Same Beliefs, Different Policies: Learning Reusable Representations for Early Stopping in LLM Reasoning

Jingwei Ni; Dmitry Abulkhanov; Vladislav Smirnov; Markus Leippold; Elliott Ash; Timothy Baldwin; Artem Shelmanov; Mrinmaya Sachan

cs.AIOct 21, 2026

Same Beliefs, Different Policies: Learning Reusable Representations for Early Stopping in LLM Reasoning

Authors: Jingwei Ni, Dmitry Abulkhanov, Vladislav Smirnov, Markus Leippold, Elliott Ash, Timothy Baldwin, Artem Shelmanov, Mrinmaya Sachan

Organizations: ETH Zürich · MBZUAI · University of Zürich · The University of Melbourne

Abstract

Reasoning LLMs often continue reasoning after reaching a correct answer, wasting computation and sometimes replacing it with an incorrect one. Deciding when to stop is a metareasoning problem: the model must weigh what further reasoning could achieve against its cost, given what it can already answer. Direct utility training learns a stopping representation at a chosen token price; supporting new prices can require additional controller training. We introduce BELT (BELief-based Termination), which separates outcome prediction from stopping control. It learns beliefs about current and future answers from a frozen LLM's hidden states, then fits a linear stopping policy on the frozen features. The prediction targets are independent of token price, so one probe supports different accuracy-cost trade-offs through CPU policy refitting or threshold adjustment. Averaged over five out-of-distribution (OOD) math benchmarks, BELT reduces generated tokens by 41% on Qwen3-1.7B and 51% on OLMo-3-7B-Think without loss of accuracy. The same math-trained BELT transfers to coding and agentic tasks without training on code or agentic data. In all three domains, its accuracy-cost trade-offs are competitive with or better than those of end-to-end controllers. On OLMo3, its probe has 21.2x fewer parameters than one checkpoint of the strongest baseline, OS-Pruner, and needs 55x fewer training GPU-hours than OS-Pruner's five price-specific checkpoints.

Figures & tables

Figure 1: BELT overview. (a) Label offline traces to supervise outcome beliefs. (b) Jointly train the BELT probe with four objectives of outcome prediction. (c) Fit stopping policies on frozen features for token prices $\lambda$ ; alternatively, fit once and vary its threshold. Both reuse one representation. B/H in panel (c) identify the feature interfaces.

Method	Probe Params	Acc. tolerance: 0 pp		Acc. tolerance: 1 pp		Acc. tolerance: 2 pp
Method	Probe Params	ID	OOD	ID	OOD	ID	OOD
CoDE-Stop ‡	0	$16.61_{\scriptscriptstyle\pm 0.50}$	$16.94_{\scriptscriptstyle\pm 0.75}$	$21.94_{\scriptscriptstyle\pm 0.51}$	$26.02_{\scriptscriptstyle\pm 0.78}$	$24.85_{\scriptscriptstyle\pm 0.51}$	$30.15_{\scriptscriptstyle\pm 0.79}$
DEER ‡	0	$12.77_{\scriptscriptstyle\pm 0.59}$	$10.36_{\scriptscriptstyle\pm 0.84}$	$20.96_{\scriptscriptstyle\pm 0.69}$	$20.07_{\scriptscriptstyle\pm 1.01}$	$24.22_{\scriptscriptstyle\pm 0.71}$	$23.19_{\scriptscriptstyle\pm 1.05}$
TC	514	$22.15_{\scriptscriptstyle\pm 0.66}$	$19.97_{\scriptscriptstyle\pm 0.15}$	$28.12_{\scriptscriptstyle\pm 0.43}$	$25.46_{\scriptscriptstyle\pm 0.27}$	$32.24_{\scriptscriptstyle\pm 0.55}$	$30.45_{\scriptscriptstyle\pm 1.50}$
ROM	5M	$3.50_{\scriptscriptstyle\pm 4.95}$	$1.22_{\scriptscriptstyle\pm 1.72}$	$21.01_{\scriptscriptstyle\pm 1.85}$	$18.76_{\scriptscriptstyle\pm 1.17}$	$26.57_{\scriptscriptstyle\pm 0.47}$	$24.00_{\scriptscriptstyle\pm 0.83}$
OSP	101M $\times 5$	$\underline{44.77}_{\scriptscriptstyle\pm 0.75}$	$25.92_{\scriptscriptstyle\pm 22.46}$	$\underline{49.72}_{\scriptscriptstyle\pm 1.09}$	$45.19_{\scriptscriptstyle\pm 1.69}$	$\underline{54.15}_{\scriptscriptstyle\pm 0.90}$	$\mathbf{50.68}_{\scriptscriptstyle\pm 2.04}$

Table 1: Token saving (%) at Qwen3 accuracy tolerances (macro averages). Cells show mean with training-seed SD, or question-bootstrap SE (

\ddagger

); bold/underline mark best/second-best means. Probe Params excludes the frozen reasoner (M: millions). R/T denote refit/threshold control. OSP/OSP-1 extensions appear in Appendix § B.8 .

Method	Probe Params	Saving: 70%		Saving: 60%		Saving: 50%		Saving: 40%
Method	Probe Params	ID	OOD	ID	OOD	ID	OOD	ID	OOD
Natural †‡	–	$53.38_{\scriptscriptstyle\pm 1.1}$	$66.66_{\scriptscriptstyle\pm 1.4}$	$53.38_{\scriptscriptstyle\pm 1.1}$	$66.66_{\scriptscriptstyle\pm 1.4}$	$53.38_{\scriptscriptstyle\pm 1.1}$	$66.66_{\scriptscriptstyle\pm 1.4}$	$53.38_{\scriptscriptstyle\pm 1.1}$	$66.66_{\scriptscriptstyle\pm 1.4}$
Fixed cap ‡	0	$31.66_{\scriptscriptstyle\pm 1.0}$	$44.05_{\scriptscriptstyle\pm 1.3}$	$37.08_{\scriptscriptstyle\pm 1.0}$	$50.37_{\scriptscriptstyle\pm 1.3}$	$42.32_{\scriptscriptstyle\pm 1.0}$	$55.38_{\scriptscriptstyle\pm 1.4}$	$46.29_{\scriptscriptstyle\pm 1.0}$	$58.89_{\scriptscriptstyle\pm 1.4}$
CoDE-Stop ‡	0	$33.29_{\scriptscriptstyle\pm 1.0}$	$46.26_{\scriptscriptstyle\pm 1.3}$	$39.46_{\scriptscriptstyle\pm 1.0}$	$53.30_{\scriptscriptstyle\pm 1.4}$	$44.71_{\scriptscriptstyle\pm 1.0}$	$57.12_{\scriptscriptstyle\pm 1.4}$	$47.32_{\scriptscriptstyle\pm 1.0}$	$60.77_{\scriptscriptstyle\pm 1.4}$
DEER ‡	0	$22.33_{\scriptscriptstyle\pm 0.8}$	$24.98_{\scriptscriptstyle\pm 1.1}$	$28.76_{\scriptscriptstyle\pm 0.8}$	$32.75_{\scriptscriptstyle\pm 1.1}$	$35.73_{\scriptscriptstyle\pm 0.9}$	$42.18_{\scriptscriptstyle\pm 1.2}$	$43.29_{\scriptscriptstyle\pm 1.0}$	$52.97_{\scriptscriptstyle\pm 1.3}$
TC	514	$31.08_{\scriptscriptstyle\pm 0.5}$	$42.86_{\scriptscriptstyle\pm 0.4}$	$39.43_{\scriptscriptstyle\pm 0.4}$	$51.53_{\scriptscriptstyle\pm 0.7}$	$44.92_{\scriptscriptstyle\pm 0.1}$	$57.42_{\scriptscriptstyle\pm 0.5}$	$48.96_{\scriptscriptstyle\pm 0.3}$	$61.83_{\scriptscriptstyle\pm 0.1}$

Table 2: Accuracy (%) at Qwen3 token-saving budgets (macro averages); conventions follow Table 1 . Natural † is the natural-completion reference; NF denotes an infeasible budget in at least one seed. OSP/OSP-1 threshold extensions and per-dataset results appear in Appendix §§ B.8 and B .

Figure 2: Micro accuracy–token cost frontiers for Qwen3 OOD and LiveCodeBench Pre. Stars mark natural completion. Additional curves: datasets ( Fig. 8 ), Qwen3 ID and code Post ( Figs. 7 and 4 ).

Method	Probe Params	Token saving (%) at tolerance			Accuracy (%) at saving
Method	Probe Params	0.00	0.01	0.02	70%	60%	50%	40%
Natural †‡	–	–	–	–	$78.63$	$78.63$	$78.63$	$78.63$
Fixed cap ‡	0	–	–	–	$53.88_{\scriptscriptstyle\pm 1.5}$	$60.26_{\scriptscriptstyle\pm 1.5}$	$65.82_{\scriptscriptstyle\pm 1.5}$	$70.67_{\scriptscriptstyle\pm 1.4}$
CoDE-Stop ‡	0	$0.00_{\scriptscriptstyle\pm 0.0}$	$8.64_{\scriptscriptstyle\pm 0.3}$	$17.27_{\scriptscriptstyle\pm 0.6}$	$49.19_{\scriptscriptstyle\pm 1.4}$	$55.71_{\scriptscriptstyle\pm 1.4}$	$61.34_{\scriptscriptstyle\pm 1.5}$	$66.23_{\scriptscriptstyle\pm 1.4}$
DEER ‡	0	$0.00_{\scriptscriptstyle\pm 0.0}$	$1.57_{\scriptscriptstyle\pm 0.3}$	$10.80_{\scriptscriptstyle\pm 0.7}$	$35.65_{\scriptscriptstyle\pm 1.2}$	$44.25_{\scriptscriptstyle\pm 1.1}$	$52.85_{\scriptscriptstyle\pm 1.2}$	$60.52_{\scriptscriptstyle\pm 1.3}$
TC	514	$24.27_{\scriptscriptstyle\pm 0.1}$	$29.16_{\scriptscriptstyle\pm 0.0}$	$33.21_{\scriptscriptstyle\pm 0.4}$	$54.02_{\scriptscriptstyle\pm 0.2}$	$63.04_{\scriptscriptstyle\pm 0.2}$	$68.50_{\scriptscriptstyle\pm 0.1}$	$74.33_{\scriptscriptstyle\pm 0.1}$

Table 3: Top: OLMo3 OOD macro averages. Bottom: math-to-code transfer to LiveCodeBench Pre. Conventions follow Tables 1 and 2 ; trained methods use three runs. OSP/OSP-1 and BELT threshold extensions appear in Appendix § B.8 .

Table 6

Method	Probe Params	ALFWorld (success)					ScienceWorld (score)
		Saving (%) at tolerance			Success at saving		Saving (%)	Score at saving
		0 pp	2 pp	5 pp	20%	40%	at 0 pt	80%	70%	60%	50%
Natural	–	–	–	–	$33.6_{\scriptscriptstyle\pm 4.2}$	$33.6_{\scriptscriptstyle\pm 4.2}$	–	$21.4_{\scriptscriptstyle\pm 3.7}$	$21.4_{\scriptscriptstyle\pm 3.7}$	$21.4_{\scriptscriptstyle\pm 3.7}$	$21.4_{\scriptscriptstyle\pm 3.7}$
OSP-1	405M	$0.0_{\scriptscriptstyle\pm 18.4}$	$11.5_{\scriptscriptstyle\pm 17.3}$	$\underline{39.5}_{\scriptscriptstyle\pm 15.8}$	$30.1_{\scriptscriptstyle\pm 3.6}$	$28.5_{\scriptscriptstyle\pm 3.6}$	$\underline{84.9}_{\scriptscriptstyle\pm 5.8}$	$25.2_{\scriptscriptstyle\pm 3.2}$	$25.7_{\scriptscriptstyle\pm 3.1}$	$\underline{26.9}_{\scriptscriptstyle\pm 3.2}$	$28.0_{\scriptscriptstyle\pm 3.4}$
BELT-B-T	19M	$\underline{12.4}_{\scriptscriptstyle\pm 13.9}$	$\underline{25.4}_{\scriptscriptstyle\pm 13.0}$	$38.0_{\scriptscriptstyle\pm 11.8}$	$\underline{32.5}_{\scriptscriptstyle\pm 3.1}$	$\underline{29.2}_{\scriptscriptstyle\pm 3.0}$	$\mathbf{85.0}_{\scriptscriptstyle\pm 8.1}$	$\underline{25.4}_{\scriptscriptstyle\pm 3.1}$	$25.9_{\scriptscriptstyle\pm 3.0}$	$26.5_{\scriptscriptstyle\pm 2.9}$	$27.0_{\scriptscriptstyle\pm 2.9}$
BELT-H-T	19M	$0.0_{\scriptscriptstyle\pm 12.1}$	$13.2_{\scriptscriptstyle\pm 13.3}$	$32.5_{\scriptscriptstyle\pm 13.8}$	$29.2_{\scriptscriptstyle\pm 3.1}$	$27.8_{\scriptscriptstyle\pm 3.0}$	$\mathbf{85.0}_{\scriptscriptstyle\pm 7.4}$	$\mathbf{25.6}_{\scriptscriptstyle\pm 3.1}$	$\underline{26.3}_{\scriptscriptstyle\pm 3.0}$	$26.6_{\scriptscriptstyle\pm 3.0}$	$26.8_{\scriptscriptstyle\pm 3.0}$

Table 6: OLMo3 agent per-turn thinking control with controllers transferred from mathematics. ALFWorld: token saving (%) at success tolerance and success (%) at saving budgets. ScienceWorld: saving at natural score and score (0–100) at saving budgets. Conventions follow Tables 1 and 2 ; subscripts are game-bootstrap SDs. Full grids appear in Appendix §§ B.5 and B.6 .

Belief set	(a) Prediction targets					(b) Policy inputs
Belief set	2	5	10	20	50	2	5	10	20	50
Now , End	0.7451	0.7307	0.7067	0.6586	0.5820	0.7447	0.7302	0.7061	0.6595	0.5831
Now , End , Later	0.7448	0.7304	0.7065	0.6585	0.5853	0.7449	0.7304	0.7063	0.6592	0.5818
Now , End , Same	0.7459	0.7313	0.7068	0.6648	0.5882	0.7456	0.7311	0.7070	0.6652	0.5838
All four	0.7454	0.7310	0.7072	0.6660	0.5851	0.7454	0.7310	0.7072	0.6660	0.5851

Table 7: Average math OOD utility for (a) prediction-target and (b) policy-input ablations (fixed four-belief probe). Columns give token prices

\lambda

10^{-6}

. Means over three BELT probe training runs. SDs and accuracy–cost curves/readings appear in Tables 52 , 10 and 53 .

Figure 3: How the BELT-B-R policy changes with the token price $\lambda$ . (a) Normalized input weights, shown as mean and sample SD over three independent BELT probe training runs; positive weights raise the stop probability. (b,c) Mean stop probability over the standardized Now – End plane at a low and a high price, from one representative training run. Gray contours show boundary counts; gray cells are empty.

Appendix figures & tables78 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Distribution	Questions	Composition
DAPO	ID	740	Held-out DAPO source questions
Numina	ID	473	Held-out NuminaMath-hard source questions
DSR	ID	481	Held-out DeepScaleR source questions
M500	OOD	295	Retained MATH-500 questions
Olymp.	OOD	366	English text-only mathematics, single answer
GK	OOD	191	Gaokao 2023 mathematics, English

Appendix

Table 8: ID/OOD test questions after filtering and deduplication. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En.

Backbone	Bank	Dataset	Rows	Unpars. %	Recovered	Still unpars. %	Nat. acc. %
Qwen3-1.7B	math_union_fit	ALL	42,408	18.0	2,918 / 2,097	9.1	48.62 $\to$ 53.57
		dapo	18,048	10.7	1,169 / 797	1.8	52.00 $\to$ 56.42
		deepscaler	12,288	11.8	837 / 661	3.5	55.87 $\to$ 61.25
		numina_hard	12,072	35.0	912 / 639	25.3	36.19 $\to$ 41.48
Qwen3-1.7B	math_union_select	ALL	10,648	15.9	581 / 418	8.7	50.93 $\to$ 54.86
		dapo	4,656	8.7	249 / 158	1.3	54.10 $\to$ 57.50

Appendix

Table 9: Natural completion per bank and dataset before and after the answer-elicitation fix. Closed = the chain ended its thinking. Unparseable = no parseable answer (math: no integer in the thinking text; code: no program). Recovered = the seeded \boxed{ request returned a parseable / correct answer. Accuracy over all rows. Code banks are graded by unit tests and have no fallback.

Population	Questions	Acc. micro (%)	Acc. macro (%)	Tokens (micro)
Qwen3-1.7B ID	1,694	49.52 $\to$ 53.92	48.85 $\to$ 53.38	6,564.0 $\to$ 6,564.5
Qwen3-1.7B OOD	1,025	65.73 $\to$ 72.73	60.58 $\to$ 66.66	5,021.8 $\to$ 5,022.1
OLMo-3-7B-Think OOD	1,025	81.83 $\to$ 82.10	78.35 $\to$ 78.63	6,868.0 $\to$ 6,868.0

Appendix

Table 10: Natural completion on the paper populations, question basis, before

\to

after the fix.

