CoT2-Meta: Budgeted Metacognitive Control for Test-Time Reasoning

Siyuan Ma; Bo Gao; Zikai Xiao; Hailong Wang; Xinlei Yu; Rui Qian; Jiayu Qian; Luqi Gong

cs.AIMar 30, 2026

CoT2-Meta: Budgeted Metacognitive Control for Test-Time Reasoning

Authors: Siyuan Ma, Bo Gao, Zikai Xiao, Hailong Wang, Xinlei Yu, Rui Qian, Jiayu Qian, Luqi Gong, +1 more

Organizations: Nanyang Technological University · Carnegie Mellon University · Zhejiang University · Sun Yat-Sen University · National University of Singapore · Fudan University · City University of Hong Kong (Dongguan) · Zhejiang Lab

Abstract

Recent test-time reasoning methods improve performance by generating more candidate chains or searching over larger reasoning trees, but they typically lack explicit control over when to expand, what to prune, how to repair, and when to abstain. We introduce CoT2-Meta, a training-free metacognitive reasoning framework that combines object-level chain-of-thought generation with meta-level control over partial reasoning trajectories. The framework integrates four components: strategy-conditioned thought generation, tree-structured search, an online process oracle for step-level reasoning evaluation, and a meta-controller that allocates computation through expansion, pruning, repair, stopping, and fallback decisions. Under matched inference budgets, CoT2-Meta consistently outperforms strong single-path, sampling-based, and search-based baselines, including ReST-MCTS. On the default backbone, it achieves 92.8 EM on MATH, 90.4 accuracy on GPQA, 98.65 EM on GSM8K, 75.8 accuracy on BBEH, 85.6 accuracy on MMMU-Pro, and 48.8 accuracy on HLE, with gains over the strongest non-CoT2-Meta baseline of +3.6, +5.2, +1.15, +2.0, +4.3, and +4.3 points, respectively. Beyond these core results, the framework remains effective across a broader 15-benchmark suite spanning knowledge and QA, multi-hop reasoning, coding, and out-of-distribution evaluation. Additional analyses show better compute scaling, improved calibration, stronger selective prediction, targeted repair behavior, and consistent gains across backbone families. These results suggest that explicit metacognitive control is a practical design principle for reliable and compute-efficient test-time reasoning systems.

Figures & tables

Figure 1: CoT 2 -Meta as reasoning over reasoning. The inner layer performs object-level reasoning, while the outer layer performs metacognitive control over partial trajectories. The meta-controller decides when to expand, prune, repair, stop, or abstain under a bounded budget.

Figure 2: CoT 2 -Meta pipeline. Strategy-conditioned generation forms a tree of partial trajectories, which are evaluated by an online process oracle using only the input and current reasoning path. The meta-controller converts oracle signals into meta-states and allocates budget through expand, prune, repair, stop, and abstain actions. The output is either the best validated trajectory or an abstention/fallback decision, together with structured decision traces.

Category	Dataset	Greedy CoT	Best-of-16	Vanilla ToT	ReST-MCTS*	Ours (CoT 2 -Meta)	Gain $\uparrow$
Mathematics	MATH ( Hendrycks et al., 2021 )	78.5	84.2	87.4	89.2	92.8	+3.6
Mathematics	GSM8K ( Cobbe et al., 2021 )	94.2	96.1	96.8	97.5	98.6	+1.1
Knowledge & QA	GPQA ( Rein et al., 2024 )	74.2	79.6	83.5	85.2	90.4	+5.2
	MMLU-Pro ( Wang et al., 2024 )	78.5	82.2	84.5	86.1	88.4	+2.3
	TruthfulQA ( Lin et al., 2022 )	81.5	85.4	88.2	90.2	92.8	+2.6
	Natural Questions ( Kwiatkowski et al., 2019 )	36.5	39.8	41.5	43.2	45.4	+2.2

Table 1: Performance across 15 core and out-of-distribution benchmarks under compute-matched evaluation. CoT 2 -Meta consistently achieves the strongest average performance across reasoning, multimodal, QA, and coding domains compared to baseline test-time scaling strategies.

