.38	0.25	1
C $-$ A	16	$-$ 7.72	0.1971	1	0.00	1	1
C $-$ D	16	$-$ 5.16	0.3455	1	0.00	1	1

Table 29: Other paired OLMo-2 input comparisons, using shared seeds. The first triplet gives mean signed-gain differences; the second gives differences in the rate of gains above ten points. Effects are percentage points.

p_{H}

corrects all 18 tests; D/E appears separately in Table 27 .

Comparison	Seeds	Difference	$p$	$p_{H}$
GSM GT: P1 3,896B $-$ P1 34B	4	+81.12	0.125	0.875
GSM GT: P2 +5B $-$ P1 3,896B	4	+17.38	0.125	0.875
GSM GT: P1 839B $-$ P1 462B	4	+25.88	0.125	0.875
GSM Random: P1 3,896B $-$ P1 34B	8	+5.81	0.0156	0.1719
GSM Random: P2 +5B $-$ P1 3,896B	8	+33.44	0.0156	0.1719
GSM Random: P1 839B $-$ P1 462B	8	0.00	1	1

Table 30: OLMo-2 developmental contrasts. Effects are later minus earlier terminal scores in percentage points, except the last row, which compares high-score frequencies. P1/P2 contrasts average checkpoints within phase and seed before pairing.

p_{H}

corrects 12 tests. Code GT has one common seed and no inferential test.

Pair	GT diff.	$p$	$p_{H}$	Random diff.	$p$	$p_{H}$
S2 $-$ S1	0.00	1	1	+0.56	0.4375	1
S3 $-$ S1	+1.25	0.5	1	+1.88	0.0156	0.4219
S4 $-$ S1	+65.25	0.5	1	+4.12	0.0078	0.25
S3 $-$ S2	+1.25	1	1	+1.31	0.3594	1
S4 $-$ S2	+65.25	0.5	1	+3.56	0.0234	0.5859
S4 $-$ S3	+64.00	0.5	1	+2.25	0.0469	1

Table 31: All checkpoint pairs within SmolLM2 and OLMo-3. Triplets give mean terminal-score differences (percentage points), raw

p

, and Holm

p

for GT and Random. GT uses two paired seeds and Random eight. Correction covers these 24 tests and the eight GT/Random tests in Table 36 .

Comparison	Mean diff.	$p$	$p_{H}$	Rate diff.	$p$	$p_{H}$
P1 3,896B	+6.70	0.3125	1	+20.00	1	1
P2 +5B	+12.90	0.1875	1	+40.00	0.5	1
P2 +50B	+19.90	0.4375	1	+40.00	0.5	1
Three-checkpoint mean	+13.17	0.3125	1	+33.33	0.125	1
GT $-$ anti steering	+35.70	0.25	1	+60.00	0.25	1

Table 32: Group-size and steering comparisons with five seed blocks. The first four rows compare

G

= 4 with

G

= 8, including an average over checkpoints; the last compares GT and inverted-GT steering. Triplets report mean-gain and jackpot-rate differences (percentage points), raw

p

, and Holm

p

over all ten tests.

Checkpoint	GT $\uparrow/\downarrow/=$	$p_{\rm dir}$	$p_{H}$	R $\uparrow/\downarrow/=$	$p_{\rm dir}$	$p_{H}$	GT $-$ R	$p$	$p_{H}$
P1 5B	1/0/3	1	1	4/0/4	0.125	1	+0.12	1	1
P1 34B	2/1/1	1	1	5/1/2	0.2188	1	$-$ 0.38	0.75	1
P1 462B	3/0/1	0.25	1	3/4/1	1	1	+1.75	0.125	1
P1 839B	4/0/0	0.125	1	3/4/1	1	1	+29.00	0.125	1
P1 1,259B	3/1/0	0.625	1	6/1/1	0.125	1	+47.12	0.25	1
P1 1,469B	4/0/0	0.125	1	1/5/2	0.2188	1	+15.62	0.125	1

Table 33: GSM changes within each checkpoint and reward condition. Tables 34 – 36 use the same layout. Up/down/tie counts use each run's initial score.

p_{\rm dir}

is a two-sided sign test; its

p_{H}

corrects 150 within-condition tests. The final triplet gives the paired GT minus Random mean score (percentage points), permutation

p

, and correction over 53 scan GT/Random tests. One-seed comparisons have no

p

value.

Checkpoint	GT $\uparrow/\downarrow/=$	$p_{\rm dir}$	$p_{H}$	R $\uparrow/\downarrow/=$	$p_{\rm dir}$	$p_{H}$	GT $-$ R	$p$	$p_{H}$
P1 5B	0/0/1	–	–	0/0/2	1	1	0.00	–	–
P1 462B	0/0/1	–	–	1/1/0	1	1	+0.50	–	–
P1 839B	0/0/1	–	–	0/0/2	1	1	0.00	–	–
P1 1,259B	0/0/1	–	–	0/2/0	0.5	1	+0.50	–	–
P1 2,098B	1/0/0	–	–	1/1/0	1	1	+9.50	–	–
P1 2,937B	1/0/0	–	–	2/0/0	0.5	1	0.00	–	–

Table 34: Code changes within each checkpoint and reward condition. Columns and tests follow Table 33 .