Id	Statement	Submitted	Random init	ReProbe init	State (random init)
H1	abstract: Qwen3 OOD token reduction "66 %" (BELT-H, 0 pp, macro)	66.18	39.11	40.65	changed
H1b	abstract: OLMo3 OOD token reduction "53 %" (BELT-H, 0 pp, macro)	53.34	51.59	51.59	changed
H2	abstract / conclusion: belief-trained $>$ raw and utility-trained features (dev-selected OOD micro utility)	BELT-H - layer16 (pp, 5 prices): 4.89, 4.66, 4.78, 5.76, 6.94; BELT-H - OSP-1 features (pp, 5 prices): 1.64, 1.62, 1.57, 1.92, 1.4	BELT-H - layer16 (pp, 5 prices): 1.14, 1.22, 1.4, 2.17, 3.61; BELT-H - OSP-1 features (pp, 5 prices): 0.68, 0.67, 0.69, 0.94, 0.32	BELT-H - layer16 (pp, 5 prices): 0.98, 1.06, 1.21, 2.11, 4.43; BELT-H - OSP-1 features (pp, 5 prices): 0.52, 0.51, 0.5, 0.88, 1.14	holds
H3	abstract: trade-offs "competitive with or better than" end-to-end controllers (34 math readings)	vs OSP-full W/T/L: 11, 20, 2; vs OSP-1 W/T/L: 19, 9, 6	vs OSP-full W/T/L: 11, 12, 11; vs OSP-1 W/T/L: 31, 0, 3	vs OSP-full W/T/L: 17, 13, 4; vs OSP-1 W/T/L: 27, 6, 1	rewritten
H4	abstract: 6.3x fewer controller parameters	6.30	6.31	6.31	holds
H4b	abstract: ~25x less training compute (five prices)	25	22	22	rewritten

Appendix

Table 11: Headline numbers of the submitted version and their corrected values. Random init = the main-line probes of this version; ReProbe init = the probes initialized from ReProbe (analysis family). State: a value holds if the corrected number rounds to the printed one; a difference holds if it also keeps its sign; “magnitude” = sign kept, magnitude changed; interpretive statements are rewritten in the text.

Setting	Qwen3	OLMo3
Optimizer	AdamW
Encoder learning rate	$5\times 10^{-5}$
Belief-head learning rate	$10^{-3}$
AdamW $(\beta_{1},\beta_{2})$ ; $\epsilon$	$(0.9,0.999)$ ; $10^{-8}$
Weight decay (all parameter groups)	$0.01$
Gradient clipping (global norm)	$1.0$

Appendix

Table 12: BELT probe training hyperparameters.

Setting	Value
Optimizer / precision	L-BFGS / float64
Learning rate / line search	1.0 / strong Wolfe
History size	20
Maximum iterations	500
Gradient / change tolerance	$10^{-9}$ / $10^{-12}$
Development evaluation	Every 25 iterations

Appendix

Table 13: Qwen3 linear stopping policy fitting hyperparameters.

Method	Probe Params	DAPO	Numina	DSR	Micro	Macro
CoDE-Stop ‡	0	$17.45_{\scriptscriptstyle\pm 0.63}$	$13.12_{\scriptscriptstyle\pm 0.94}$	$19.27_{\scriptscriptstyle\pm 1.00}$	$16.29_{\scriptscriptstyle\pm 0.37}$	$16.82_{\scriptscriptstyle\pm 0.50}$
DEER ‡	0	$11.08_{\scriptscriptstyle\pm 0.76}$	$9.22_{\scriptscriptstyle\pm 1.08}$	$18.01_{\scriptscriptstyle\pm 1.23}$	$11.41_{\scriptscriptstyle\pm 0.45}$	$11.26_{\scriptscriptstyle\pm 0.57}$
TC	514	$18.86_{\scriptscriptstyle\pm 0.59}$	$23.40_{\scriptscriptstyle\pm 1.03}$	$24.19_{\scriptscriptstyle\pm 0.37}$	$21.03_{\scriptscriptstyle\pm 0.54}$	$20.79_{\scriptscriptstyle\pm 0.35}$
ROM	5M	$3.02_{\scriptscriptstyle\pm 4.27}$	$2.95_{\scriptscriptstyle\pm 4.17}$	$4.54_{\scriptscriptstyle\pm 6.41}$	$2.67_{\scriptscriptstyle\pm 3.78}$	$2.08_{\scriptscriptstyle\pm 2.93}$
OSP	101M $\times 5$	$\underline{43.11}_{\scriptscriptstyle\pm 1.50}$	$49.72_{\scriptscriptstyle\pm 1.33}$	$41.49_{\scriptscriptstyle\pm 0.59}$	$39.07_{\scriptscriptstyle\pm 7.13}$	$32.99_{\scriptscriptstyle\pm 13.86}$
OSP-1	101M	$31.80_{\scriptscriptstyle\pm 5.00}$	$44.94_{\scriptscriptstyle\pm 5.69}$	$40.82_{\scriptscriptstyle\pm 6.01}$	$28.64_{\scriptscriptstyle\pm 7.76}$	$20.36_{\scriptscriptstyle\pm 11.06}$

Appendix

Table 14: Token saving given the accuracy tolerance, reported by dataset. Reporting convention follows Table 1 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 0 pp tolerance, ID and pooled averages.

Method	Probe Params	M500	Olymp.	GK	AIME	AMC
CoDE-Stop ‡	0	$24.11_{\scriptscriptstyle\pm 1.90}$	$11.48_{\scriptscriptstyle\pm 0.93}$	$16.02_{\scriptscriptstyle\pm 1.49}$	$15.22_{\scriptscriptstyle\pm 1.49}$	$17.88_{\scriptscriptstyle\pm 2.16}$
DEER ‡	0	$23.82_{\scriptscriptstyle\pm 2.23}$	$7.73_{\scriptscriptstyle\pm 1.11}$	$9.72_{\scriptscriptstyle\pm 1.66}$	$2.79_{\scriptscriptstyle\pm 1.47}$	$7.72_{\scriptscriptstyle\pm 2.49}$
TC	514	$22.97_{\scriptscriptstyle\pm 0.54}$	$21.86_{\scriptscriptstyle\pm 0.63}$	$23.16_{\scriptscriptstyle\pm 0.16}$	$13.15_{\scriptscriptstyle\pm 0.52}$	$18.73_{\scriptscriptstyle\pm 0.00}$
ROM	5M	$2.61_{\scriptscriptstyle\pm 3.69}$	$1.03_{\scriptscriptstyle\pm 1.46}$	$1.53_{\scriptscriptstyle\pm 2.15}$	$0.27_{\scriptscriptstyle\pm 0.38}$	$0.66_{\scriptscriptstyle\pm 0.93}$
OSP	101M $\times 5$	$24.89_{\scriptscriptstyle\pm 21.69}$	$31.06_{\scriptscriptstyle\pm 26.92}$	$22.58_{\scriptscriptstyle\pm 19.55}$	$25.15_{\scriptscriptstyle\pm 21.98}$	$25.90_{\scriptscriptstyle\pm 22.51}$
OSP-1	101M	$10.89_{\scriptscriptstyle\pm 18.87}$	$10.89_{\scriptscriptstyle\pm 18.85}$	$9.57_{\scriptscriptstyle\pm 16.57}$	$5.93_{\scriptscriptstyle\pm 10.27}$	$8.08_{\scriptscriptstyle\pm 13.99}$

Appendix

Table 15: Token saving given the accuracy tolerance, reported by dataset. Reporting convention follows Table 1 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 0 pp tolerance, OOD.

Method	Probe Params	DAPO	Numina	DSR	Micro	Macro
CoDE-Stop ‡	0	$23.83_{\scriptscriptstyle\pm 0.65}$	$18.08_{\scriptscriptstyle\pm 0.99}$	$23.90_{\scriptscriptstyle\pm 1.00}$	$22.95_{\scriptscriptstyle\pm 0.39}$	$24.49_{\scriptscriptstyle\pm 0.52}$
DEER ‡	0	$18.05_{\scriptscriptstyle\pm 0.90}$	$18.77_{\scriptscriptstyle\pm 1.35}$	$26.07_{\scriptscriptstyle\pm 1.32}$	$20.05_{\scriptscriptstyle\pm 0.53}$	$20.40_{\scriptscriptstyle\pm 0.68}$
TC	514	$24.42_{\scriptscriptstyle\pm 0.26}$	$29.91_{\scriptscriptstyle\pm 0.21}$	$30.03_{\scriptscriptstyle\pm 0.82}$	$26.82_{\scriptscriptstyle\pm 0.41}$	$26.46_{\scriptscriptstyle\pm 0.33}$
ROM	5M	$18.14_{\scriptscriptstyle\pm 1.51}$	$19.49_{\scriptscriptstyle\pm 1.56}$	$25.38_{\scriptscriptstyle\pm 2.48}$	$19.55_{\scriptscriptstyle\pm 1.44}$	$19.60_{\scriptscriptstyle\pm 1.43}$
OSP	101M $\times 5$	$\mathbf{47.74}_{\scriptscriptstyle\pm 1.36}$	$55.44_{\scriptscriptstyle\pm 2.37}$	$45.99_{\scriptscriptstyle\pm 0.49}$	$\underline{48.86}_{\scriptscriptstyle\pm 1.45}$	$\underline{46.89}_{\scriptscriptstyle\pm 1.46}$
OSP-1	101M	$38.60_{\scriptscriptstyle\pm 2.47}$	$53.64_{\scriptscriptstyle\pm 1.62}$	$\mathbf{47.54}_{\scriptscriptstyle\pm 2.56}$	$43.11_{\scriptscriptstyle\pm 1.68}$	$40.53_{\scriptscriptstyle\pm 0.85}$

Appendix

Table 16: Token saving given the accuracy tolerance, reported by dataset. Reporting convention follows Table 1 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 1 pp tolerance, ID and pooled averages.

Method	Probe Params	M500	Olymp.	GK	AIME	AMC
CoDE-Stop ‡	0	$32.42_{\scriptscriptstyle\pm 1.86}$	$20.20_{\scriptscriptstyle\pm 1.08}$	$25.10_{\scriptscriptstyle\pm 1.69}$	$25.05_{\scriptscriptstyle\pm 1.67}$	$27.31_{\scriptscriptstyle\pm 2.20}$
DEER ‡	0	$35.18_{\scriptscriptstyle\pm 2.41}$	$19.77_{\scriptscriptstyle\pm 1.42}$	$23.07_{\scriptscriptstyle\pm 2.35}$	$6.44_{\scriptscriptstyle\pm 1.68}$	$15.88_{\scriptscriptstyle\pm 3.04}$
TC	514	$28.62_{\scriptscriptstyle\pm 0.74}$	$27.84_{\scriptscriptstyle\pm 0.86}$	$28.91_{\scriptscriptstyle\pm 0.42}$	$17.87_{\scriptscriptstyle\pm 1.01}$	$24.06_{\scriptscriptstyle\pm 0.17}$
ROM	5M	$29.77_{\scriptscriptstyle\pm 2.00}$	$16.95_{\scriptscriptstyle\pm 0.13}$	$22.57_{\scriptscriptstyle\pm 2.01}$	$8.15_{\scriptscriptstyle\pm 0.45}$	$16.38_{\scriptscriptstyle\pm 2.16}$
OSP	101M $\times 5$	$40.10_{\scriptscriptstyle\pm 2.31}$	$\mathbf{53.82}_{\scriptscriptstyle\pm 2.52}$	$38.43_{\scriptscriptstyle\pm 0.72}$	$\underline{47.94}_{\scriptscriptstyle\pm 6.31}$	$\underline{45.68}_{\scriptscriptstyle\pm 2.72}$
OSP-1	101M	$37.98_{\scriptscriptstyle\pm 3.82}$	$45.51_{\scriptscriptstyle\pm 3.07}$	$33.41_{\scriptscriptstyle\pm 2.66}$	$34.37_{\scriptscriptstyle\pm 5.21}$	$33.20_{\scriptscriptstyle\pm 1.01}$

Appendix

Table 17: Token saving given the accuracy tolerance, reported by dataset. Reporting convention follows Table 1 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 1 pp tolerance, OOD.

Method	Probe Params	DAPO	Numina	DSR	Micro	Macro
CoDE-Stop ‡	0	$27.23_{\scriptscriptstyle\pm 0.65}$	$21.35_{\scriptscriptstyle\pm 0.99}$	$25.96_{\scriptscriptstyle\pm 0.96}$	$26.34_{\scriptscriptstyle\pm 0.39}$	$28.16_{\scriptscriptstyle\pm 0.53}$
DEER ‡	0	$20.76_{\scriptscriptstyle\pm 0.95}$	$22.66_{\scriptscriptstyle\pm 1.39}$	$29.26_{\scriptscriptstyle\pm 1.34}$	$23.27_{\scriptscriptstyle\pm 0.55}$	$23.58_{\scriptscriptstyle\pm 0.71}$
TC	514	$28.44_{\scriptscriptstyle\pm 0.30}$	$34.19_{\scriptscriptstyle\pm 0.61}$	$34.10_{\scriptscriptstyle\pm 0.73}$	$31.21_{\scriptscriptstyle\pm 0.87}$	$31.12_{\scriptscriptstyle\pm 1.14}$
ROM	5M	$22.75_{\scriptscriptstyle\pm 0.22}$	$26.00_{\scriptscriptstyle\pm 0.15}$	$30.97_{\scriptscriptstyle\pm 1.05}$	$24.90_{\scriptscriptstyle\pm 0.41}$	$24.97_{\scriptscriptstyle\pm 0.69}$
OSP	101M $\times 5$	$\mathbf{51.72}_{\scriptscriptstyle\pm 0.79}$	$\underline{60.37}_{\scriptscriptstyle\pm 2.55}$	$50.34_{\scriptscriptstyle\pm 0.54}$	$\mathbf{53.55}_{\scriptscriptstyle\pm 1.26}$	$\mathbf{51.98}_{\scriptscriptstyle\pm 1.57}$
OSP-1	101M	$43.47_{\scriptscriptstyle\pm 2.08}$	$59.10_{\scriptscriptstyle\pm 1.00}$	$\mathbf{51.92}_{\scriptscriptstyle\pm 1.86}$	$48.86_{\scriptscriptstyle\pm 0.95}$	$47.04_{\scriptscriptstyle\pm 0.06}$

Appendix

Table 18: Token saving given the accuracy tolerance, reported by dataset. Reporting convention follows Table 1 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 2 pp tolerance, ID and pooled averages.

Method	Probe Params	M500	Olymp.	GK	AIME	AMC
CoDE-Stop ‡	0	$32.77_{\scriptscriptstyle\pm 1.84}$	$24.20_{\scriptscriptstyle\pm 1.08}$	$28.20_{\scriptscriptstyle\pm 1.85}$	$32.83_{\scriptscriptstyle\pm 1.70}$	$32.77_{\scriptscriptstyle\pm 2.19}$
DEER ‡	0	$38.20_{\scriptscriptstyle\pm 2.42}$	$24.02_{\scriptscriptstyle\pm 1.48}$	$27.62_{\scriptscriptstyle\pm 2.48}$	$7.82_{\scriptscriptstyle\pm 1.78}$	$18.31_{\scriptscriptstyle\pm 3.15}$
TC	514	$34.42_{\scriptscriptstyle\pm 1.86}$	$32.85_{\scriptscriptstyle\pm 2.07}$	$33.85_{\scriptscriptstyle\pm 1.18}$	$22.52_{\scriptscriptstyle\pm 1.30}$	$28.60_{\scriptscriptstyle\pm 1.10}$
ROM	5M	$35.94_{\scriptscriptstyle\pm 1.94}$	$22.89_{\scriptscriptstyle\pm 0.29}$	$28.40_{\scriptscriptstyle\pm 1.20}$	$11.03_{\scriptscriptstyle\pm 0.30}$	$21.76_{\scriptscriptstyle\pm 1.58}$
OSP	101M $\times 5$	$42.88_{\scriptscriptstyle\pm 2.86}$	$\mathbf{59.41}_{\scriptscriptstyle\pm 2.67}$	$43.57_{\scriptscriptstyle\pm 0.51}$	$\underline{54.92}_{\scriptscriptstyle\pm 6.90}$	$\mathbf{52.60}_{\scriptscriptstyle\pm 3.60}$
OSP-1	101M	$43.28_{\scriptscriptstyle\pm 3.30}$	$53.32_{\scriptscriptstyle\pm 1.57}$	$39.78_{\scriptscriptstyle\pm 2.68}$	$43.79_{\scriptscriptstyle\pm 5.48}$	$41.67_{\scriptscriptstyle\pm 1.98}$

Appendix

Table 19: Token saving given the accuracy tolerance, reported by dataset. Reporting convention follows Table 1 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 2 pp tolerance, OOD.

Method	Probe Params	DAPO	Numina	DSR	Micro	Macro
Natural †‡	–	$56.59_{\scriptscriptstyle\pm 1.51}$	$42.77_{\scriptscriptstyle\pm 2.02}$	$60.77_{\scriptscriptstyle\pm 1.90}$	$61.01_{\scriptscriptstyle\pm 0.77}$	$61.68_{\scriptscriptstyle\pm 0.95}$
Fixed cap ‡	0	$26.63_{\scriptscriptstyle\pm 1.32}$	$29.48_{\scriptscriptstyle\pm 1.78}$	$38.86_{\scriptscriptstyle\pm 1.93}$	$38.43_{\scriptscriptstyle\pm 0.73}$	$39.40_{\scriptscriptstyle\pm 0.89}$
CoDE-Stop ‡	0	$30.82_{\scriptscriptstyle\pm 1.32}$	$30.81_{\scriptscriptstyle\pm 1.75}$	$38.23_{\scriptscriptstyle\pm 1.85}$	$40.52_{\scriptscriptstyle\pm 0.73}$	$41.40_{\scriptscriptstyle\pm 0.92}$
DEER ‡	0	$19.47_{\scriptscriptstyle\pm 0.96}$	$22.06_{\scriptscriptstyle\pm 1.46}$	$25.45_{\scriptscriptstyle\pm 1.50}$	$24.41_{\scriptscriptstyle\pm 0.60}$	$23.99_{\scriptscriptstyle\pm 0.71}$
TC	514	$27.10_{\scriptscriptstyle\pm 0.40}$	$30.27_{\scriptscriptstyle\pm 0.74}$	$35.88_{\scriptscriptstyle\pm 0.22}$	$37.52_{\scriptscriptstyle\pm 0.56}$	$38.44_{\scriptscriptstyle\pm 0.43}$
ROM	5M	$26.67_{\scriptscriptstyle\pm 1.18}$	$27.34_{\scriptscriptstyle\pm 1.27}$	$33.48_{\scriptscriptstyle\pm 2.11}$	$35.12_{\scriptscriptstyle\pm 2.50}$	$35.80_{\scriptscriptstyle\pm 2.38}$

Appendix

Table 20: Acc. controlling the token budget, reported by dataset. Reporting convention follows Table 2 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 70% token saving, ID and pooled averages.