Backbone	Strategy	MATH	GPQA	BBEH	MMMU-Pro	$\Delta$ (Avg)
Claude-4.5	Greedy CoT ( Wei et al., 2022 )	78.5	74.2	65.4	68.4	–
	Best-of-16 ( Wang et al., 2022 )	84.2	79.6	68.9	73.1	+4.8
	Vanilla ToT ( Yao et al., 2023 )	87.4	83.5	71.2	77.8	+8.3
	Ours (CoT 2 -Meta)	92.8	90.4	75.8	85.6	+14.5
DeepSeek-V3.2	Greedy CoT ( Wei et al., 2022 )	70.8	82.4	74.4	64.6	–
	Best-of-16 ( Wang et al., 2022 )	75.3	85.1	75.9	68.2	+3.0

Table 2: Cross-backbone comparison of CoT 2 -Meta. Evaluation across distinct LLM families demonstrates consistent improvements over inference-time baselines on MATH ( Hendrycks et al., 2021 ) , GPQA ( Rein et al., 2024 ) , BBEH ( Kazemi et al., 2025 ) , and MMMU-Pro ( Yue et al., 2025 ) .

\Delta

(Avg) denotes the average improvement over the corresponding Greedy CoT baseline. Bold indicates the best result within each backbone block.

Variant Configuration	MATH	GPQA	BBEH	Avg Calls	Avg Tokens
Full CoT 2 -Meta (Ours)	92.8	90.4	75.8	15.2	5,280
w/o Thought Tree (Single-chain)	78.5	74.2	65.4	1.0	1,929
w/o Process Evaluator	86.5	82.3	69.3	16.0	6,120
w/o Metacognitive Pruning	88.2	84.1	71.3	16.0	9,850
Process-only Controller ( $v=v_{\mathrm{proc}}$ )	89.5	86.2	72.5	16.0	5,840
Outcome-only Controller ( $v=v_{\mathrm{out}}$ )	87.8	84.5	70.1	16.0	6,430

Table 3: Comprehensive ablation study of CoT 2 -Meta on core reasoning benchmarks under a fixed maximum inference budget (

C=16

Figure 3: Compute–accuracy scaling on MATH. Accuracy versus total inference budget. CoT 2 -Meta is strongest in the low-budget regime and reaches the highest ceiling at larger budgets.

Figure 4: Calibration and selective prediction on MATH (

C=16

). (a) Selective accuracy versus coverage. (b) Reliability diagram. CoT 2 -Meta shows the best calibration and selective accuracy among the compared methods.

Figure 5: Decision trace interpretability of CoT 2 -Meta on the curated hard subset of MATH . (a) Action distribution over search depth, showing early expansion, mid-stage pruning, and late repair/stop behavior. (b) Pruning audit across datasets, comparing prune precision, false-prune rate, and oracle gap. (c) Root-to-outcome flow on the MATH hard subset.

Appendix figures & tables28 assets

Supplementary material from the paper’s appendix.

Appendix

Signal	Meaning	Parsing Rule
Semantic	Semantic consistency with the problem and the current reasoning goal	Numeric score in $[0,1]$
Logical	Whether the current step is internally coherent and supports subsequent reasoning	Numeric score in $[0,1]$
Fix	Whether the step explicitly detects, revises, rejects, or corrects a previous incorrect intermediate claim	Numeric score in $[0,1]$

Appendix

Table 4: Definitions of the three oracle sub-signals used in online process evaluation.

Method	Gen. Calls	Eval. Calls	Verify Calls	Repair Calls	Selection Overhead	Fallback Calls	Total Budgeted
Greedy CoT	1	0	0	0	0	0	1
Best-of-16	16	0	0	0	0	0	16
Vanilla ToT	branch exp.	0	0	0	0	0	budget-matched
ReST-MCTS	branch exp.	value/reward q.	0	0	controller q.	0	budget-matched
CoT 2 -Meta	branch exp.	process-oracle q.	verify-style conf. q.	selective repair q.	controller q.	optional fallback q.	budget-matched

Appendix

Table 5: Per-method budget accounting protocol. All quantities denote model-side calls counted toward the total inference budget.