Checkpoint	GT $\uparrow/\downarrow/=$	$p_{\rm dir}$	$p_{H}$	R $\uparrow/\downarrow/=$	$p_{\rm dir}$	$p_{H}$	GT $-$ R	$p$	$p_{H}$
P1 5B	0/0/2	1	1	0/0/8	1	1	0.00	1	1
P1 462B	2/0/0	0.5	1	5/3/0	0.7266	1	+83.25	0.5	1
P1 839B	2/0/0	0.5	1	2/5/1	0.4531	1	+78.50	0.5	1
P1 1,259B	2/0/0	0.5	1	2/6/0	0.2891	1	+77.75	0.5	1
P1 2,098B	2/0/0	0.5	1	0/8/0	0.0078	1	+87.50	0.5	1
P1 2,937B	2/0/0	0.5	1	3/5/0	0.7266	1	+97.75	0.5	1

Table 35: Lookup changes within each checkpoint and reward condition. Columns and tests follow Table 33 .

Checkpoint	GT $\uparrow/\downarrow/=$	$p_{\rm dir}$	$p_{H}$	R $\uparrow/\downarrow/=$	$p_{\rm dir}$	$p_{H}$	GT $-$ R	$p$	$p_{H}$
O1	0/2/0	0.5	1	0/5/3	0.0625	1	0.00	1	1
O2	2/0/0	0.5	1	0/8/0	0.0078	1	+79.25	0.5	1
O3	2/0/0	0.5	1	3/5/0	0.7266	1	+49.25	0.5	1
O4	2/0/0	0.5	1	4/4/0	1	1	+44.50	0.5	1
S1	1/0/1	1	1	1/0/7	1	1	+0.25	1	1
S2	2/0/0	0.5	1	6/0/2	0.0312	1	+0.25	1	1

Table 36: SmolLM2 and OLMo-3 changes within each checkpoint and reward condition. Columns follow Table 33 , except that the GT minus Random correction covers 32 additional-model tests.

Cell	Mean gain	$\uparrow/\downarrow/=$	$p_{\rm dir}$	$p_{H}$	Gain: count [95% CI]
OLMo-2 B	$-$ 6.64	6/25/1	$8.78\!\times\!10^{-4}$	0.1317	3/32 [2.0, 25.0]
OLMo-2 C	$-$ 4.94	6/9/1	0.6072	1	2/16 [1.6, 38.3]
OLMo-2 A	+2.78	7/9/0	0.8036	1	2/16 [1.6, 38.3]
OLMo-2 D	$-$ 3.00	13/19/0	0.3771	1	4/32 [3.5, 29.0]
OLMo-2 E	$-$ 0.14	13/13/6	1	1	0/32 [0.0, 10.9]
Qwen2.5 D	+6.56	4/4/0	1	1	4/8 [15.7, 84.3]

Table 37: Changes within each input-control and earlier-study condition. Mean gains are percentage points. Sign tests compare upward with downward nonzero changes, with Holm correction over 150 tests. The final column gives the jackpot count and its marginal 95% interval (%).

Checkpoint / task	Gain: count [95% CI]	High score: count [95% CI]
GSM P1 5B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]
GSM P1 34B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]
GSM P1 462B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]
GSM P1 839B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]
GSM P1 1,259B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]
GSM P1 1,469B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]

Table 38: Random-run success rates, part 1. Entries give success count / seed count and marginal 95% Clopper–Pearson intervals (%). Gain means

\Delta>10

points and high score means terminal accuracy 50% or more. Each checkpoint is estimated separately.

Checkpoint / task	Gain: count [95% CI]	High score: count [95% CI]
GSM P1 3,896B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]
GSM P2 +5B	4/8 [15.7, 84.3]	4/8 [15.7, 84.3]
GSM P2 +9B	2/8 [3.2, 65.1]	2/8 [3.2, 65.1]
GSM P2 +13B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]
GSM P2 +17B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]
GSM P2 +21B	3/8 [8.5, 75.5]	3/8 [8.5, 75.5]

Table 39: Random-run success rates, part 2. Entries give success count / seed count and marginal 95% Clopper–Pearson intervals (%). Gain means

\Delta>10

points and high score means terminal accuracy 50% or more. Each checkpoint is estimated separately.

Checkpoint / task	Gain: count [95% CI]	High score: count [95% CI]
CODE P2 +50B	1/8 [0.3, 52.7]	0/8 [0.0, 36.9]
Lookup P1 5B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]
Lookup P1 462B	2/8 [3.2, 65.1]	0/8 [0.0, 36.9]
Lookup P1 839B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]
Lookup P1 1,259B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]
Lookup P1 2,098B	0/8 [0.0, 36.9]	0/8 [0.0, 36.9]

Table 40: Random-run success rates, part 3. Entries give success count / seed count and marginal 95% Clopper–Pearson intervals (%). Gain means

\Delta>10

points and high score means terminal accuracy 50% or more. Each checkpoint is estimated separately.

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