Method	Probe Params	M500	Olymp.	GK	AIME	AMC
Natural †‡	–	$92.33_{\scriptscriptstyle\pm 1.19}$	$65.88_{\scriptscriptstyle\pm 2.20}$	$80.96_{\scriptscriptstyle\pm 2.52}$	$29.46_{\scriptscriptstyle\pm 3.71}$	$64.67_{\scriptscriptstyle\pm 4.51}$
Fixed cap ‡	0	$74.67_{\scriptscriptstyle\pm 1.96}$	$40.66_{\scriptscriptstyle\pm 2.07}$	$62.42_{\scriptscriptstyle\pm 3.02}$	$6.17_{\scriptscriptstyle\pm 1.93}$	$36.32_{\scriptscriptstyle\pm 4.61}$
CoDE-Stop ‡	0	$74.88_{\scriptscriptstyle\pm 1.84}$	$46.20_{\scriptscriptstyle\pm 2.15}$	$59.70_{\scriptscriptstyle\pm 3.02}$	$8.98_{\scriptscriptstyle\pm 2.43}$	$41.57_{\scriptscriptstyle\pm 4.71}$
DEER ‡	0	$39.47_{\scriptscriptstyle\pm 2.14}$	$23.29_{\scriptscriptstyle\pm 1.61}$	$37.50_{\scriptscriptstyle\pm 2.80}$	$5.17_{\scriptscriptstyle\pm 1.22}$	$19.47_{\scriptscriptstyle\pm 3.35}$
TC	514	$65.96_{\scriptscriptstyle\pm 1.62}$	$44.26_{\scriptscriptstyle\pm 0.67}$	$57.74_{\scriptscriptstyle\pm 0.61}$	$9.70_{\scriptscriptstyle\pm 0.29}$	$36.63_{\scriptscriptstyle\pm 0.56}$
ROM	5M	$64.40_{\scriptscriptstyle\pm 8.13}$	$38.80_{\scriptscriptstyle\pm 3.18}$	$52.79_{\scriptscriptstyle\pm 4.77}$	$9.16_{\scriptscriptstyle\pm 2.16}$	$33.76_{\scriptscriptstyle\pm 0.55}$

Appendix

Table 21: Acc. controlling the token budget, reported by dataset. Reporting convention follows Table 2 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 70% token saving, OOD.

Method	Probe Params	DAPO	Numina	DSR	Micro	Macro
Natural †‡	–	$56.59_{\scriptscriptstyle\pm 1.51}$	$42.77_{\scriptscriptstyle\pm 2.02}$	$60.77_{\scriptscriptstyle\pm 1.90}$	$61.01_{\scriptscriptstyle\pm 0.77}$	$61.68_{\scriptscriptstyle\pm 0.95}$
Fixed cap ‡	0	$33.82_{\scriptscriptstyle\pm 1.34}$	$33.45_{\scriptscriptstyle\pm 1.82}$	$43.98_{\scriptscriptstyle\pm 1.89}$	$44.61_{\scriptscriptstyle\pm 0.73}$	$45.39_{\scriptscriptstyle\pm 0.92}$
CoDE-Stop ‡	0	$39.77_{\scriptscriptstyle\pm 1.38}$	$34.55_{\scriptscriptstyle\pm 1.80}$	$44.05_{\scriptscriptstyle\pm 1.81}$	$47.54_{\scriptscriptstyle\pm 0.73}$	$48.11_{\scriptscriptstyle\pm 0.94}$
DEER ‡	0	$28.37_{\scriptscriptstyle\pm 1.15}$	$26.06_{\scriptscriptstyle\pm 1.57}$	$31.85_{\scriptscriptstyle\pm 1.64}$	$31.98_{\scriptscriptstyle\pm 0.65}$	$31.25_{\scriptscriptstyle\pm 0.75}$
TC	514	$38.09_{\scriptscriptstyle\pm 0.41}$	$36.26_{\scriptscriptstyle\pm 0.38}$	$43.96_{\scriptscriptstyle\pm 0.39}$	$46.45_{\scriptscriptstyle\pm 0.46}$	$46.99_{\scriptscriptstyle\pm 0.56}$
ROM	5M	$34.53_{\scriptscriptstyle\pm 1.17}$	$32.15_{\scriptscriptstyle\pm 1.20}$	$39.51_{\scriptscriptstyle\pm 1.59}$	$42.01_{\scriptscriptstyle\pm 1.85}$	$42.57_{\scriptscriptstyle\pm 1.84}$

Appendix

Table 22: Acc. controlling the token budget, reported by dataset. Reporting convention follows Table 2 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 60% token saving, ID and pooled averages.

Method	Probe Params	M500	Olymp.	GK	AIME	AMC
Natural †‡	–	$92.33_{\scriptscriptstyle\pm 1.19}$	$65.88_{\scriptscriptstyle\pm 2.20}$	$80.96_{\scriptscriptstyle\pm 2.52}$	$29.46_{\scriptscriptstyle\pm 3.71}$	$64.67_{\scriptscriptstyle\pm 4.51}$
Fixed cap ‡	0	$81.55_{\scriptscriptstyle\pm 1.74}$	$49.46_{\scriptscriptstyle\pm 2.13}$	$67.78_{\scriptscriptstyle\pm 2.96}$	$9.03_{\scriptscriptstyle\pm 2.31}$	$44.04_{\scriptscriptstyle\pm 4.82}$
CoDE-Stop ‡	0	$83.48_{\scriptscriptstyle\pm 1.63}$	$54.29_{\scriptscriptstyle\pm 2.17}$	$68.09_{\scriptscriptstyle\pm 2.91}$	$11.39_{\scriptscriptstyle\pm 2.66}$	$49.26_{\scriptscriptstyle\pm 5.03}$
DEER ‡	0	$49.57_{\scriptscriptstyle\pm 2.04}$	$33.61_{\scriptscriptstyle\pm 1.78}$	$44.28_{\scriptscriptstyle\pm 2.66}$	$9.01_{\scriptscriptstyle\pm 1.69}$	$27.29_{\scriptscriptstyle\pm 3.61}$
TC	514	$76.47_{\scriptscriptstyle\pm 0.77}$	$53.90_{\scriptscriptstyle\pm 0.30}$	$65.67_{\scriptscriptstyle\pm 0.65}$	$15.81_{\scriptscriptstyle\pm 0.22}$	$45.79_{\scriptscriptstyle\pm 1.33}$
ROM	5M	$72.61_{\scriptscriptstyle\pm 5.90}$	$47.21_{\scriptscriptstyle\pm 1.37}$	$58.95_{\scriptscriptstyle\pm 3.05}$	$13.94_{\scriptscriptstyle\pm 0.64}$	$41.64_{\scriptscriptstyle\pm 1.07}$

Appendix

Table 23: Acc. controlling the token budget, reported by dataset. Reporting convention follows Table 2 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 60% token saving, OOD.

Method	Probe Params	DAPO	Numina	DSR	Micro	Macro
Natural †‡	–	$56.59_{\scriptscriptstyle\pm 1.51}$	$42.77_{\scriptscriptstyle\pm 2.02}$	$60.77_{\scriptscriptstyle\pm 1.90}$	$61.01_{\scriptscriptstyle\pm 0.77}$	$61.68_{\scriptscriptstyle\pm 0.95}$
Fixed cap ‡	0	$40.77_{\scriptscriptstyle\pm 1.46}$	$37.28_{\scriptscriptstyle\pm 1.96}$	$48.92_{\scriptscriptstyle\pm 1.94}$	$50.09_{\scriptscriptstyle\pm 0.77}$	$50.48_{\scriptscriptstyle\pm 0.96}$
CoDE-Stop ‡	0	$46.08_{\scriptscriptstyle\pm 1.45}$	$38.33_{\scriptscriptstyle\pm 1.89}$	$49.73_{\scriptscriptstyle\pm 1.82}$	$52.49_{\scriptscriptstyle\pm 0.75}$	$52.47_{\scriptscriptstyle\pm 0.96}$
DEER ‡	0	$37.45_{\scriptscriptstyle\pm 1.27}$	$31.61_{\scriptscriptstyle\pm 1.68}$	$38.14_{\scriptscriptstyle\pm 1.71}$	$40.45_{\scriptscriptstyle\pm 0.69}$	$39.76_{\scriptscriptstyle\pm 0.81}$
TC	514	$45.21_{\scriptscriptstyle\pm 0.47}$	$39.69_{\scriptscriptstyle\pm 0.13}$	$49.87_{\scriptscriptstyle\pm 0.19}$	$52.31_{\scriptscriptstyle\pm 0.26}$	$52.73_{\scriptscriptstyle\pm 0.37}$
ROM	5M	$41.45_{\scriptscriptstyle\pm 0.66}$	$36.17_{\scriptscriptstyle\pm 1.51}$	$45.16_{\scriptscriptstyle\pm 0.82}$	$48.02_{\scriptscriptstyle\pm 1.11}$	$48.48_{\scriptscriptstyle\pm 1.22}$

Appendix

Table 24: Acc. controlling the token budget, reported by dataset. Reporting convention follows Table 2 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 50% token saving, ID and pooled averages.

Method	Probe Params	M500	Olymp.	GK	AIME	AMC
Natural †‡	–	$92.33_{\scriptscriptstyle\pm 1.19}$	$65.88_{\scriptscriptstyle\pm 2.20}$	$80.96_{\scriptscriptstyle\pm 2.52}$	$29.46_{\scriptscriptstyle\pm 3.71}$	$64.67_{\scriptscriptstyle\pm 4.51}$
Fixed cap ‡	0	$86.26_{\scriptscriptstyle\pm 1.60}$	$56.90_{\scriptscriptstyle\pm 2.22}$	$71.45_{\scriptscriptstyle\pm 2.94}$	$11.89_{\scriptscriptstyle\pm 2.72}$	$50.40_{\scriptscriptstyle\pm 5.05}$
CoDE-Stop ‡	0	$87.52_{\scriptscriptstyle\pm 1.49}$	$59.05_{\scriptscriptstyle\pm 2.22}$	$72.54_{\scriptscriptstyle\pm 2.91}$	$13.40_{\scriptscriptstyle\pm 2.81}$	$53.09_{\scriptscriptstyle\pm 5.11}$
DEER ‡	0	$61.92_{\scriptscriptstyle\pm 1.94}$	$44.26_{\scriptscriptstyle\pm 1.98}$	$54.78_{\scriptscriptstyle\pm 2.62}$	$13.57_{\scriptscriptstyle\pm 2.16}$	$36.35_{\scriptscriptstyle\pm 3.97}$
TC	514	$83.51_{\scriptscriptstyle\pm 0.55}$	$59.86_{\scriptscriptstyle\pm 0.20}$	$70.71_{\scriptscriptstyle\pm 0.00}$	$21.52_{\scriptscriptstyle\pm 0.39}$	$51.48_{\scriptscriptstyle\pm 1.45}$
ROM	5M	$79.07_{\scriptscriptstyle\pm 3.33}$	$54.42_{\scriptscriptstyle\pm 0.06}$	$64.93_{\scriptscriptstyle\pm 1.15}$	$18.69_{\scriptscriptstyle\pm 0.88}$	$47.98_{\scriptscriptstyle\pm 1.34}$

Appendix

Table 25: Acc. controlling the token budget, reported by dataset. Reporting convention follows Table 2 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 50% token saving, OOD.

Method	Probe Params	DAPO	Numina	DSR	Micro	Macro
Natural †‡	–	$56.59_{\scriptscriptstyle\pm 1.51}$	$42.77_{\scriptscriptstyle\pm 2.02}$	$60.77_{\scriptscriptstyle\pm 1.90}$	$61.01_{\scriptscriptstyle\pm 0.77}$	$61.68_{\scriptscriptstyle\pm 0.95}$
Fixed cap ‡	0	$46.04_{\scriptscriptstyle\pm 1.49}$	$40.10_{\scriptscriptstyle\pm 1.95}$	$52.72_{\scriptscriptstyle\pm 1.90}$	$54.00_{\scriptscriptstyle\pm 0.76}$	$54.16_{\scriptscriptstyle\pm 0.96}$
CoDE-Stop ‡	0	$47.82_{\scriptscriptstyle\pm 1.48}$	$40.56_{\scriptscriptstyle\pm 1.96}$	$53.58_{\scriptscriptstyle\pm 1.88}$	$55.33_{\scriptscriptstyle\pm 0.75}$	$55.73_{\scriptscriptstyle\pm 0.95}$
DEER ‡	0	$46.44_{\scriptscriptstyle\pm 1.40}$	$37.37_{\scriptscriptstyle\pm 1.83}$	$46.05_{\scriptscriptstyle\pm 1.78}$	$49.59_{\scriptscriptstyle\pm 0.74}$	$49.34_{\scriptscriptstyle\pm 0.88}$
TC	514	$50.64_{\scriptscriptstyle\pm 0.03}$	$42.16_{\scriptscriptstyle\pm 0.50}$	$54.09_{\scriptscriptstyle\pm 0.37}$	$56.49_{\scriptscriptstyle\pm 0.20}$	$57.01_{\scriptscriptstyle\pm 0.18}$
ROM	5M	$47.50_{\scriptscriptstyle\pm 0.34}$	$39.96_{\scriptscriptstyle\pm 1.20}$	$50.56_{\scriptscriptstyle\pm 0.42}$	$53.24_{\scriptscriptstyle\pm 0.63}$	$53.67_{\scriptscriptstyle\pm 0.71}$

Appendix

Table 26: Acc. controlling the token budget, reported by dataset. Reporting convention follows Table 2 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 40% token saving, ID and pooled averages.

Method	Probe Params	M500	Olymp.	GK	AIME	AMC
Natural †‡	–	$92.33_{\scriptscriptstyle\pm 1.19}$	$65.88_{\scriptscriptstyle\pm 2.20}$	$80.96_{\scriptscriptstyle\pm 2.52}$	$29.46_{\scriptscriptstyle\pm 3.71}$	$64.67_{\scriptscriptstyle\pm 4.51}$
Fixed cap ‡	0	$89.00_{\scriptscriptstyle\pm 1.44}$	$61.33_{\scriptscriptstyle\pm 2.20}$	$73.75_{\scriptscriptstyle\pm 2.87}$	$15.48_{\scriptscriptstyle\pm 3.01}$	$54.88_{\scriptscriptstyle\pm 5.03}$
CoDE-Stop ‡	0	$89.94_{\scriptscriptstyle\pm 1.32}$	$63.10_{\scriptscriptstyle\pm 2.20}$	$75.40_{\scriptscriptstyle\pm 2.78}$	$18.41_{\scriptscriptstyle\pm 3.12}$	$57.02_{\scriptscriptstyle\pm 4.91}$
DEER ‡	0	$76.49_{\scriptscriptstyle\pm 1.76}$	$54.15_{\scriptscriptstyle\pm 2.11}$	$66.92_{\scriptscriptstyle\pm 2.65}$	$19.31_{\scriptscriptstyle\pm 2.73}$	$47.97_{\scriptscriptstyle\pm 4.27}$
TC	514	$87.69_{\scriptscriptstyle\pm 0.22}$	$63.28_{\scriptscriptstyle\pm 0.16}$	$74.82_{\scriptscriptstyle\pm 0.16}$	$25.87_{\scriptscriptstyle\pm 0.19}$	$57.51_{\scriptscriptstyle\pm 0.23}$
ROM	5M	$84.11_{\scriptscriptstyle\pm 1.04}$	$59.55_{\scriptscriptstyle\pm 0.49}$	$70.58_{\scriptscriptstyle\pm 0.09}$	$23.35_{\scriptscriptstyle\pm 1.91}$	$53.75_{\scriptscriptstyle\pm 0.40}$

Appendix

Table 27: Acc. controlling the token budget, reported by dataset. Reporting convention follows Table 2 . Micro and Macro aggregate ID and OOD. DAPO, Numina, DSR, M500, Olymp., and GK denote DAPO-Math, NuminaMath, DeepScaleR, MATH-500, OlympiadBench, and Gaokao2023-Math-En, respectively. Panel: 40% token saving, OOD.

Method	Probe Params	0 pp	1 pp	2 pp
CoDE-Stop ‡	0	$0.00_{\scriptscriptstyle\pm 0.00}$	$8.43_{\scriptscriptstyle\pm 0.27}$	$16.86_{\scriptscriptstyle\pm 0.53}$
DEER ‡	0	$0.00_{\scriptscriptstyle\pm 0.00}$	$1.71_{\scriptscriptstyle\pm 0.29}$	$10.98_{\scriptscriptstyle\pm 0.64}$
TC	514	$24.23_{\scriptscriptstyle\pm 0.07}$	$29.32_{\scriptscriptstyle\pm 0.19}$	$32.88_{\scriptscriptstyle\pm 0.18}$
ROM	8M	$27.58_{\scriptscriptstyle\pm 3.79}$	$34.15_{\scriptscriptstyle\pm 1.16}$	$36.34_{\scriptscriptstyle\pm 0.87}$
OSP	405M $\times 5$	$47.32_{\scriptscriptstyle\pm 0.53}$	$51.31_{\scriptscriptstyle\pm 0.29}$	$55.15_{\scriptscriptstyle\pm 0.47}$
OSP+T	405M $\times 5$	$49.58_{\scriptscriptstyle\pm 3.37}$	$51.69_{\scriptscriptstyle\pm 1.72}$	$56.09_{\scriptscriptstyle\pm 0.45}$

Appendix

Table 28: Token saving and accuracy on OLMo3, reported as Micro and Macro averages. Reporting convention follows Table 3 . Panel: token saving, micro averages. OSP-1+T retains the native stochastic policy alongside its existing 13 source-validation thresholds. OSP+T adds 13 source-validation thresholds to each of the five OSP checkpoints and keeps their native policies.