Active Backbone	Role	Backend / Access Mode	Temp.	Max Tokens	Usage
Claude-4.5	Object-level generator	Official API	0.4	2048	Direct / decompose / verify generation
Claude-4.5	Online process oracle	Official API	0.0	512	Structured step-level process evaluation
Claude-4.5	Fallback policy	Official API	0.4	2048	Best-of-16 voting fallback
DeepSeek-V3.2	Object-level generator, oracle, and fallback	Official API	task-dependent	task-dependent	Cross-backbone evaluation
Qwen2.5-VL-7B-Instruct	Object-level generator, oracle, and fallback	vLLM local deployment	task-dependent	task-dependent	Cross-backbone evaluation

Appendix

Table 6: Runtime configuration and model endpoints used in our experiments.

Category	Dataset	Metric	Baseline	Ours (CoT 2 -Meta)	Gain	$p$ -value
Reasoning	MATH	EM	87.40 $\pm$ 0.9	92.80 $\pm$ 0.4	+5.40	$<0.001$
	GPQA	Acc.	83.50 $\pm$ 1.2	90.40 $\pm$ 0.6	+6.90	$<0.001$
	BBEH	Acc.	71.20 $\pm$ 0.8	75.80 $\pm$ 0.5	+4.60	0.002
Multimodal	MMMU-Pro	Acc.	77.80 $\pm$ 1.1	85.60 $\pm$ 0.7	+7.80	$<0.001$
Multimodal	HLE	Acc.	42.10 $\pm$ 1.5	48.80 $\pm$ 0.9	+6.70	0.002

Appendix

Table 7: Statistical robustness across random seeds. We report mean

\pm

standard deviation over 3 independent runs under the same fixed inference budget (

C=16

). The baseline column shows the strongest replicated baseline under this seed-controlled protocol.

Figure 6: Cross-model macro-metric summary under a fixed inference budget (

C=16

). For each backbone, we report three aggregated dimensions: Math-EM (average of GSM8K and MATH), QA-F1 (HotpotQA F1), and Gen-Acc (macro-average over GPQA, MMLU-Pro, BBEH, TruthfulQA, and LiveCodeBench). In each bar group, the base segment corresponds to the strongest budget-matched baseline score, and the top segment indicates the additional gain of CoT 2 -Meta over that baseline.

Dataset	Method	$C=4$	$C=8$	$C=16$	$\Delta$ at $C=4$ (vs. ToT)
GPQA	Best-of-16	79.60	82.10	84.20	–
	Vanilla ToT	76.80	80.10	83.50	–
	ReST-MCTS	78.40	81.50	85.20	+1.60
	CoT 2 -Meta	82.30	87.40	90.40	+5.50
BBEH	Best-of-16	65.80	67.50	68.90	–
	Vanilla ToT	68.20	71.40	72.80	–

Appendix

Table 8: Compute scaling on GPQA and BBEH . We report task accuracy (%) under incremental inference budgets. The final column shows the low-budget gain of CoT 2 -Meta over Vanilla ToT at

C=4

Dataset	$C=4$		$C=8$		$C=16$
Dataset	SBMB	Ours	SBMB	Ours	SBMB	Ours
GSM8K	95.10	97.20	96.20	98.10	96.80	98.60
MATH	81.20	86.50	84.50	90.20	87.40	92.80
GPQA	76.80	82.30	80.10	87.40	83.50	90.40
MMLU-Pro	82.10	84.50	84.20	86.90	86.10	88.40
TruthfulQA	86.40	89.20	88.50	91.40	90.20	92.80

Appendix

Table 9: Efficiency profiles across inference budgets. SBMB denotes the strongest budget-matched baseline.

Dataset (Radar Axis)	Context Trunc. (S4)		Noisy Format (S5)
Dataset (Radar Axis)	S.B.M.B.	Ours	S.B.M.B.	Ours
GSM8K (EM)	90.20	94.50	91.20	97.80
MATH (EM)	78.40	88.10	79.50	91.50
GPQA (Acc.)	72.40	85.20	75.60	89.20
MMLU-Pro (Acc.)	79.50	84.30	80.20	87.50
TQA (Binary BvI Acc.)	84.10	89.10	82.10	92.10

Appendix

Table 10: Robustness profiles under technical perturbations at

N=16

budget. S4: 50% context truncation of intermediate scratchpads. S5: formatting noise including shuffled options and LaTeX jitter.