Method	Probe Params	0 pp	1 pp	2 pp
CoDE-Stop ‡	0	$0.00_{\scriptscriptstyle\pm 0.00}$	$8.64_{\scriptscriptstyle\pm 0.31}$	$17.27_{\scriptscriptstyle\pm 0.63}$
DEER ‡	0	$0.00_{\scriptscriptstyle\pm 0.00}$	$1.57_{\scriptscriptstyle\pm 0.25}$	$10.80_{\scriptscriptstyle\pm 0.68}$
TC	514	$24.27_{\scriptscriptstyle\pm 0.07}$	$29.16_{\scriptscriptstyle\pm 0.04}$	$33.21_{\scriptscriptstyle\pm 0.41}$
ROM	8M	$29.17_{\scriptscriptstyle\pm 3.83}$	$35.63_{\scriptscriptstyle\pm 1.17}$	$37.72_{\scriptscriptstyle\pm 0.94}$
OSP	405M $\times 5$	$45.92_{\scriptscriptstyle\pm 0.64}$	$49.86_{\scriptscriptstyle\pm 0.27}$	$53.73_{\scriptscriptstyle\pm 0.50}$
OSP+T	405M $\times 5$	$48.09_{\scriptscriptstyle\pm 3.49}$	$50.36_{\scriptscriptstyle\pm 1.68}$	$54.37_{\scriptscriptstyle\pm 0.53}$

Appendix

Table 29: Token saving and accuracy on OLMo3, reported as Micro and Macro averages. Reporting convention follows Table 3 . Panel: token saving, macro averages.

Method	Probe Params	70%	60%	50%	40%
Natural †‡	–	$82.10$	$82.10$	$82.10$	$82.10$
Fixed cap ‡	0	$61.60_{\scriptscriptstyle\pm 1.22}$	$68.00_{\scriptscriptstyle\pm 1.20}$	$72.69_{\scriptscriptstyle\pm 1.14}$	$76.63_{\scriptscriptstyle\pm 1.07}$
CoDE-Stop ‡	0	$56.94_{\scriptscriptstyle\pm 1.22}$	$63.68_{\scriptscriptstyle\pm 1.17}$	$69.34_{\scriptscriptstyle\pm 1.15}$	$73.50_{\scriptscriptstyle\pm 1.09}$
DEER ‡	0	$40.28_{\scriptscriptstyle\pm 1.05}$	$49.13_{\scriptscriptstyle\pm 1.00}$	$57.97_{\scriptscriptstyle\pm 1.05}$	$65.55_{\scriptscriptstyle\pm 1.04}$
TC	514	$60.67_{\scriptscriptstyle\pm 0.19}$	$69.24_{\scriptscriptstyle\pm 0.11}$	$74.12_{\scriptscriptstyle\pm 0.08}$	$78.36_{\scriptscriptstyle\pm 0.06}$
ROM	8M	$34.13_{\scriptscriptstyle\pm 1.03}$	$45.03_{\scriptscriptstyle\pm 0.80}$	$62.60_{\scriptscriptstyle\pm 1.10}$	$77.90_{\scriptscriptstyle\pm 0.69}$

Appendix

Table 30: Token saving and accuracy on OLMo3, reported as Micro and Macro averages. Reporting convention follows Table 3 . Panel: accuracy, micro averages.

Method	Probe Params	70%	60%	50%	40%
Natural †‡	–	$78.63$	$78.63$	$78.63$	$78.63$
Fixed cap ‡	0	$53.88_{\scriptscriptstyle\pm 1.47}$	$60.26_{\scriptscriptstyle\pm 1.52}$	$65.82_{\scriptscriptstyle\pm 1.47}$	$70.67_{\scriptscriptstyle\pm 1.43}$
CoDE-Stop ‡	0	$49.19_{\scriptscriptstyle\pm 1.40}$	$55.71_{\scriptscriptstyle\pm 1.44}$	$61.34_{\scriptscriptstyle\pm 1.47}$	$66.23_{\scriptscriptstyle\pm 1.41}$
DEER ‡	0	$35.65_{\scriptscriptstyle\pm 1.15}$	$44.25_{\scriptscriptstyle\pm 1.13}$	$52.85_{\scriptscriptstyle\pm 1.22}$	$60.52_{\scriptscriptstyle\pm 1.25}$
TC	514	$54.02_{\scriptscriptstyle\pm 0.21}$	$63.04_{\scriptscriptstyle\pm 0.19}$	$68.50_{\scriptscriptstyle\pm 0.08}$	$74.33_{\scriptscriptstyle\pm 0.11}$
ROM	8M	$32.17_{\scriptscriptstyle\pm 1.22}$	$43.08_{\scriptscriptstyle\pm 0.37}$	$59.79_{\scriptscriptstyle\pm 0.49}$	$74.78_{\scriptscriptstyle\pm 0.57}$

Appendix

Table 31: Token saving and accuracy on OLMo3, reported as Micro and Macro averages. Reporting convention follows Table 3 . Panel: accuracy, macro averages.

Method	Probe Params	0 pp	1 pp	2 pp
Natural †	–	–	–	–
OSP	101M $\times 5$	$\mathbf{54.58}_{\scriptscriptstyle\pm 2.30}$	$\mathbf{62.38}_{\scriptscriptstyle\pm 2.01}$	$\mathbf{70.23}_{\scriptscriptstyle\pm 0.20}$
OSP-1	101M	$14.07_{\scriptscriptstyle\pm 24.37}$	$53.92_{\scriptscriptstyle\pm 5.13}$	$66.75_{\scriptscriptstyle\pm 1.65}$
OSP-1+T	101M	$14.07_{\scriptscriptstyle\pm 24.37}$	$53.92_{\scriptscriptstyle\pm 5.13}$	$66.87_{\scriptscriptstyle\pm 1.53}$
BELT-B-R	16M	$\underline{53.51}_{\scriptscriptstyle\pm 2.49}$	$\underline{61.21}_{\scriptscriptstyle\pm 1.07}$	$\underline{68.77}_{\scriptscriptstyle\pm 2.91}$
BELT-H-R	16M	$50.48_{\scriptscriptstyle\pm 5.43}$	$58.76_{\scriptscriptstyle\pm 3.94}$	$65.80_{\scriptscriptstyle\pm 2.33}$

Appendix

Table 32: Token saving given the accuracy tolerance and accuracy given the token budget on LiveCodeBench Post (from August 2024). Reporting convention follows Table 3 . Panel: token saving. OSP-1+T retains the native stochastic policy alongside its existing 13 source-validation thresholds.

Method	Probe Params	70%	60%	50%	40%
Natural †	–	$28.67$	$28.67$	$28.67$	$28.67$
OSP	101M $\times 5$	$\mathbf{26.71}_{\scriptscriptstyle\pm 0.04}$	$\mathbf{27.96}_{\scriptscriptstyle\pm 0.29}$	$\mathbf{29.53}_{\scriptscriptstyle\pm 0.54}$	$\underline{30.05}_{\scriptscriptstyle\pm 0.51}$
OSP-1	101M	$26.20_{\scriptscriptstyle\pm 0.14}$	$27.27_{\scriptscriptstyle\pm 0.21}$	$27.94_{\scriptscriptstyle\pm 0.35}$	$28.33_{\scriptscriptstyle\pm 0.41}$
OSP-1+T	101M	$26.27_{\scriptscriptstyle\pm 0.07}$	$27.27_{\scriptscriptstyle\pm 0.21}$	$27.94_{\scriptscriptstyle\pm 0.35}$	$28.33_{\scriptscriptstyle\pm 0.41}$
BELT-B-R	16M	$\underline{26.44}_{\scriptscriptstyle\pm 0.41}$	$\underline{27.85}_{\scriptscriptstyle\pm 0.18}$	$28.99_{\scriptscriptstyle\pm 0.22}$	$29.84_{\scriptscriptstyle\pm 0.16}$
BELT-H-R	16M	$25.98_{\scriptscriptstyle\pm 0.42}$	$27.57_{\scriptscriptstyle\pm 0.46}$	$28.72_{\scriptscriptstyle\pm 0.55}$	$29.47_{\scriptscriptstyle\pm 0.34}$

Appendix

Table 33: Token saving given the accuracy tolerance and accuracy given the token budget on LiveCodeBench Post (from August 2024). Reporting convention follows Table 3 . Panel: accuracy.

Figure 4: LiveCodeBench Post native frontiers. Mean curves use common seed cost support; their endpoints are not per-method maximum accuracies. Other conventions follow the LiveCodeBench Pre panel of Fig. 2 .

Method	Probe Params	Token saving (%) at tolerance			Accuracy (%) at saving
Method	Probe Params	0.00	0.01	0.02	70%	60%	50%	40%
Natural †	–	–	–	–	$69.35$	$69.35$	$69.35$	$69.35$
OSP	405M $\times 5$	$9.10_{\scriptscriptstyle\pm 15.8}$	$27.58_{\scriptscriptstyle\pm 5.6}$	$34.01_{\scriptscriptstyle\pm 0.5}$	$52.83_{\scriptscriptstyle\pm 0.6}$	$57.79_{\scriptscriptstyle\pm 0.8}$	$62.04_{\scriptscriptstyle\pm 0.8}$	$65.50_{\scriptscriptstyle\pm 0.4}$
OSP-1	405M	$6.94_{\scriptscriptstyle\pm 12.0}$	$21.94_{\scriptscriptstyle\pm 7.3}$	$27.67_{\scriptscriptstyle\pm 6.0}$	$52.44_{\scriptscriptstyle\pm 0.5}$	$57.56_{\scriptscriptstyle\pm 0.5}$	$61.53_{\scriptscriptstyle\pm 0.6}$	$64.61_{\scriptscriptstyle\pm 0.8}$
BELT-B-T	19M	$21.84_{\scriptscriptstyle\pm 4.2}$	$27.33_{\scriptscriptstyle\pm 4.7}$	$32.69_{\scriptscriptstyle\pm 3.8}$	$53.01_{\scriptscriptstyle\pm 0.7}$	$58.32_{\scriptscriptstyle\pm 1.2}$	$62.62_{\scriptscriptstyle\pm 1.1}$	$65.69_{\scriptscriptstyle\pm 1.0}$
BELT-H-T	19M	$\underline{24.39}_{\scriptscriptstyle\pm 5.2}$	$29.91_{\scriptscriptstyle\pm 4.9}$	$\underline{34.10}_{\scriptscriptstyle\pm 4.9}$	$\underline{54.06}_{\scriptscriptstyle\pm 1.1}$	$\underline{59.20}_{\scriptscriptstyle\pm 1.3}$	$\underline{62.98}_{\scriptscriptstyle\pm 1.2}$	$65.96_{\scriptscriptstyle\pm 1.2}$

Appendix

Table 34: OLMo3 math-to-code transfer on LiveCodeBench problems before August 2024 (549 problems). Conventions follow Tables 3 and 3 ; policies are trained on mathematics only and read on the code test frontier. Natural † is the natural-completion reference.

Method	Probe Params	Token saving (%) at tolerance			Accuracy (%) at saving
Method	Probe Params	0.00	0.01	0.02	70%	60%	50%	40%
Natural †	–	–	–	–	$45.25$	$45.25$	$45.25$	$45.25$
OSP	405M $\times 5$	$29.59_{\scriptscriptstyle\pm 25.6}$	$\mathbf{48.17}_{\scriptscriptstyle\pm 1.7}$	$\underline{50.95}_{\scriptscriptstyle\pm 1.5}$	$\mathbf{36.32}_{\scriptscriptstyle\pm 0.4}$	$\underline{40.55}_{\scriptscriptstyle\pm 0.2}$	$\mathbf{43.63}_{\scriptscriptstyle\pm 0.6}$	$\underline{45.45}_{\scriptscriptstyle\pm 0.9}$
OSP-1	405M	$9.11_{\scriptscriptstyle\pm 15.8}$	$35.35_{\scriptscriptstyle\pm 6.8}$	$45.06_{\scriptscriptstyle\pm 5.8}$	$\underline{36.29}_{\scriptscriptstyle\pm 0.2}$	$40.09_{\scriptscriptstyle\pm 0.4}$	$42.40_{\scriptscriptstyle\pm 1.0}$	$43.76_{\scriptscriptstyle\pm 0.7}$
BELT-B-T	19M	$37.67_{\scriptscriptstyle\pm 0.8}$	$44.22_{\scriptscriptstyle\pm 2.2}$	$48.23_{\scriptscriptstyle\pm 1.9}$	$35.47_{\scriptscriptstyle\pm 1.0}$	$39.73_{\scriptscriptstyle\pm 0.8}$	$42.78_{\scriptscriptstyle\pm 0.5}$	$44.87_{\scriptscriptstyle\pm 0.2}$
BELT-H-T	19M	$35.95_{\scriptscriptstyle\pm 6.5}$	$43.78_{\scriptscriptstyle\pm 2.3}$	$48.61_{\scriptscriptstyle\pm 1.7}$	$35.84_{\scriptscriptstyle\pm 0.5}$	$40.01_{\scriptscriptstyle\pm 0.6}$	$42.93_{\scriptscriptstyle\pm 0.3}$	$44.98_{\scriptscriptstyle\pm 0.6}$

Appendix

Table 35: OLMo3 math-to-code transfer on LiveCodeBench problems from August 2024 (416 problems). Conventions follow Tables 3 and 3 ; policies are trained on mathematics only and read on the code test frontier. Natural † is the natural-completion reference.

Figure 5: Label-free stop diagnostic on the unseen natural trajectories. Left: share of a candidate’s projected stops whose forced action equals the natural action; right: projected stops that change the action, per game; both against projected token saving. Lines: threshold grids (one per probe run); points: refit policies; open circles: the calibrated members.

Figure 6: ALFWorld test frontiers on the unseen games: success versus generated tokens per episode. Multi-run methods show the mean of the runs’ frontiers over their common cost range; single-run methods show the frontier through their members. Star: natural completion; diamond: no thinking.

Method	Probe Params	Token saving (%) at tolerance			Success (%) at saving
Method	Probe Params	0 pp	2 pp	5 pp	50%	40%	30%	20%
Natural	–	–	–	–	$33.6_{\scriptscriptstyle\pm 4.2}$	$33.6_{\scriptscriptstyle\pm 4.2}$	$33.6_{\scriptscriptstyle\pm 4.2}$	$33.6_{\scriptscriptstyle\pm 4.2}$
OSP-1	405M	$0.0_{\scriptscriptstyle\pm 18.4}$	$11.5_{\scriptscriptstyle\pm 17.3}$	$\underline{39.5}_{\scriptscriptstyle\pm 15.8}$	$\underline{26.5}_{\scriptscriptstyle\pm 3.5}$	$28.5_{\scriptscriptstyle\pm 3.6}$	$29.4_{\scriptscriptstyle\pm 3.6}$	$30.1_{\scriptscriptstyle\pm 3.6}$
BELT-B-T	19M	$\underline{12.4}_{\scriptscriptstyle\pm 13.9}$	$\underline{25.4}_{\scriptscriptstyle\pm 13.0}$	$38.0_{\scriptscriptstyle\pm 11.8}$	$25.9_{\scriptscriptstyle\pm 2.9}$	$\underline{29.2}_{\scriptscriptstyle\pm 3.0}$	$\underline{31.1}_{\scriptscriptstyle\pm 3.0}$	$\underline{32.5}_{\scriptscriptstyle\pm 3.1}$
BELT-H-T	19M	$0.0_{\scriptscriptstyle\pm 12.1}$	$13.2_{\scriptscriptstyle\pm 13.3}$	$32.5_{\scriptscriptstyle\pm 13.8}$	$26.4_{\scriptscriptstyle\pm 2.9}$	$27.8_{\scriptscriptstyle\pm 3.0}$	$28.6_{\scriptscriptstyle\pm 3.0}$	$29.2_{\scriptscriptstyle\pm 3.1}$
BELT-B-R	19M	$0.0_{\scriptscriptstyle\pm 17.5}$	$13.4_{\scriptscriptstyle\pm 16.3}$	$34.2_{\scriptscriptstyle\pm 15.5}$	$26.4_{\scriptscriptstyle\pm 3.7}$	$27.8_{\scriptscriptstyle\pm 3.6}$	$29.2_{\scriptscriptstyle\pm 3.5}$	$30.6_{\scriptscriptstyle\pm 3.6}$

Appendix

Table 36: ALFWorld per-turn thinking control with OLMo3 (134 unseen games, at most 50 steps, one generation seed). Token saving (%) relative to natural completion at a success tolerance of 0/2/5 pp, and success (%) at token-saving budgets. Probes, fitted directions and the OSP-1 checkpoint are transferred from mathematics ( § A.12 ). BELT-T and OSP-1 thresholds are calibrated on the seen split without outcome labels; BELT-R uses the refit policies selected by the same rule. Subscripts: SD of the reading over 5,000 game resamples (methods with several probe training runs are averaged over runs inside each resample); the probe-run spread is in § E.7 .

Method	Run	Success / realised saving [projected saving] (%)
OSP-1	s0	29.9 / 21.5 [20.0]	29.1 / 36.8 [35.0]	26.1 / 51.9 [50.0]	22.4 / 65.6 [65.0]
BELT-B-T	s0	35.1 / 21.5 [20.0]	33.6 / 37.1 [35.0]	23.1 / 52.3 [50.0]	20.9 / 68.3 [65.0]
BELT-B-T	s1	32.1 / 21.3 [20.0]	29.9 / 35.7 [35.0]	26.1 / 50.1 [50.0]	19.4 / 63.3 [65.0]
BELT-B-T	s2	25.4 / 18.4 [20.0]	27.6 / 34.7 [35.0]	26.9 / 51.8 [50.0]	24.6 / 68.2 [65.0]
BELT-H-T	s0	24.6 / 13.9 [20.0]	24.6 / 29.4 [35.0]	29.9 / 45.8 [50.0]	22.4 / 63.5 [65.0]
BELT-H-T	s1	29.9 / 19.7 [20.0]	29.1 / 33.5 [35.0]	26.9 / 50.2 [50.0]	17.9 / 64.3 [65.0]

Appendix

Table 37: ALFWorld members on the 134 unseen games: success (%) / realised token saving (%), with the projected saving on the seen calibration split in brackets. Thresholded and refit rows: the four members calibrated for 20/35/50/65% projected saving. Natural completion: 33.6% success at 227k tokens per episode.

Comparison	Saving difference at tolerance			Success difference at saving
Comparison	0 pp	2 pp	5 pp	50%	40%	30%	20%
BELT-B-T vs. OSP-1	$+12.4$ [ $-28.5$ , $+37.5$ ]	$+13.9$ [ $-21.7$ , $+28.6$ ]	$-1.5$ [ $-19.0$ , $+25.5$ ]	$-0.6$ [ $-5.7$ , $+5.2$ ]	$+0.8$ [ $-4.5$ , $+6.2$ ]	$+1.7$ [ $-3.5$ , $+6.3$ ]	$+2.4$ [ $-3.1$ , $+6.4$ ]
BELT-H-T vs. OSP-1	$+0.0$ [ $-40.8$ , $+27.2$ ]	$+1.7$ [ $-31.1$ , $+20.1$ ]	$-7.0$ [ $-27.6$ , $+20.0$ ]	$-0.1$ [ $-6.1$ , $+5.1$ ]	$-0.7$ [ $-5.8$ , $+4.3$ ]	$-0.8$ [ $-5.7$ , $+3.6$ ]	$-0.9$ [ $-5.7$ , $+3.5$ ]
BELT-H-R vs. OSP-1	$+20.0$ [ $-39.8$ , $+52.7$ ]	$+18.7$ [ $-31.4$ , $+40.0$ ]	$+5.9$ [ $-28.6$ , $+32.6$ ]	$+1.2$ [ $-7.5$ , $+8.5$ ]	$+1.2$ [ $-7.4$ , $+7.8$ ]	$+2.2$ [ $-7.2$ , $+8.5$ ]	$+3.5$ [ $-5.4$ , $+9.8$ ]

Appendix

Table 38: ALFWorld paired bootstrap over the 134 unseen games (5,000 resamples; both methods re-read on every resample, multi-run methods averaged inside it): difference in token saving (pp) at each tolerance and in success (pp) at each budget, with the 95% percentile interval. The resampling covers games, not probe or training seeds.

Method	Probe Params	Token saving (%) at tolerance			Score at saving
Method	Probe Params	0 pt	2 pt	5 pt	80%	70%	60%	50%
Natural	–	–	–	–	$21.4_{\scriptscriptstyle\pm 3.7}$	$21.4_{\scriptscriptstyle\pm 3.7}$	$21.4_{\scriptscriptstyle\pm 3.7}$	$21.4_{\scriptscriptstyle\pm 3.7}$
OSP-1	405M	$\underline{84.9}_{\scriptscriptstyle\pm 5.8}$	$\underline{84.9}_{\scriptscriptstyle\pm 2.6}$	$\underline{84.9}_{\scriptscriptstyle\pm 1.6}$	$25.2_{\scriptscriptstyle\pm 3.2}$	$25.7_{\scriptscriptstyle\pm 3.1}$	$\underline{26.9}_{\scriptscriptstyle\pm 3.2}$	$28.0_{\scriptscriptstyle\pm 3.4}$
BELT-B-T	19M	$\mathbf{85.0}_{\scriptscriptstyle\pm 8.1}$	$\mathbf{85.0}_{\scriptscriptstyle\pm 3.5}$	$\mathbf{85.0}_{\scriptscriptstyle\pm 1.7}$	$\underline{25.4}_{\scriptscriptstyle\pm 3.1}$	$25.9_{\scriptscriptstyle\pm 3.0}$	$26.5_{\scriptscriptstyle\pm 2.9}$	$27.0_{\scriptscriptstyle\pm 2.9}$
BELT-H-T	19M	$\mathbf{85.0}_{\scriptscriptstyle\pm 7.4}$	$\mathbf{85.0}_{\scriptscriptstyle\pm 2.9}$	$\mathbf{85.0}_{\scriptscriptstyle\pm 1.7}$	$\mathbf{25.6}_{\scriptscriptstyle\pm 3.1}$	$\underline{26.3}_{\scriptscriptstyle\pm 3.0}$	$26.6_{\scriptscriptstyle\pm 3.0}$	$26.8_{\scriptscriptstyle\pm 3.0}$
BELT-B-R	19M	$\underline{84.9}_{\scriptscriptstyle\pm 6.6}$	$\underline{84.9}_{\scriptscriptstyle\pm 3.0}$	$\underline{84.9}_{\scriptscriptstyle\pm 1.6}$	$25.0_{\scriptscriptstyle\pm 3.1}$	$25.4_{\scriptscriptstyle\pm 3.0}$	$26.7_{\scriptscriptstyle\pm 3.0}$	$\underline{28.2}_{\scriptscriptstyle\pm 3.1}$

Appendix

Table 39: ScienceWorld per-turn thinking control with OLMo3 (90 test games, three per task type, at most 50 steps, one generation seed). Token saving (%) relative to natural completion at a score tolerance of 0/2/5 points, and mean score (0–100) at token-saving budgets. Conventions follow Table 36 : probes, fitted directions and the OSP-1 checkpoint are transferred from mathematics; BELT-T and OSP-1 thresholds and the BELT-R refit policies are calibrated on the disjoint development variations without outcome labels, at projected savings of 20–95% ( § A.13 ). Subscripts: SD of the reading over 5,000 game resamples (methods with several probe training runs are averaged over runs inside each resample); the probe-run spread is in § E.7 .

Method	Run	Score / realised saving [projected saving] (%)
OSP-1	s0	20.7 / 10.9 [20.0]	23.3 / 26.3 [35.0]	27.3 / 34.9 [50.0]	28.0 / 50.2 [65.0]	25.7 / 69.9 [80.0]	25.0 / 83.0 [90.0]	24.8 / 84.9 [91.5]
BELT-B-T	s0	26.5 / 13.9 [20.0]	28.0 / 28.9 [35.0]	26.2 / 38.1 [50.0]	26.0 / 49.1 [65.0]	26.5 / 65.4 [80.0]	23.9 / 83.6 [90.0]	24.8 / 84.9 [91.5]
BELT-B-T	s1	23.8 / 12.8 [20.0]	22.1 / 21.4 [35.0]	18.5 / 34.9 [50.0]	23.4 / 54.3 [65.0]	24.4 / 67.6 [80.0]	24.4 / 81.8 [90.0]	24.8 / 84.9 [91.5]
BELT-B-T	s2	24.6 / 10.3 [20.0]	25.2 / 23.9 [35.0]	28.3 / 38.2 [50.0]	29.2 / 48.1 [65.0]	26.9 / 69.6 [80.0]	24.8 / 83.0 [90.0]	25.7 / 85.0 [91.5]
BELT-H-T	s0	18.1 / 15.6 [20.0]	27.4 / 27.4 [35.0]	26.5 / 40.6 [50.0]	28.2 / 55.5 [65.0]	22.6 / 74.5 [80.0]	21.4 / 83.2 [90.0]	24.8 / 84.9 [91.5]
BELT-H-T	s1	22.5 / 12.9 [20.0]	25.6 / 23.8 [35.0]	24.2 / 39.9 [50.0]	26.9 / 53.8 [65.0]	27.4 / 70.0 [80.0]	26.5 / 83.1 [90.0]	24.8 / 85.0 [91.5]

Appendix

Table 40: ScienceWorld members on the 90 test games: score / realised token saving (%), with the projected saving on the development calibration split in brackets. Each row lists the seven members calibrated for 20/35/50/65/80/90/95% projected saving; the 95% target lies above the 91.5% ceiling of the label-free rule on the calibration split, so every family’s 95% member is the first-probe stop (the same actions in 87 of 90 games across families). Natural completion: score 21.4 at 144k tokens per episode; no thinking: 8.9.

Comparison	Saving difference at tolerance			Score difference at saving
Comparison	0 pt	2 pt	5 pt	80%	70%	60%	50%
BELT-H-T vs. OSP-1	$+0.0$ [ $-13.0$ , $+0.4$ ]	$+0.0$ [ $-0.6$ , $+0.2$ ]	$+0.0$ [ $-0.0$ , $+0.1$ ]	$+0.5$ [ $-2.0$ , $+1.4$ ]	$+0.5$ [ $-3.9$ , $+2.2$ ]	$-0.2$ [ $-4.5$ , $+2.7$ ]	$-1.2$ [ $-5.4$ , $+2.9$ ]
BELT-B-T vs. OSP-1	$+0.0$ [ $-20.0$ , $+0.5$ ]	$+0.0$ [ $-0.3$ , $+0.0$ ]	$+0.0$ [ $-0.0$ , $+0.0$ ]	$+0.2$ [ $-2.0$ , $+1.4$ ]	$+0.2$ [ $-3.8$ , $+2.1$ ]	$-0.4$ [ $-4.9$ , $+2.6$ ]	$-1.0$ [ $-6.4$ , $+3.2$ ]
BELT-H-R vs. OSP-1	$+0.0$ [ $-11.0$ , $+2.3$ ]	$+0.0$ [ $-0.3$ , $+0.1$ ]	$+0.0$ [ $+0.0$ , $+0.0$ ]	$+0.2$ [ $-2.8$ , $+0.9$ ]	$+0.8$ [ $-4.8$ , $+2.4$ ]	$+0.7$ [ $-5.2$ , $+3.9$ ]	$+0.5$ [ $-6.1$ , $+5.2$ ]
BELT-B-R vs. OSP-1	$+0.0$ [ $-10.3$ , $+1.1$ ]	$+0.0$ [ $-0.5$ , $+0.0$ ]	$+0.0$ [ $+0.0$ , $+0.0$ ]	$-0.2$ [ $-2.5$ , $+0.8$ ]	$-0.4$ [ $-4.5$ , $+2.4$ ]	$-0.2$ [ $-4.9$ , $+3.9$ ]	$+0.2$ [ $-5.8$ , $+5.3$ ]

Appendix

Table 41: ScienceWorld paired bootstrap over the 90 test games (5,000 resamples shared across pairs; both methods re-read on every resample, multi-run methods averaged inside it): difference in token saving (pp) at each tolerance and in score (points) at each budget, with the 95% percentile interval. The resampling covers games, not probe or training runs.

Math ID
Method	Utility by $\lambda$ ( $\times 10^{-6}$ )
	2	5	10	20	50
OSP	$0.5464_{\scriptscriptstyle\pm 0.0030}$	$0.5340_{\scriptscriptstyle\pm 0.0026}$	$\mathbf{0.5133}_{\scriptscriptstyle\pm 0.0019}$	$\mathbf{0.4719}_{\scriptscriptstyle\pm 0.0008}$	$\mathbf{0.3698}_{\scriptscriptstyle\pm 0.0010}$
OSP-1	$0.5363_{\scriptscriptstyle\pm 0.0089}$	$0.5238_{\scriptscriptstyle\pm 0.0046}$	$0.4989_{\scriptscriptstyle\pm 0.0045}$	$0.4552_{\scriptscriptstyle\pm 0.0089}$	$0.3584_{\scriptscriptstyle\pm 0.0031}$
BELT-B-R	$\mathbf{0.5619}_{\scriptscriptstyle\pm 0.0011}$	$\mathbf{0.5430}_{\scriptscriptstyle\pm 0.0009}$	$\underline{0.5116}_{\scriptscriptstyle\pm 0.0007}$	$0.4654_{\scriptscriptstyle\pm 0.0049}$	$\underline{0.3614}_{\scriptscriptstyle\pm 0.0052}$
BELT-H-R	$\underline{0.5513}_{\scriptscriptstyle\pm 0.0017}$	$\underline{0.5356}_{\scriptscriptstyle\pm 0.0014}$	$0.5109_{\scriptscriptstyle\pm 0.0011}$	$\underline{0.4670}_{\scriptscriptstyle\pm 0.0015}$	$0.3610_{\scriptscriptstyle\pm 0.0021}$

Appendix

Table 42: Utility of dev-selected policies. Reporting convention follows Table 1 ; all values are micro averages.

ID
Method	0 pp	1 pp	2 pp
OSP	$44.77_{\scriptscriptstyle\pm 0.75}$	$49.72_{\scriptscriptstyle\pm 1.09}$	$54.15_{\scriptscriptstyle\pm 0.90}$
OSP+T	$45.58_{\scriptscriptstyle\pm 0.99}$	$49.78_{\scriptscriptstyle\pm 3.28}$	$\mathbf{54.82}_{\scriptscriptstyle\pm 0.24}$
OSP-1+T	$39.19_{\scriptscriptstyle\pm 5.46}$	$46.59_{\scriptscriptstyle\pm 2.01}$	$51.50_{\scriptscriptstyle\pm 1.49}$
BELT-B-T	$25.21_{\scriptscriptstyle\pm 1.08}$	$29.57_{\scriptscriptstyle\pm 1.50}$	$33.54_{\scriptscriptstyle\pm 1.74}$
BELT-H-T	$\underline{46.52}_{\scriptscriptstyle\pm 1.51}$	$\mathbf{50.64}_{\scriptscriptstyle\pm 0.98}$	$54.29_{\scriptscriptstyle\pm 0.79}$

Appendix

Table 43: Original and threshold-expanded families on Qwen3 (Macro). Reporting convention follows Table 1 . NF denotes an infeasible budget in at least one seed. Panel: token saving. OSP-1+T retains the native stochastic policy alongside its existing 13 source-validation thresholds.

ID
Method	70%	60%	50%	40%
OSP	$44.83_{\scriptscriptstyle\pm 0.15}$	$49.24_{\scriptscriptstyle\pm 0.26}$	$\underline{52.44}_{\scriptscriptstyle\pm 0.26}$	$54.41_{\scriptscriptstyle\pm 0.20}$
OSP+T	$\underline{45.46}_{\scriptscriptstyle\pm 0.32}$	$\underline{49.64}_{\scriptscriptstyle\pm 0.28}$	$52.40_{\scriptscriptstyle\pm 0.77}$	$\mathbf{54.57}_{\scriptscriptstyle\pm 0.27}$
OSP-1+T	$44.74_{\scriptscriptstyle\pm 0.10}$	$48.73_{\scriptscriptstyle\pm 0.12}$	$51.33_{\scriptscriptstyle\pm 0.35}$	$53.20_{\scriptscriptstyle\pm 0.39}$
BELT-B-T	$37.46_{\scriptscriptstyle\pm 2.1}$	$43.11_{\scriptscriptstyle\pm 1.4}$	$47.14_{\scriptscriptstyle\pm 0.8}$	$50.09_{\scriptscriptstyle\pm 0.5}$
BELT-H-T	$43.81_{\scriptscriptstyle\pm 0.3}$	$49.34_{\scriptscriptstyle\pm 0.4}$	$\mathbf{52.49}_{\scriptscriptstyle\pm 0.2}$	$\underline{54.42}_{\scriptscriptstyle\pm 0.2}$

Appendix

Table 44: Original and threshold-expanded families on Qwen3 (Macro). Reporting convention follows Table 1 . NF denotes an infeasible budget in at least one seed. Panel: accuracy.

Pre
Method	0 pp	1 pp	2 pp
OSP	$34.76_{\scriptscriptstyle\pm 3.37}$	$38.25_{\scriptscriptstyle\pm 2.80}$	$41.74_{\scriptscriptstyle\pm 2.78}$
OSP+T	$36.78_{\scriptscriptstyle\pm 2.14}$	$41.35_{\scriptscriptstyle\pm 1.74}$	$43.73_{\scriptscriptstyle\pm 0.47}$
OSP-1+T	$0.03_{\scriptscriptstyle\pm 0.06}$	$15.25_{\scriptscriptstyle\pm 7.45}$	$23.71_{\scriptscriptstyle\pm 7.40}$
BELT-B-T	$\mathbf{42.96}_{\scriptscriptstyle\pm 0.36}$	$\mathbf{47.65}_{\scriptscriptstyle\pm 1.07}$	$\underline{51.40}_{\scriptscriptstyle\pm 1.14}$
BELT-H-T	$33.40_{\scriptscriptstyle\pm 6.35}$	$38.25_{\scriptscriptstyle\pm 6.81}$	$42.22_{\scriptscriptstyle\pm 6.02}$

Appendix

Table 45: Original and threshold-expanded families on LiveCodeBench. Reporting convention follows Table 1 . Panel: token saving. OSP-1+T retains the native stochastic policy alongside its existing 13 source-validation thresholds.

Pre
Method	70%	60%	50%	40%
OSP	NF	$55.10_{\scriptscriptstyle\pm 0.74}$	$58.39_{\scriptscriptstyle\pm 0.81}$	$60.99_{\scriptscriptstyle\pm 0.70}$
OSP+T	$54.04_{\scriptscriptstyle\pm 0.34}$	$56.88_{\scriptscriptstyle\pm 0.82}$	$58.96_{\scriptscriptstyle\pm 0.72}$	$61.95_{\scriptscriptstyle\pm 0.49}$
OSP-1+T	$50.84_{\scriptscriptstyle\pm 0.30}$	$54.40_{\scriptscriptstyle\pm 0.41}$	$56.76_{\scriptscriptstyle\pm 0.24}$	$58.74_{\scriptscriptstyle\pm 0.90}$
BELT-B-T	$\underline{54.76}_{\scriptscriptstyle\pm 0.32}$	$\underline{57.84}_{\scriptscriptstyle\pm 0.39}$	$\underline{60.89}_{\scriptscriptstyle\pm 0.29}$	$\mathbf{63.09}_{\scriptscriptstyle\pm 0.06}$
BELT-H-T	$52.22_{\scriptscriptstyle\pm 0.57}$	$55.81_{\scriptscriptstyle\pm 0.84}$	$58.67_{\scriptscriptstyle\pm 1.23}$	$61.08_{\scriptscriptstyle\pm 1.36}$

Appendix

Table 46: Original and threshold-expanded families on LiveCodeBench. Reporting convention follows Table 1 . Panel: accuracy.