Textified Vision Axis (S6)	S.B.M.B.	Ours
MMMU-Pro (Overall)	52.40	79.20
HLE-Vision (Overall)	28.30	44.60
Science (Bio, Chem, Phys)	32.50	65.20
Health & Medicine	38.40	68.30
Tech & Engineering	22.10	55.10
Art & Design	45.10	71.40

Appendix

Table 11: Performance on textified vision benchmarks under OCR and image degradation. Overall scores represent macro-averages across all available domains in MMMU-Pro and HLE.

Figure 7: Radar profiles of CoT 2 -Meta across efficiency, robustness, and vision degradation. S1–S3 evaluate budget scaling under Calls=

\{4,8,16\}

on eight benchmarks. S4 applies context truncation to intermediate scratchpads (50% truncation). S5 injects formatting noise. S6 evaluates textified vision benchmarks under OCR and image degradation. In each subplot, S.B.M.B. denotes the strongest budget-matched baseline.

Figure 8: Latent memory visualization. Left: across all benchmarks (GSM8K, MATH, GPQA, BBEH, etc.). Middle: GPQA subset. Right: GSM8K subset. The visualizations show clustering of different datasets or latent groups, revealing how different reasoning regimes occupy distinct regions in latent space.

Total LLM Calls	Greedy CoT	Best-of- $N$	Vanilla ToT	Q* Search	ReST- MCTS	rStar- Math	CoT 2 - Meta
1	50.2	50.2	50.2	50.2	50.2	50.2	50.2
2	–	53.4	52.8	53.5	53.1	53.8	54.5
4	–	56.8	58.5	59.2	58.8	60.1	62.4
8	–	59.5	64.2	66.1	67.4	68.2	70.5
16	–	62.1	68.7	71.5	74.2	74.8	77.1
32	–	64.3	71.4	75.2	79.5	80.3	82.4

Appendix

Table 12: Compute–accuracy scaling comparison on the MATH dataset (

N=500

). Performance is reported as mean accuracy (%). “Total LLM Calls” includes generation, process-evaluation, value-estimation, and meta-control queries under a unified compute budget. Bold indicates the best result at each budget level.

Method	Avg In Tok.	Avg Out Tok.	Avg Total Tok.	Acc. (%)	Acc. / 1K Tok.	$\Delta$ Tok. / +1% Acc
Greedy CoT ( $C=1$ )	1,244	685	1,929	50.2	26.02	–
Self-Consistency (Maj@16)	1,984	7,488	9,472	62.1	6.56	634
Vanilla ToT	4,350	2,120	6,470	68.7	10.62	245
ReST-MCTS	7,840	6,150	13,990	74.2	5.30	502
CoT 2 -Meta (Ours)	3,820	1,460	5,280	77.1	14.60	124

Appendix

Table 13: Token cost analysis on MATH under the compute-scaling protocol (

N=500

). We report token statistics at the

C=16

operating point. Acc / 1K Tokens measures absolute efficiency, while

\Delta

Tokens / +1% Acc measures the marginal token cost required to gain one percentage point of accuracy over the Greedy baseline.

Difficulty	Count ( $N$ )	Greedy ( $C=1$ )	Vanilla ToT	ReST-MCTS	CoT 2 -Meta
Easy (L1–L2)	150	81.3%	93.3%	94.0%	95.3%
Medium (L3–L4)	200	42.5%	65.0%	74.5%	79.5%
Hard (L5)	150	29.3%	48.6%	54.0%	55.3%
Overall	500	50.2%	68.7%	74.2%	77.1%

Appendix

Table 14: Performance breakdown by MATH difficulty level at

C=16

. We categorize the 500 test problems into Easy (Levels 1–2), Medium (Levels 3–4), and Hard (Level 5).

Figure 9: OOD generalization across diverse benchmarks. (a) Task-standard scores (%) on ARC-Challenge , BBH Unseen , HumanEval , MBPP , and Natural Questions . (b) Per-task gain of CoT 2 -Meta over the strongest task-specific baseline, with an average gain of +2.4 points across all OOD tasks.