Qwen3 ID (Macro)
Method	$2\times 10^{-6}$	$5\times 10^{-6}$	$10^{-5}$	$2\times 10^{-5}$	$5\times 10^{-5}$
OSP	$0.5421_{\scriptscriptstyle\pm 0.0022}$	$0.5300_{\scriptscriptstyle\pm 0.0018}$	$0.5098_{\scriptscriptstyle\pm 0.0012}$	$\underline{0.4695}_{\scriptscriptstyle\pm 0.0004}$	$\underline{0.3714}_{\scriptscriptstyle\pm 0.0009}$
OSP+T	$\underline{0.5493}_{\scriptscriptstyle\pm 0.0033}$	$\underline{0.5342}_{\scriptscriptstyle\pm 0.0030}$	$\mathbf{0.5110}_{\scriptscriptstyle\pm 0.0012}$	$\mathbf{0.4704}_{\scriptscriptstyle\pm 0.0014}$	$\mathbf{0.3746}_{\scriptscriptstyle\pm 0.0020}$
BELT-B-R	$\mathbf{0.5564}_{\scriptscriptstyle\pm 0.0007}$	$\mathbf{0.5379}_{\scriptscriptstyle\pm 0.0006}$	$0.5092_{\scriptscriptstyle\pm 0.0023}$	$0.4669_{\scriptscriptstyle\pm 0.0018}$	$0.3678_{\scriptscriptstyle\pm 0.0010}$
BELT-B-R+T	$\mathbf{0.5564}_{\scriptscriptstyle\pm 0.0007}$	$\mathbf{0.5379}_{\scriptscriptstyle\pm 0.0006}$	$\underline{0.5107}_{\scriptscriptstyle\pm 0.0015}$	$0.4670_{\scriptscriptstyle\pm 0.0018}$	$0.3691_{\scriptscriptstyle\pm 0.0015}$
BELT-H-R	$0.5460_{\scriptscriptstyle\pm 0.0018}$	$0.5308_{\scriptscriptstyle\pm 0.0014}$	$0.5070_{\scriptscriptstyle\pm 0.0013}$	$0.4646_{\scriptscriptstyle\pm 0.0018}$	$0.3657_{\scriptscriptstyle\pm 0.0012}$

Appendix

Table 47: Utility of the original and threshold-expanded families. Qwen3 uses Macro averages. Reporting convention follows Table 1 .

Method	70%	60%	50%	40%
OSP	$43.52_{\scriptscriptstyle\pm 0.48}$	$46.91_{\scriptscriptstyle\pm 1.52}$	$51.43_{\scriptscriptstyle\pm 0.78}$	$51.43_{\scriptscriptstyle\pm 0.78}$
OSP-1	$43.39_{\scriptscriptstyle\pm 0.86}$	$46.94_{\scriptscriptstyle\pm 1.19}$	$50.31_{\scriptscriptstyle\pm 1.19}$	$52.67_{\scriptscriptstyle\pm 0.64}$
BELT-B-T	$37.11_{\scriptscriptstyle\pm 1.98}$	$41.98_{\scriptscriptstyle\pm 1.48}$	$45.70_{\scriptscriptstyle\pm 0.85}$	$48.43_{\scriptscriptstyle\pm 0.65}$
BELT-H-T	$40.60_{\scriptscriptstyle\pm 1.73}$	$46.88_{\scriptscriptstyle\pm 1.08}$	$51.57_{\scriptscriptstyle\pm 0.48}$	$53.93_{\scriptscriptstyle\pm 0.34}$
BELT-B-R	$41.39_{\scriptscriptstyle\pm 0.82}$	$47.33_{\scriptscriptstyle\pm 1.94}$	$50.83_{\scriptscriptstyle\pm 0.88}$	$53.22_{\scriptscriptstyle\pm 0.69}$
BELT-H-R	$43.11_{\scriptscriptstyle\pm 0.93}$	$47.24_{\scriptscriptstyle\pm 1.18}$	$51.40_{\scriptscriptstyle\pm 0.21}$	$53.92_{\scriptscriptstyle\pm 0.29}$

Appendix

Table 48: Macro accuracy (%) under single-member token budgets. At each target, the most accurate feasible member is selected without interpolation. Entries are mean ±SD across seeds; NF denotes an infeasible target. Panel: ID.

Method	70%	60%	50%	40%
OSP	$58.45_{\scriptscriptstyle\pm 0.98}$	$60.32_{\scriptscriptstyle\pm 1.44}$	$64.35_{\scriptscriptstyle\pm 0.61}$	$65.09_{\scriptscriptstyle\pm 1.49}$
OSP-1	$57.78_{\scriptscriptstyle\pm 0.76}$	$61.64_{\scriptscriptstyle\pm 0.55}$	$63.89_{\scriptscriptstyle\pm 0.39}$	$65.51_{\scriptscriptstyle\pm 0.29}$
BELT-B-T	$47.14_{\scriptscriptstyle\pm 3.10}$	$53.39_{\scriptscriptstyle\pm 1.52}$	$59.77_{\scriptscriptstyle\pm 0.73}$	$61.92_{\scriptscriptstyle\pm 0.56}$
BELT-H-T	$52.72_{\scriptscriptstyle\pm 2.30}$	$59.65_{\scriptscriptstyle\pm 1.38}$	$64.26_{\scriptscriptstyle\pm 0.37}$	$66.14_{\scriptscriptstyle\pm 0.43}$
BELT-B-R	$56.56_{\scriptscriptstyle\pm 0.94}$	$61.19_{\scriptscriptstyle\pm 1.22}$	$64.20_{\scriptscriptstyle\pm 0.63}$	$66.31_{\scriptscriptstyle\pm 0.48}$
BELT-H-R	$56.61_{\scriptscriptstyle\pm 0.21}$	$61.38_{\scriptscriptstyle\pm 1.15}$	$64.08_{\scriptscriptstyle\pm 0.14}$	$66.60_{\scriptscriptstyle\pm 0.08}$

Appendix

Table 49: Macro accuracy (%) under single-member token budgets. At each target, the most accurate feasible member is selected without interpolation. Entries are mean ±SD across seeds; NF denotes an infeasible target. Panel: OOD.

Figure 7: Qwen3 ID frontiers; the inset enlarges the central accuracy–cost trade-off. The presentation follows Fig. 2 .

Figure 8: Dataset-level frontiers. The presentation follows Fig. 2 . Axes differ across panels. Each inset enlarges the same 38–52% token-saving cost range to distinguish nearby frontiers.

Figure 9: OLMo3 OOD frontiers. The presentation follows Fig. 2 . Single-direction and OSP-1+T comparisons also appear in Tables 28 – 31 .

Comparison	Policy fitting / development selection	Test grading	SD across
Qwen3 BELT utility controls	Numeric-equality labels	Strict math	3 independent probe training runs
Fixed feature controls	Numeric-equality labels	Strict math	5 policy-fitting starts
BELT-L	Numeric-equality labels	Strict math	3 probe training seeds
Matched OSP-1/BELT features	Shared fitter; strict math development utility	Strict math; code unit tests	3 independent training runs per method
OLMo3 BELT	Loose-label utility (refit)	Strict math	3 independent probe training runs

Appendix

Table 50: Grading and variability in utility comparisons. The 320-question bank first selects the BELT probe checkpoint; its disjoint 160-question halves then fit and select policies. The last column gives the source of sample SD.

(a) Features for a linear stopping policy
Control	Utility by $\lambda$ ( $\times 10^{-6}$ )
	2	5	10	20	50
Pretrained ReProbe representation	$0.719_{\scriptscriptstyle\pm 0.004}$	$0.713_{\scriptscriptstyle\pm 0.001}$	$0.691_{\scriptscriptstyle\pm 0.001}$	$0.650_{\scriptscriptstyle\pm 0.000}$	$0.550_{\scriptscriptstyle\pm 0.000}$
Reasoner state (layer 16)	$0.724_{\scriptscriptstyle\pm 0.000}$	$0.711_{\scriptscriptstyle\pm 0.001}$	$0.688_{\scriptscriptstyle\pm 0.001}$	$0.647_{\scriptscriptstyle\pm 0.001}$	$0.550_{\scriptscriptstyle\pm 0.002}$
Reasoner state (PCA, 768-d)	$0.724_{\scriptscriptstyle\pm 0.000}$	$0.710_{\scriptscriptstyle\pm 0.000}$	$0.688_{\scriptscriptstyle\pm 0.000}$	$0.645_{\scriptscriptstyle\pm 0.001}$	$0.549_{\scriptscriptstyle\pm 0.004}$
Mean-pooled reasoner states	$0.717_{\scriptscriptstyle\pm 0.000}$	$0.702_{\scriptscriptstyle\pm 0.000}$	$0.677_{\scriptscriptstyle\pm 0.000}$	$0.627_{\scriptscriptstyle\pm 0.000}$	$0.527_{\scriptscriptstyle\pm 0.002}$

Appendix

Table 51: Utility comparing features (a) and controller capacity (b). Mean and SD are reported over five policy-fitting starts for the first four feature controls, and three training seeds for OSP-1, BELT-L and BELT policies.

BELT probe training targets
Control	Utility by $\lambda$ ( $\times 10^{-6}$ )
	2	5	10	20	50
Now , End	$\underline{0.745}_{\scriptscriptstyle\pm 0.001}$	$\mathbf{0.731}_{\scriptscriptstyle\pm 0.001}$	$0.7067_{\scriptscriptstyle\pm 0.0006}$	$0.659_{\scriptscriptstyle\pm 0.001}$	$0.582_{\scriptscriptstyle\pm 0.002}$
Now , End , Later	$\underline{0.745}_{\scriptscriptstyle\pm 0.000}$	$\underline{0.730}_{\scriptscriptstyle\pm 0.000}$	$0.7065_{\scriptscriptstyle\pm 0.0004}$	$0.659_{\scriptscriptstyle\pm 0.001}$	$\underline{0.585}_{\scriptscriptstyle\pm 0.004}$
Now , End , Same	$\mathbf{0.746}_{\scriptscriptstyle\pm 0.001}$	$\mathbf{0.731}_{\scriptscriptstyle\pm 0.001}$	$\underline{0.7068}_{\scriptscriptstyle\pm 0.0010}$	$\underline{0.665}_{\scriptscriptstyle\pm 0.002}$	$\mathbf{0.588}_{\scriptscriptstyle\pm 0.003}$
All four	$\underline{0.745}_{\scriptscriptstyle\pm 0.001}$	$\mathbf{0.731}_{\scriptscriptstyle\pm 0.000}$	$\mathbf{0.7072}_{\scriptscriptstyle\pm 0.0006}$	$\mathbf{0.666}_{\scriptscriptstyle\pm 0.002}$	$\underline{0.585}_{\scriptscriptstyle\pm 0.004}$

Appendix

Table 52: Utility comparing belief training targets and policy inputs across five token prices. Mean and sample SD cover three BELT probe training seeds. The two panels respectively retrain the BELT probe and refit policies on a fixed four-target BELT probe. Values at

\lambda=10^{-5}

retain four decimal places.

(a) Retrain prediction targets
	Token saving (%) at tolerance			Accuracy (%) at saving
Belief set	0 pp	1 pp	2 pp	70%	60%	50%	40%
Now, End	$33.26_{\scriptscriptstyle\pm 1.35}$	$41.54_{\scriptscriptstyle\pm 2.30}$	$47.49_{\scriptscriptstyle\pm 1.30}$	$58.46_{\scriptscriptstyle\pm 0.72}$	$62.04_{\scriptscriptstyle\pm 0.87}$	$64.59_{\scriptscriptstyle\pm 0.37}$	$66.19_{\scriptscriptstyle\pm 0.39}$
Now, End, Later	$35.06_{\scriptscriptstyle\pm 3.87}$	$43.52_{\scriptscriptstyle\pm 1.29}$	$49.16_{\scriptscriptstyle\pm 0.67}$	NF	$62.38_{\scriptscriptstyle\pm 0.33}$	$64.92_{\scriptscriptstyle\pm 0.26}$	$66.41_{\scriptscriptstyle\pm 0.29}$
Now, End, Same	$36.19_{\scriptscriptstyle\pm 0.58}$	$43.28_{\scriptscriptstyle\pm 0.75}$	$49.17_{\scriptscriptstyle\pm 0.96}$	$58.36_{\scriptscriptstyle\pm 0.82}$	$62.61_{\scriptscriptstyle\pm 0.23}$	$64.82_{\scriptscriptstyle\pm 0.29}$	$66.58_{\scriptscriptstyle\pm 0.11}$
All four	$37.14_{\scriptscriptstyle\pm 1.95}$	$43.76_{\scriptscriptstyle\pm 1.77}$	$48.74_{\scriptscriptstyle\pm 0.62}$	$58.42_{\scriptscriptstyle\pm 0.23}$	$62.53_{\scriptscriptstyle\pm 0.29}$	$64.93_{\scriptscriptstyle\pm 0.12}$	$66.66_{\scriptscriptstyle\pm 0.38}$

Appendix

Table 53: Corrected OOD accuracy–cost readings for prediction-target retraining and policy-input removal. Both retain prefix length. Each seed contributes 17 native policies plus natural completion, with the main adjacent-member interpolation protocol. Constraints are imposed on micro accuracy/cost; the selected members are then reported as macro token savings or accuracy, as in the main math tables. Entries are mean and sample SD over three probe training runs. NF means that at least one seed cannot reach the budget. All-four is the same BELT-B-R family in both panels and in the main tables. These are exploratory test-family readings, not development-selected deployment guarantees.

	Now, End, Same targets		All four targets
$\lambda$ ( $\times 10^{-6}$ )	Accuracy (%)	Total tokens	Accuracy (%)	Total tokens
2	$75.56_{\scriptscriptstyle\pm 0.12}$	$4864.3_{\scriptscriptstyle\pm 9.9}$	$75.50_{\scriptscriptstyle\pm 0.07}$	$4807.3_{\scriptscriptstyle\pm 42.3}$
5	$75.56_{\scriptscriptstyle\pm 0.12}$	$4864.3_{\scriptscriptstyle\pm 9.9}$	$75.50_{\scriptscriptstyle\pm 0.07}$	$4807.3_{\scriptscriptstyle\pm 42.3}$
10	$75.52_{\scriptscriptstyle\pm 0.06}$	$4836.0_{\scriptscriptstyle\pm 58.5}$	$75.49_{\scriptscriptstyle\pm 0.06}$	$4778.3_{\scriptscriptstyle\pm 33.9}$
20	$73.65_{\scriptscriptstyle\pm 0.63}$	$3583.7_{\scriptscriptstyle\pm 385.2}$	$73.67_{\scriptscriptstyle\pm 0.95}$	$3535.2_{\scriptscriptstyle\pm 561.7}$
50	$69.84_{\scriptscriptstyle\pm 0.57}$	$2204.6_{\scriptscriptstyle\pm 130.3}$	$71.46_{\scriptscriptstyle\pm 1.31}$	$2590.2_{\scriptscriptstyle\pm 325.2}$

Appendix

Table 54: Accuracy and cost underlying the development-selected utilities for Now/End/Same and all-four target training in Table 7 a. Test accuracy is strict and micro averaged; cost is mean total tokens per question. Entries are mean and sample SD over three separately trained probes. At each price,

U_{\lambda}=A/100-\lambda C

. These selected points use the archived development selector, rather than test-family constraint selection.

Figure 10: Corrected strict OOD accuracy–cost curves for (a) prediction-target retraining and (b) policy-input removal with the four-target probe fixed; prefix length is retained in every arm. Curves show mean interpolated micro accuracy over three probe training runs on each family’s common cost support. Insets enlarge the region around the one-point accuracy tolerance (dotted line); stars mark natural completion. Table 53 reports the selected members with macro aggregation.

	Token saving (%) at tolerance			Accuracy (%) at saving
Scope	0 pp	1 pp	2 pp	70%	60%	50%	40%
Qwen3 ID	$30.92_{\scriptscriptstyle\pm 1.25}$	$36.24_{\scriptscriptstyle\pm 0.27}$	$39.94_{\scriptscriptstyle\pm 0.80}$	$40.39_{\scriptscriptstyle\pm 0.68}$	$44.79_{\scriptscriptstyle\pm 0.56}$	$48.65_{\scriptscriptstyle\pm 0.22}$	$51.53_{\scriptscriptstyle\pm 0.21}$
Qwen3 OOD	$27.31_{\scriptscriptstyle\pm 0.59}$	$33.72_{\scriptscriptstyle\pm 0.81}$	$38.71_{\scriptscriptstyle\pm 0.46}$	NF	$58.28_{\scriptscriptstyle\pm 0.36}$	$61.78_{\scriptscriptstyle\pm 0.40}$	$64.45_{\scriptscriptstyle\pm 0.16}$
Code Pre	$38.90_{\scriptscriptstyle\pm 2.43}$	$43.52_{\scriptscriptstyle\pm 1.94}$	$48.14_{\scriptscriptstyle\pm 1.45}$	$55.06_{\scriptscriptstyle\pm 0.14}$	$57.96_{\scriptscriptstyle\pm 0.10}$	$60.14_{\scriptscriptstyle\pm 0.28}$	$62.32_{\scriptscriptstyle\pm 0.53}$
Code Post	$47.94_{\scriptscriptstyle\pm 2.86}$	$55.07_{\scriptscriptstyle\pm 2.21}$	$62.19_{\scriptscriptstyle\pm 1.58}$	$25.57_{\scriptscriptstyle\pm 0.07}$	$26.99_{\scriptscriptstyle\pm 0.26}$	$28.40_{\scriptscriptstyle\pm 0.39}$	$29.81_{\scriptscriptstyle\pm 0.52}$
OLMo3 OOD	–	–	–	–	–	–	–

Appendix

Table 55: BELT-L operating points under the primary accuracy–cost protocol. Token saving and accuracy are percentages; means and sample SDs cover three corrected probe-training seeds. Math uses macro averages and code uses its single-dataset average. The probe has 8,196 parameters, with a six-parameter policy and 17 fitted prices. NF is an infeasible budget in at least one seed; – denotes the unevaluated OLMo3 setting. Rows describe different populations and are not ranked against one another.

Figure 11: Corrected Qwen3 OOD and LiveCodeBench Pre frontiers including the BELT-L control. Stars mark natural completion; math here is micro averaged, whereas Table 55 reports macro averages. BELT-L on ID, individual math datasets and Code Post also appears in Figs. 7 , 8 and 4 .