Failure Type	Defining Characteristics (Automated Rule)	Count	% of Fails	Avg Calls
F1: Search Not Converged	Budget exhausted ( $C=16$ ); top- $k$ leaves show high entropy; no path surpasses the confidence threshold.	21	18.3%	16.0
F2: Evaluator Misjudgment	Oracle assigns a higher process reward to an incorrect leaf than to a generated correct leaf.	16	13.9%	16.0
F3: Over-pruning	Shadow run (looser pruning bounds) solves the case; the correct trajectory was pruned too early.	11	9.6%	10.4
F4: Base-Model Limit	All $C=16$ baselines fail; no valid reasoning trace is generated; the case appears to require knowledge absent in the LLM.	67	58.2%	13.5
Total Failures	–	115	100%	–

Appendix

Table 15: Taxonomy of failure modes for CoT 2 -Meta . We perform automated heuristic triage followed by human verification on the failed cases at budget

C=16

Method	Topology subset ( $n=50$ )	Phys/Chem subset ( $n=75$ )	Combinatorics subset ( $n=100$ )
Greedy CoT ( $T=0$ )	35.4 $\pm$ 0.0	41.5 $\pm$ 0.0	45.4 $\pm$ 0.0
Self-Consistency (Best-of-16)	37.1 $\pm$ 0.5	43.2 $\pm$ 0.4	47.8 $\pm$ 0.6
Vanilla ToT ( $C=16$ )	39.7 $\pm$ 1.0	47.5 $\pm$ 0.9	50.7 $\pm$ 0.6
ReST-MCTS ( $C=16$ )	42.3 $\pm$ 0.8	50.1 $\pm$ 0.7	53.5 $\pm$ 0.5
CoT 2 -Meta (Ours, $C=16$ )	44.4 $\pm$ 0.7	52.8 $\pm$ 0.3	55.9 $\pm$ 0.4

Appendix

Table 16: Targeted subset accuracy on curated hard reasoning subsets. Results are mean

\pm

standard deviation over 3 seeds. Greedy CoT uses deterministic decoding (

T=0

Figure 10: Qualitative failure cases of CoT 2 -Meta on MATH . Panel 1 shows evaluator misjudgment, where a fluent but flawed derivation is ranked above a simpler correct path. Panel 2 shows over-pruning, where a rare-domain argument is discarded early but recovered by a relaxed-pruning shadow run. Panel 3 shows a base-model limit, where all high-value branches identify the correct strategy yet collapse to the same final arithmetic error.

Method	Accuracy ( $\uparrow$ )	ECE ( $\downarrow$ )	Brier Score ( $\downarrow$ )	AURC ( $\downarrow$ )
Best-of-16	62.1%	0.158 [0.142, 0.175]	0.235 [0.215, 0.255]	0.278 [0.258, 0.298]
Vanilla ToT	68.7%	0.124 [0.109, 0.140]	0.205 [0.188, 0.222]	0.224 [0.205, 0.243]
ReST-MCTS ∗	74.2%	0.092 [0.078, 0.107]	0.154 [0.140, 0.169]	0.162 [0.147, 0.178]
Pure CoT 2 -Meta (Standalone, No Fallback)	77.1%	0.035 [0.026, 0.045]	0.126 [0.112, 0.141]	0.108 [0.095, 0.122]

Appendix

Table 17: Calibration and selective-risk metrics on the MATH compute-scaling evaluation subset (

N=500

C=16

). We report mean values together with 95% bootstrap confidence intervals computed from 1000 resamples. CoT 2 -Meta shows consistently better calibration and lower selective risk than all baselines.

Verify Prompt Variant	ECE ( $\downarrow$ )	Brier Score ( $\downarrow$ )	AURC ( $\downarrow$ )
Variant A (Current default)	0.035 [0.026, 0.045]	0.126 [0.112, 0.141]	0.108 [0.095, 0.122]
Variant B (Wording change)	0.037 [0.028, 0.047]	0.129 [0.114, 0.144]	0.110 [0.096, 0.125]
Variant C (Constrained numeric)	0.032 [0.023, 0.042]	0.124 [0.110, 0.138]	0.106 [0.093, 0.120]

Appendix

Table 18: Sensitivity of calibration quality to verify-prompt wording. All variants are evaluated on the same set of final reasoning trajectories generated by CoT 2 -Meta on the MATH compute-scaling evaluation subset. Calibration quality remains stable across moderate wording and output-format changes.