Features / fitting	2	5	10	20	50
OSP-1 features	$0.5409_{\scriptscriptstyle\pm 0.0011}$	$0.5252_{\scriptscriptstyle\pm 0.0013}$	$0.4994_{\scriptscriptstyle\pm 0.0021}$	$0.4582_{\scriptscriptstyle\pm 0.0018}$	$0.3612_{\scriptscriptstyle\pm 0.0021}$
OSP-1 features (four starts)	$0.5424_{\scriptscriptstyle\pm 0.0015}$	$0.5252_{\scriptscriptstyle\pm 0.0013}$	$0.4994_{\scriptscriptstyle\pm 0.0021}$	$0.4580_{\scriptscriptstyle\pm 0.0017}$	$0.3611_{\scriptscriptstyle\pm 0.0021}$
BELT-H-R	$0.5513_{\scriptscriptstyle\pm 0.0017}$	$0.5356_{\scriptscriptstyle\pm 0.0014}$	$0.5109_{\scriptscriptstyle\pm 0.0011}$	$\underline{0.4670}_{\scriptscriptstyle\pm 0.0015}$	$0.3610_{\scriptscriptstyle\pm 0.0021}$
BELT-H-R (four starts)	$0.5511_{\scriptscriptstyle\pm 0.0016}$	$0.5356_{\scriptscriptstyle\pm 0.0014}$	$0.5108_{\scriptscriptstyle\pm 0.0008}$	$\mathbf{0.4671}_{\scriptscriptstyle\pm 0.0015}$	$\mathbf{0.3619}_{\scriptscriptstyle\pm 0.0083}$
BELT-B-R	$\underline{0.5619}_{\scriptscriptstyle\pm 0.0011}$	$\underline{0.5430}_{\scriptscriptstyle\pm 0.0009}$	$\underline{0.5116}_{\scriptscriptstyle\pm 0.0007}$	$0.4654_{\scriptscriptstyle\pm 0.0049}$	$\underline{0.3614}_{\scriptscriptstyle\pm 0.0052}$
BELT-B-R (four starts)	$\mathbf{0.5620}_{\scriptscriptstyle\pm 0.0012}$	$\mathbf{0.5431}_{\scriptscriptstyle\pm 0.0010}$	$\mathbf{0.5117}_{\scriptscriptstyle\pm 0.0008}$	$0.4647_{\scriptscriptstyle\pm 0.0040}$	$0.3611_{\scriptscriptstyle\pm 0.0026}$

Appendix

Table 56: Utility under matched fitting and selection budgets. Reporting convention follows Table 1 ; all values are micro averages. Panel: Math ID. Token prices

\lambda

are in

10^{-6}

units.

Features / fitting	2	5	10	20	50
OSP-1 features	$0.7284_{\scriptscriptstyle\pm 0.0033}$	$0.7160_{\scriptscriptstyle\pm 0.0036}$	$0.6955_{\scriptscriptstyle\pm 0.0039}$	$0.6594_{\scriptscriptstyle\pm 0.0012}$	$0.5829_{\scriptscriptstyle\pm 0.0060}$
OSP-1 features (four starts)	$0.7294_{\scriptscriptstyle\pm 0.0024}$	$0.7160_{\scriptscriptstyle\pm 0.0036}$	$0.6956_{\scriptscriptstyle\pm 0.0039}$	$0.6594_{\scriptscriptstyle\pm 0.0012}$	$0.5829_{\scriptscriptstyle\pm 0.0059}$
BELT-H-R	$0.7352_{\scriptscriptstyle\pm 0.0015}$	$0.7227_{\scriptscriptstyle\pm 0.0012}$	$0.7024_{\scriptscriptstyle\pm 0.0015}$	$\mathbf{0.6689}_{\scriptscriptstyle\pm 0.0011}$	$\underline{0.5861}_{\scriptscriptstyle\pm 0.0020}$
BELT-H-R (four starts)	$0.7340_{\scriptscriptstyle\pm 0.0011}$	$0.7219_{\scriptscriptstyle\pm 0.0007}$	$0.7021_{\scriptscriptstyle\pm 0.0013}$	$\underline{0.6688}_{\scriptscriptstyle\pm 0.0011}$	$\mathbf{0.5873}_{\scriptscriptstyle\pm 0.0047}$
BELT-B-R	$\mathbf{0.7454}_{\scriptscriptstyle\pm 0.0006}$	$\mathbf{0.7310}_{\scriptscriptstyle\pm 0.0005}$	$\mathbf{0.7072}_{\scriptscriptstyle\pm 0.0006}$	$0.6660_{\scriptscriptstyle\pm 0.0024}$	$0.5851_{\scriptscriptstyle\pm 0.0042}$
BELT-B-R (four starts)	$\underline{0.7451}_{\scriptscriptstyle\pm 0.0004}$	$\underline{0.7307}_{\scriptscriptstyle\pm 0.0003}$	$\underline{0.7066}_{\scriptscriptstyle\pm 0.0001}$	$0.6662_{\scriptscriptstyle\pm 0.0026}$	$0.5843_{\scriptscriptstyle\pm 0.0045}$

Appendix

Table 57: Utility under matched fitting and selection budgets. Reporting convention follows Table 1 ; all values are micro averages. Panel: Math OOD. Token prices

\lambda

are in

10^{-6}

units.

Features / fitting	2	5	10	20	50
OSP-1 features	$0.6170_{\scriptscriptstyle\pm 0.0129}$	$0.6013_{\scriptscriptstyle\pm 0.0103}$	$0.5756_{\scriptscriptstyle\pm 0.0067}$	$0.5240_{\scriptscriptstyle\pm 0.0060}$	$0.4169_{\scriptscriptstyle\pm 0.0069}$
OSP-1 features (four starts)	$0.6185_{\scriptscriptstyle\pm 0.0137}$	$0.6015_{\scriptscriptstyle\pm 0.0104}$	$0.5757_{\scriptscriptstyle\pm 0.0067}$	$0.5240_{\scriptscriptstyle\pm 0.0062}$	$0.4170_{\scriptscriptstyle\pm 0.0070}$
BELT-H-R	$0.6308_{\scriptscriptstyle\pm 0.0066}$	$0.6148_{\scriptscriptstyle\pm 0.0051}$	$0.5795_{\scriptscriptstyle\pm 0.0188}$	$\underline{0.5332}_{\scriptscriptstyle\pm 0.0138}$	$\mathbf{0.4206}_{\scriptscriptstyle\pm 0.0125}$
BELT-H-R (four starts)	$0.6290_{\scriptscriptstyle\pm 0.0112}$	$0.6131_{\scriptscriptstyle\pm 0.0095}$	$0.5802_{\scriptscriptstyle\pm 0.0195}$	$0.5329_{\scriptscriptstyle\pm 0.0136}$	$\underline{0.4201}_{\scriptscriptstyle\pm 0.0131}$
BELT-B-R	$\mathbf{0.6699}_{\scriptscriptstyle\pm 0.0014}$	$\mathbf{0.6502}_{\scriptscriptstyle\pm 0.0015}$	$\mathbf{0.6179}_{\scriptscriptstyle\pm 0.0010}$	$0.5325_{\scriptscriptstyle\pm 0.0134}$	$0.4064_{\scriptscriptstyle\pm 0.0192}$
BELT-B-R (four starts)	$\underline{0.6698}_{\scriptscriptstyle\pm 0.0020}$	$\underline{0.6501}_{\scriptscriptstyle\pm 0.0021}$	$\underline{0.6172}_{\scriptscriptstyle\pm 0.0024}$	$\mathbf{0.5339}_{\scriptscriptstyle\pm 0.0148}$	$0.4069_{\scriptscriptstyle\pm 0.0183}$

Appendix

Table 58: Utility under matched fitting and selection budgets. Reporting convention follows Table 1 ; all values are micro averages. Panel: LiveCodeBench Pre. Token prices

\lambda

are in

10^{-6}

units.

Features / fitting	2	5	10	20	50
OSP-1 features	$0.2834_{\scriptscriptstyle\pm 0.0037}$	$0.2635_{\scriptscriptstyle\pm 0.0018}$	$0.2312_{\scriptscriptstyle\pm 0.0083}$	$\underline{0.1898}_{\scriptscriptstyle\pm 0.0105}$	$\underline{0.1130}_{\scriptscriptstyle\pm 0.0088}$
OSP-1 features (four starts)	$0.2844_{\scriptscriptstyle\pm 0.0043}$	$0.2636_{\scriptscriptstyle\pm 0.0019}$	$0.2311_{\scriptscriptstyle\pm 0.0082}$	$0.1896_{\scriptscriptstyle\pm 0.0106}$	$\mathbf{0.1131}_{\scriptscriptstyle\pm 0.0092}$
BELT-H-R	$0.2857_{\scriptscriptstyle\pm 0.0046}$	$0.2656_{\scriptscriptstyle\pm 0.0019}$	$\mathbf{0.2335}_{\scriptscriptstyle\pm 0.0044}$	$\mathbf{0.1899}_{\scriptscriptstyle\pm 0.0092}$	$0.1036_{\scriptscriptstyle\pm 0.0386}$
BELT-H-R (four starts)	$0.2848_{\scriptscriptstyle\pm 0.0066}$	$0.2648_{\scriptscriptstyle\pm 0.0041}$	$\underline{0.2331}_{\scriptscriptstyle\pm 0.0037}$	$0.1885_{\scriptscriptstyle\pm 0.0115}$	$0.1108_{\scriptscriptstyle\pm 0.0360}$
BELT-B-R	$\mathbf{0.3073}_{\scriptscriptstyle\pm 0.0007}$	$\mathbf{0.2791}_{\scriptscriptstyle\pm 0.0009}$	$0.2316_{\scriptscriptstyle\pm 0.0030}$	$0.1506_{\scriptscriptstyle\pm 0.0384}$	$0.0583_{\scriptscriptstyle\pm 0.0794}$
BELT-B-R (four starts)	$\underline{0.3071}_{\scriptscriptstyle\pm 0.0015}$	$\underline{0.2789}_{\scriptscriptstyle\pm 0.0017}$	$0.2320_{\scriptscriptstyle\pm 0.0025}$	$0.1522_{\scriptscriptstyle\pm 0.0409}$	$0.0526_{\scriptscriptstyle\pm 0.0718}$

Appendix

Table 59: Utility under matched fitting and selection budgets. Reporting convention follows Table 1 ; all values are micro averages. Panel: LiveCodeBench Post. Token prices

\lambda

are in

10^{-6}

units.

Features / fitting / seed	Utility by $\lambda$ ( $\times 10^{-6}$ )
	2	5	10	20	50
OSP-1 features, main, s0	0.6027	0.5897	0.5679	0.5202	0.4172
OSP-1 features, main, s1	0.6278	0.6090	0.5788	0.5207	0.4099
OSP-1 features, main, s2	0.6204	0.6053	0.5800	0.5309	0.4237
OSP-1 features, four starts, s0	0.6027	0.5897	0.5679	0.5202	0.4177
OSP-1 features, four starts, s1	0.6278	0.6090	0.5788	0.5207	0.4097

Appendix

Table 60: Individual training runs for the matched feature comparison on LiveCodeBench Pre. Seed indices are not paired across methods.

Features / fitting / seed	Utility by $\lambda$ ( $\times 10^{-6}$ )
	2	5	10	20	50
OSP-1 features, main, s0	0.2794	0.2640	0.2383	0.2015	0.1223
OSP-1 features, main, s1	0.2867	0.2615	0.2221	0.1811	0.1120
OSP-1 features, main, s2	0.2841	0.2651	0.2333	0.1867	0.1047
OSP-1 features, four starts, s0	0.2794	0.2640	0.2383	0.2015	0.1228
OSP-1 features, four starts, s1	0.2867	0.2615	0.2221	0.1811	0.1116

Appendix

Table 61: Individual training runs for the matched feature comparison on LiveCodeBench Post. Seed indices are not paired across methods.

Scope	Interface	Refit	Threshold	Threshold $-$ Refit	Positive seeds
Math ID	H	$48.58_{\scriptscriptstyle\pm 0.44}$	$50.64_{\scriptscriptstyle\pm 0.98}$	$+2.07_{\scriptscriptstyle\pm 0.59}$	3/3
Math OOD	H	$45.25_{\scriptscriptstyle\pm 1.12}$	$45.98_{\scriptscriptstyle\pm 0.60}$	$+0.74_{\scriptscriptstyle\pm 1.44}$	2/3
Code Pre	B	$42.87_{\scriptscriptstyle\pm 3.28}$	$47.65_{\scriptscriptstyle\pm 1.07}$	$+4.77_{\scriptscriptstyle\pm 2.22}$	3/3
Code Post	B	$61.21_{\scriptscriptstyle\pm 1.07}$	$59.08_{\scriptscriptstyle\pm 1.13}$	$-2.12_{\scriptscriptstyle\pm 1.04}$	0/3

Appendix

Table 62: Illustrative corrected comparisons at a 1 pp accuracy tolerance: token savings (%) and sample SD over three probe training seeds. Differences (pp, computed before rounding) pair the two families by shared probe seed; the last column counts positive within-seed differences. Math uses the main macro reading; code its single-dataset reading. The four rows illustrate both directions of the comparison rather than forming a new evaluation setting.

Setting	B/H	Contrast	Save 0	Save 1	Save 2	Acc 0.3	Acc 0.4	Acc 0.5	Acc 0.6
Qwen3 ID	B	R+T $-$ T	$+20.44_{\scriptscriptstyle\pm 0.79}^{\dagger}$	$+19.85_{\scriptscriptstyle\pm 0.93}^{\dagger}$	$+20.56_{\scriptscriptstyle\pm 1.11}^{\dagger}$	$+6.95_{\scriptscriptstyle\pm 2.13}^{\dagger}$	$+6.11_{\scriptscriptstyle\pm 1.52}^{\dagger}$	$+5.02_{\scriptscriptstyle\pm 0.50}^{\dagger}$	$+4.03_{\scriptscriptstyle\pm 0.48}^{\dagger}$
	B	R $-$ T	$+19.24_{\scriptscriptstyle\pm 1.95}^{\dagger}$	$+19.18_{\scriptscriptstyle\pm 0.60}^{\dagger}$	$+19.04_{\scriptscriptstyle\pm 1.14}^{\dagger}$	$+7.20_{\scriptscriptstyle\pm 2.15}^{\dagger}$	$+6.21_{\scriptscriptstyle\pm 1.40}^{\dagger}$	$+4.77_{\scriptscriptstyle\pm 0.65}^{\dagger}$	$+3.90_{\scriptscriptstyle\pm 0.13}^{\dagger}$
	H	R+T $-$ T	$+0.09_{\scriptscriptstyle\pm 0.90}$	$-0.58_{\scriptscriptstyle\pm 1.12}$	$+0.41_{\scriptscriptstyle\pm 0.22}$	$+1.72_{\scriptscriptstyle\pm 0.49}^{\dagger}$	$+0.35_{\scriptscriptstyle\pm 0.34}$	$-0.08_{\scriptscriptstyle\pm 0.28}$	$-0.16_{\scriptscriptstyle\pm 0.19}$
	H	R $-$ T	$-2.76_{\scriptscriptstyle\pm 0.89}$	$-2.07_{\scriptscriptstyle\pm 0.59}$	$-2.01_{\scriptscriptstyle\pm 0.54}$	$+1.58_{\scriptscriptstyle\pm 0.18}^{\dagger}$	$-0.07_{\scriptscriptstyle\pm 0.43}$	$-0.37_{\scriptscriptstyle\pm 0.20}$	$-0.37_{\scriptscriptstyle\pm 0.09}$
Qwen3 OOD	B	R+T $-$ T	$+11.01_{\scriptscriptstyle\pm 1.80}^{\dagger}$	$+13.88_{\scriptscriptstyle\pm 0.86}^{\dagger}$	$+13.22_{\scriptscriptstyle\pm 1.72}^{\dagger}$	$+7.91_{\scriptscriptstyle\pm 3.35}^{\dagger}$	$+6.29_{\scriptscriptstyle\pm 1.10}^{\dagger}$	$+4.96_{\scriptscriptstyle\pm 0.66}^{\dagger}$	$+3.97_{\scriptscriptstyle\pm 0.30}^{\dagger}$
	B	R $-$ T	$+11.92_{\scriptscriptstyle\pm 0.95}^{\dagger}$	$+12.89_{\scriptscriptstyle\pm 0.85}^{\dagger}$	$+12.73_{\scriptscriptstyle\pm 0.72}^{\dagger}$	$+8.41_{\scriptscriptstyle\pm 2.47}^{\dagger}$	$+6.33_{\scriptscriptstyle\pm 0.95}^{\dagger}$	$+4.89_{\scriptscriptstyle\pm 0.64}^{\dagger}$	$+3.86_{\scriptscriptstyle\pm 0.09}^{\dagger}$

Appendix

Table 63: Refit plus thresholds (R+T) and refit (R) minus the threshold family (T), paired by probe seed, random-initialized probes: mean

\pm

SD over three seed-paired differences (pp); † at least 1 pp on all three seeds. OLMo3’s registered BELT family is already refit

\times

threshold, so its R

-

T row is not defined. – = infeasible for one side.

Architecture / prediction targets	Utility by $\lambda$ ( $\times 10^{-6}$ )
	2	5	10	20	50
BELT probe / Now , End	$\mathbf{0.7451}_{\scriptscriptstyle\pm 0.0005}$	$\mathbf{0.7307}_{\scriptscriptstyle\pm 0.0005}$	$\mathbf{0.7067}_{\scriptscriptstyle\pm 0.0006}$	$\underline{0.6586}_{\scriptscriptstyle\pm 0.0008}$	$0.5820_{\scriptscriptstyle\pm 0.0021}$
Top-two branch / Now , End	$0.7447_{\scriptscriptstyle\pm 0.0007}$	$0.7301_{\scriptscriptstyle\pm 0.0004}$	$0.7061_{\scriptscriptstyle\pm 0.0003}$	$\mathbf{0.6589}_{\scriptscriptstyle\pm 0.0014}$	$\underline{0.5851}_{\scriptscriptstyle\pm 0.0027}$
BELT probe / Now , End , Later	$\underline{0.7448}_{\scriptscriptstyle\pm 0.0002}$	$\underline{0.7304}_{\scriptscriptstyle\pm 0.0003}$	$\underline{0.7065}_{\scriptscriptstyle\pm 0.0004}$	$0.6585_{\scriptscriptstyle\pm 0.0007}$	$\mathbf{0.5853}_{\scriptscriptstyle\pm 0.0043}$
Top-two branch / Now , End , Later	$\underline{0.7448}_{\scriptscriptstyle\pm 0.0008}$	$0.7303_{\scriptscriptstyle\pm 0.0008}$	$0.7062_{\scriptscriptstyle\pm 0.0008}$	$0.6580_{\scriptscriptstyle\pm 0.0009}$	$0.5834_{\scriptscriptstyle\pm 0.0030}$

Appendix

Table 64: Utility comparing BELT probe architectures with belief inputs. Reporting convention follows Table 52 .