Variant	Accuracy	Avg Tokens	Rescue Rate ( $\uparrow$ )	Hurt Rate ( $\downarrow$ )	Net Trajectory Gain
No Repair	73.4%	4,650	–	–	–
Repair-All (No selective trigger)	74.8%	6,920	18.2%	31.4%	+107
Full Selective Repair (Ours)	77.1%	5,280	42.5%	11.6%	+275

Appendix

Table 19: Trajectory-level repair statistics and system accuracy on MATH (

N=500

C=16

). Rescue and hurt rates are computed over repair-triggered trajectories. Net trajectory gain reports the absolute difference between rescued and harmed trajectories over the accumulated trajectory pool.

System setup	Accuracy	Avg. tokens	$\Delta$ Acc. vs. matched
No Repair (Base)	73.4%	4,650	–
No Repair + Extra Expansion (Matched)	74.6%	5,290	–
Full Selective Repair (Ours)	77.1%	5,280	+2.5%

Appendix

Table 20: Token-matched evaluation on MATH (

N=500

C=16

). Even when the no-repair baseline is allowed to spend an equivalent token budget on extra search expansions, it fails to match the performance of full selective repair.

System setup	Fallback	Avg. calls	Accuracy	$\Delta$ vs. best standalone
ReST-MCTS only	100%	16.0	74.2%	-2.9%
CoT 2 -Meta only	0%	16.0	77.1%	0.0%
Hybrid cascade ( $\tau=0.6$ )	20%	19.2	79.1%	+2.0%
Hybrid cascade ( $\tau=0.8$ )	40%	22.4	79.4%	+2.3%

Appendix

Table 21: Reliability and cascade fallback analysis on MATH under adaptive compute (

N=500

Field	Content
Role	You are a reasoning-process evaluator.
Inputs	• a problem • a multi-step reasoning trajectory
Instruction	Evaluate each reasoning step independently.
Returned scores	• Semantic : semantic alignment with the problem and current reasoning goal • Logical : logical coherence and inferential validity • Fix : whether the step explicitly detects, revises, rejects, or corrects a previous incorrect intermediate claim
Output format	Step 1: Semantic=0.xx, Logical=0.xx, Fix=0.xx
Step 2: Semantic=0.xx, Logical=0.xx, Fix=0.xx

Appendix

Table 22: Structured prompt template for the online process oracle.

Field	Content
Purpose	Re-issue a format-constrained request when the original oracle output cannot be parsed.
Instruction	Return the scores again using exactly the following format.
Required format	Step 1: Semantic=0.xx, Logical=0.xx, Fix=0.xx Step 2: Semantic=0.xx, Logical=0.xx, Fix=0.xx ...
Constraint	Output only the step-wise scores. Do not include any other text.

Appendix

Table 23: Constrained re-prompt used after parse failure.

Signal	Meaning	Parsing rule
Semantic	Semantic consistency with the problem and the current reasoning goal	Numeric score in $[0,1]$
Logical	Whether the current step is internally coherent and supports subsequent reasoning	Numeric score in $[0,1]$
Fix	Whether the step explicitly detects, revises, rejects, or corrects a previous incorrect intermediate claim	Numeric score in $[0,1]$

Appendix

Table 24: Definitions of the three oracle sub-signals used in online process evaluation.

Metric	Value	Description
First-pass success	98.2%	Step-aligned numeric fields were extracted successfully on the first attempt.
Re-prompt recovery	1.7%	Formatting errors were corrected by a single constrained re-prompt.
Final failure	0.1%	The oracle still failed after retry; these cases were assigned conservative default scores.

Appendix

Table 25: Oracle formatting parse robustness. Evaluated over 1,500 online process-oracle calls randomly sampled from the MATH (

N=500

) evaluation under the

C=16

budget.

Algorithm 2: CoT 2 -Meta inference loop

Input: problem

x

, budget

C

, backbone

G_{\theta}

Initialize: root node

r

, frontier

\mathcal{F}\leftarrow\{r\}

while budget remains and

\mathcal{F}\neq\emptyset

Select frontier node

n

with highest controller score

Generate candidate child thoughts under Direct / Decompose / Verify modes

For each child trajectory

\tau

Appendix

Figure 11: Pseudo-code summary of the CoT 2 -Meta controller used in our experiments.

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