Setting	Policy	Save 1	$U$ 2e-6	$U$ 5e-6	$U$ 1e-5	$U$ 2e-5	$U$ 5e-5
Qwen3 OOD	Now + length	$31.96$	$0.7440$	$0.7295$	$0.7050$	$0.6569$	$0.5535$
	Now , Same + length	$41.01$	$0.7451$	$0.7306$	$0.7065$	$0.6657$	$0.5807$
	All four + length (B-R)	$43.76$	$0.7454$	$0.7310$	$0.7072$	$0.6660$	$0.5851$
	Dense forced cap	$20.62$	$0.7260$	$0.7118$	$0.6880$	$0.6398$	$0.5138$
	BELT-H-R	$45.25$	$0.7352$	$0.7227$	$0.7024$	$0.6689$	$0.5861$
	OSP	$45.19$	$0.7270$	$0.7177$	$0.7022$	$0.6713$	$0.5914$

Appendix

Table 66: Parsimonious input controls next to BELT and OSP (random-initialized probes; OLMo3 has one probe family). Save 1 = token saving (%) at a 1 pp accuracy tolerance on the test frontier (Qwen3 and OLMo3 macro, LiveCodeBench single dataset);

U

= development-selected utility at price

\lambda

(micro), the protocol of Tables 65 and 42 . Mean over three seeds; the dense cap is deterministic. All four + length is BELT-B-R (on OLMo3 without the length input); the OLMo3 dense cap row is the union of the dense block caps and the published token caps ( Table 67 ).

Setting	Cap family	Save 0	Save 1	Save 2	$U$ 1e-5	$U$ 2e-5	$U$ 5e-5
Qwen3 ID	Published	$0.00$	$8.41$	$16.82$	$0.4735$	$0.4079$	$0.2684$
	Dense	$11.73$	$16.57$	$21.26$	$0.4841$	$0.4241$	$0.2587$
Qwen3 OOD	Published	$0.00$	$13.71$	$27.23$	$0.6771$	$0.6269$	$0.5196$
	Dense	$15.14$	$20.62$	$27.25$	$0.6880$	$0.6398$	$0.5138$
OLMo3	Published	$0.00$	$15.10$	$26.90$	$0.7536$	$0.7033$	$0.5616$
	Dense	$15.14$	$18.37$	$21.77$	$0.7710$	$0.7066$	$0.5284$

Appendix

Table 67: Fixed caps with the forced answer at the cap. Published = the five block caps of the main tables (OLMo3: eight thinking-token caps); dense = a cap at every probe boundary (block

4k

), so the published Qwen3 and LiveCodeBench caps (16, 32, 64, 96, 128 blocks) are members of the dense grid; OLMo3 dense

\cup

published adds the token caps. Save

k

= token saving (%) at a

k

pp tolerance (test frontier);

U

= development-selected utility. Deterministic.

$\lambda$	2.00	18.91	108.58
Length $\hat{w}_{d}$	$0.931_{\scriptscriptstyle\pm 0.021}$	$0.155_{\scriptscriptstyle\pm 0.158}$	$-0.116_{\scriptscriptstyle\pm 0.167}$
Threshold $t_{\lambda}$	$2.164_{\scriptscriptstyle\pm 0.047}$	$0.993_{\scriptscriptstyle\pm 0.288}$	$-0.326_{\scriptscriptstyle\pm 0.118}$
Scale $a_{\lambda}$	$6.233_{\scriptscriptstyle\pm 0.100}$	$6.131_{\scriptscriptstyle\pm 0.940}$	$7.774_{\scriptscriptstyle\pm 0.492}$

Appendix

Table 68: Policy geometry at three token prices;

\lambda

is in

10^{-6}

units. Mean ±SD over three BELT probe seeds; probe size 16M.

Figure 12: Within- and across-question belief AUROC, shown as mean and sample SD over three independent BELT probe training runs.

	Qwen ID		Qwen OOD		LCB Pre		LCB Post
Reading	H-R	B-R	H-R	B-R	H-R	B-R	H-R	B-R
Save 0	$+1.6^{\dagger}$	$+0.6$	$+1.5^{\dagger}$	$+0.4$	$+2.8$	$-0.2$	$+2.2$	$-2.3$
Save 1	$+1.6^{\dagger}$	$+1.9^{\dagger}$	$+3.2^{\dagger}$	$+1.1^{\dagger}$	$+3.2^{\dagger}$	$+0.4$	$+2.8$	$-1.7$
Save 2	$+2.2^{\dagger}$	$+1.8^{\dagger}$	$+3.7^{\dagger}$	$+1.0$	$+4.2^{\dagger}$	$-0.1$	$+3.6$	$-2.1$
Acc 0.3	$+0.4^{\dagger}$	$+0.7^{\dagger}$	$+0.7$	$+0.7^{\dagger}$	$+1.5^{\dagger}$	$+1.1^{\dagger}$	$+0.5$	$-0.5$
Acc 0.4	$+0.8^{\dagger}$	$+0.2^{\dagger}$	$+0.6^{\dagger}$	$+0.2$	$+1.3^{\dagger}$	$+0.4$	$+0.3$	$-0.2$

Appendix

Table 69: Initialization pushes further. Seed-paired change from random to ReProbe initialization (ReProbe

-

random, mean

\pm

SD over three seed pairs; † all three pairs share a sign). Saving in pp at a

k

pp tolerance; accuracy in pp at a token budget; utility

\times 10^{-4}

, development-selected. OLMo-3 is excluded: its ReProbe-initialized probes exist only at layer 24.

Backbone	BELT	OSP
Qwen3	15,957,765	100,676,098
OLMo3	19,103,494	404,791,298

Appendix

Table 70: Trainable controller parameters, excluding the frozen reasoner. OSP counts one two-block branch and its stopping head.

Method	Qwen3 ID/OOD (2,719)					OLMo3 OOD (1,025)					LiveCodeBench (549 + 416)
Method	Probe Params	$n$	med.	max $\|\cdot\|$	SD	Probe Params	$n$	med.	max $\|\cdot\|$	SD	Probe Params	$n$	med.	max $\|\cdot\|$	SD
Natural	–	1	+0.000	0.00	0.01	–	–	–	–	–	–	–	–	–	–
Fixed cap	0	5	+0.000	1.13	1.07	0	8	+0.002	1.20	0.66	–	–	–	–	–
CoDE-Stop	0	225	+0.001	1.52	1.07	0	75	+0.001	0.79	0.65	–	–	–	–	–
DEER	0	54	+0.001	1.31	1.05	0	18	+0.000	0.99	0.61	–	–	–	–	–
TC	514	300	+0.002	0.86	1.00	514	150	+0.002	0.93	1.14	–	–	–	–	–

Appendix

Table 71: Split-half sensitivity of test-frontier readings.

n

: members per seed; med.: median shrinkage; max

|\cdot|

: largest absolute shrinkage; SD: median cross-reading SD, pooled over splits and seeds for LiveCodeBench. Tolerance and budget entries use percentage points; price entries use utility units. Summaries pool these protocols and are descriptive.

Setting	Comparator	BELT	Comparator	$\Delta$	Verdict	Pooled interval; 2 SE
Qwen3 ID	full OSP	50.64	49.72	$+0.92$	T	–
Qwen3 ID	OSP-1	50.64	46.59	$+4.05$	W	$[-0.05,+9.67]$ ; no
Qwen3 OOD	full OSP	45.98	45.19	$+0.79$	T	–
Qwen3 OOD	OSP-1	45.98	36.89	$+9.09$	W	$[+3.30,+16.22]$ ; yes
LiveCodeBench Pre	full OSP	38.25	38.25	$-0.00$	T	–
LiveCodeBench Pre	OSP-1	38.25	15.25	$+23.00$	W	$[+10.23,+36.37]$ ; yes

Appendix

Table 72: BELT against full OSP and OSP-1 at a 1 pp accuracy tolerance (token saving, %; random-initialized probes; mean over three seeds). BELT is the recommended configuration, BELT-H-T ( § 5 ); the other three variants are in Tables 73 and 74 . Verdict: W/L when the difference of seed means exceeds the larger seed SD, else T. Interval: pooled question

\times

seed 95% bootstrap interval, with whether

|\Delta|>2

SE; shown for the pre-registered comparisons ( Table 76 ).

Variant	Comparator	Qwen3 ID	Qwen3 OOD	LCB Pre	LCB Post	OLMo3	OLMo3 LCB before 8/2024	OLMo3 LCB from 8/2024	Total
H-R	full OSP	1/1/5 1/2	0/6/1 0/0	1/5/0 0/0 (NF 1)	0/3/4 0/2	6/1/0 6/0	5/2/0 1/0	0/6/1 0/0	13/24/11 8/4
	OSP-1	4/3/0 4/0	5/2/0 5/0	7/0/0 7/0	3/4/0 1/0	7/0/0 6/0	7/0/0 5/0	5/2/0 3/0	38/11/0 31/0
	OSP+T	0/3/4 0/4	0/1/6 0/3	0/5/2 0/1	0/0/7 0/7	5/2/0 4/0	–	–	5/11/19 4/15
H-T	full OSP	1/5/1 0/1	1/4/2 1/2	0/6/0 0/0 (NF 1)	0/6/1 0/0	2/4/1 1/0	2/5/0 0/0	0/4/3 0/1	6/34/8 2/4
	OSP-1	6/0/1 6/1	5/0/2 5/1	7/0/0 7/0	2/5/0 0/0	5/1/1 3/0	7/0/0 2/0	3/4/0 2/0	35/10/4 25/2
	OSP+T	0/6/1 0/1	0/3/4 0/3	0/5/2 0/1	0/0/7 0/1	1/5/1 0/0	–	–	1/19/15 0/6

Appendix

Table 73: Single-variant win/tie/loss counts over the seven constrained readings per setting, random-initialized probes (main line). Cells: W/T/L with robust W/L counts as superscript. One row per variant; no count combines variants. OSP+T is not evaluated on the OLMo3 LiveCodeBench sets (–); its Total covers the other five settings.

Setting	Variant	Comparator	Save 0	Save 1	Save 2	Acc 0.3	Acc 0.4	Acc 0.5	Acc 0.6
Qwen3 ID	H-R	full OSP	$-1.01$ L e	$-1.14$ L e	$-1.86$ L r	$+0.56$ W r	$+0.03$ T	$-0.32$ L e	$-0.35$ L r
	H-R	OSP-1	$+4.58$ T	$+1.98$ T	$+0.79$ T	$+0.66$ W r	$+0.55$ W r	$+0.79$ W r	$+0.86$ W r
	H-R	OSP+T	$-1.82$ L r	$-1.20$ T	$-2.54$ L r	$-0.07$ T	$-0.37$ L r	$-0.28$ T	$-0.52$ L r
	H-T	full OSP	$+1.75$ W e	$+0.92$ T	$+0.15$ T	$-1.02$ L r	$+0.10$ T	$+0.05$ T	$+0.02$ T
	H-T	OSP-1	$+7.34$ W r	$+4.05$ W r	$+2.80$ W r	$-0.92$ L r	$+0.62$ W r	$+1.16$ W r	$+1.23$ W r
	H-T	OSP+T	$+0.95$ T	$+0.87$ T	$-0.53$ T	$-1.65$ L r	$-0.30$ T	$+0.09$ T	$-0.15$ T

Appendix

Table 74: Single-variant margins (BELT minus comparator, difference of seed means; pp) and verdicts, random-initialized probes. r robust, e threshold edge (verdict changes when one seed is left out).

Variant	Comparator	Qwen3 ID	Qwen3 OOD	LCB Pre	LCB Post	OLMo3	OLMo3 LCB before 8/2024	OLMo3 LCB from 8/2024	Total
H-R	full OSP	2/5/0 2/0	3/4/0 3/0	6/0/0 5/0 (NF 1)	0/7/0 0/0	6/1/0 6/0	5/2/0 1/0	0/6/1 0/0	22/25/1 17/0
	OSP-1	7/0/0 7/0	7/0/0 6/0	7/0/0 7/0	7/0/0 6/0	7/0/0 6/0	7/0/0 5/0	5/2/0 3/0	47/2/0 40/0
	OSP+T	1/5/1 0/0	1/4/2 1/1	3/4/0 2/0	0/0/7 0/1	5/2/0 4/0	–	–	10/15/10 7/2
H-T	full OSP	2/5/0 1/0	0/6/1 0/1	5/1/0 4/0 (NF 1)	0/7/0 0/0	2/4/1 1/0	2/5/0 0/0	0/4/3 0/1	11/32/5 6/2
	OSP-1	5/2/0 5/0	4/2/1 4/1	7/0/0 7/0	2/5/0 2/0	5/1/1 3/0	7/0/0 2/0	3/4/0 2/0	33/14/2 25/1
	OSP+T	1/6/0 0/0	0/5/2 0/1	2/4/1 1/0	0/6/1 0/0	1/5/1 0/0	–	–	4/26/5 1/1

Appendix

Table 75: Single-variant win/tie/loss counts as in Table 73 , ReProbe-initialized probes (analysis family; Table 69 ). OLMo3 has one probe family and repeats the main-line cells, also on its two LiveCodeBench sets.

Seed	Recipe	Near-never-stop checkpoints	Saving (%) $\uparrow$ at 1 pp	Acc. (%) $\uparrow$ at 0.5 cost	Acc. (%) $\uparrow$ at 0.3 cost	Utility $\uparrow$
0	Direct	2/4	6.47	62.98	52.36	0.627
0	Belief warm-up $\to$ direct	0/4	5.89	61.83	52.46	0.627
1	Direct	0/2	4.75	62.12	53.72	0.627
1	Belief warm-up $\to$ direct	0/2	2.94	54.05	41.32	0.627
2	Direct	0/1	7.35	65.48	56.45	0.627
2	Belief warm-up $\to$ direct	1/1	1.21	34.39	25.25	0.627

Appendix

Table 87: Belief warm-up and direct utility training on the OOD test frontier. Each family combines retained checkpoints, source-validation thresholds, and native stochastic policies. Near-never-stop counts use native token saving below 0.5%. Paired differences are within training seed; frozen BELT-B-R / BELT-H-R references show mean

\pm

SD over three probe seeds. Utility uses

\lambda=2\times 10^{-5}

. Natural completion achieves 72.73% accuracy at 5,022 total tokens.

Run	Source-val $e5\to e7$	Native OOD $e5\to e7$	OOD $\Delta$ [95% CI]
Seed 1	$.4856\to.4186$	$.6672\to.6102$	$-.0570$ [ $-.0701,-.0453$ ]
Seed 2	$.4450\to.2099$	$.6365\to.4268$	$-.2097$ [ $-.2310,-.1893$ ]

Appendix

Table 89: Separately scoped ReProbe-initialization sensitivity. Source-validation utility is chain weighted; OOD utility is question-micro on the corrected 1,025-question roster. Intervals use 2,000 paired, dataset-stratified question resamples and are conditional on the trained policies, not training-seed intervals. An earlier inconsistent validation-score export was excluded; this diagnostic uses verified repeats with unchanged bank and trainer controls.

Early Stopping for Language Model Reasoning

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Settle: Learning When to Stop Reasoning
Sep 30, 2026Ryan Brown, Zihao Fu, Chris RussellEarly Stopping for Language Model Reasoning Efficient Language Model Reasoning
Oxford Internet Institute, University of Oxford · Department of Linguistics and Modern Languages, The Chinese University of Hong Kong
When Does Learning to Stop Help? A Cost-Aware Study of Early Exits in Reasoning Models
Jun 29, 2026Zhe Dong, Fang Qin, Manish ShahEfficient Language Model Reasoning Cost-Aware Inference
University of Maine at Presque Isle · Stanford University · Independent Researcher
MetaCtrl: Your Large Language Models Can Reason Better and More Concisely with a Metacognitive Controller
Sep 29, 2026Zhibin Wen, Tao Han, Lei Bai +2Adaptive Reasoning in Language Models RL for Language Model Reasoning
Southern University of Science and Technology · The Hong Kong University of Science and Technology · Shanghai Artificial Intelligence Laboratory +1

Sep 30, 2026J/K move · Enter open · S save

Settle: Learning When to Stop Reasoning

Ryan Brown, Zihao Fu, Chris Russell

Oxford Internet Institute, University of Oxford · Department of Linguistics and Modern Languages, The Chinese University of Hong Kong

Early Stopping for Language Model Reasoning Efficient Language Model Reasoning Overthinking in Language Models

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Reasoning models often continue generating after their answers have settled. Settle learns when to stop from answer stability in completed traces. It trains the existing end-of-reasoning token while keeping other predictions close to the base model, and requires only ordinary decoding at inference. On MATH-500 with Qwen3-4B, Settle reduces token count by 40% with a 0.5-percentage-point decrease in accuracy. It gains 6.16 percentage points over supervised fine-tuning on the same traces shortened at their first stable answer, at nearly identical token counts. Its stopping score predicts whether a correct answer will remain correct. Settle extends the accuracy-token-count Pareto frontier of the evaluated stopping methods.

Abstract

Early Stopping for Language Model Reasoning

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Settle: Learning When to Stop Reasoning

When Does Learning to Stop Help? A Cost-Aware Study of Early Exits in Reasoning Models

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