Reasoning Concentrates Errors, and Self-Consistency Never Notices

Asaad Althoubi

cs.AISep 25, 2026

Reasoning Concentrates Errors, and Self-Consistency Never Notices

Authors: Asaad Althoubi

Organizations: Department of Computer Science Oklahoma State University Stillwater, OK, USA

Abstract

Self-consistency assumes that independent samples disagree when a model is unsure, so agreement is evidence of correctness. Holding weights fixed and toggling only a reasoning mode, over five benchmarks and 74,944 samples, we show that reasoning concentrates a model's errors: the probability that two independently drawn wrong answers coincide rises in all ten dataset-scale comparisons (p = 0.00098), and in nine of nine after restricting both arms to the problems each gets wrong. Where the answer space is unbounded, reasoning cuts the distinct answers produced to 0.43-0.65 of the non-reasoning count; where it is bounded, both arms hold an identical option set and reasoning concentrates mass on it instead, which no positional prior can explain at fixed weights. The aggregate cost is smaller than the mechanism predicts, because reasoning also shrinks the set of problems where answer diversity can decide anything, in ten of ten cells and by 2.7x; normalized for available headroom, both arms convert a quarter of it in domain. Confidence weighting does not recover what is left. Across 280 method-dataset-model combinations on eight models and five benchmarks, not one beats plain majority voting after correction; weighted voting agrees with it on 98.5% of problem-method pairs and is right 56.3% of the time on the rest; and a signal's direction can invert within fixed weights, with answer log-probability predicting correctness when reasoning is off and error when it is on. A learned six-signal combination gains nothing out of domain. Confidence signals should be evaluated on decisions, not on discrimination.

Figures & tables

dataset	pair	reasoning on	reasoning off	$\Delta$	distinct keys (on / off)
MATH-500	8B	0.509	0.379	$+0.130$	339 / 522
MATH-500	32B	0.487	0.297	$+0.190$	334 / 542
AIME	8B	0.263	0.141	$+0.122$	143 / 292
AIME	32B	0.193	0.096	$+0.097$	152 / 324
AMC	8B	0.253	0.218	$+0.035$	67 / 106
AMC	32B	0.594	0.163	$+0.431$	52 / 122

Table 1: Collision among wrong answers, within fixed weights. Every one of the ten dataset–scale comparisons favors reasoning (exact sign test

p=2^{-10}=0.00098

). The last column is the number of distinct answer keys each arm produces.

Figure 1: Reasoning shrinks the set of problems voting can decide: 10 of 10 below the diagonal, exact sign test

p=0.00098

. The danger zone with reasoning on against off, log–log, one point per dataset–scale cell. The upper line is

y=x

, so a point below it is a cell where reasoning shrank the zone; the lower line is the mean ratio,

2.72\times

. Per-cell values are in Table 8 .

Appendix figures & tables24 assets

Supplementary material from the paper’s appendix.

Appendix

short name	model	family	reasoning	params
qwen_small	Qwen3-8B (thinking)	Qwen3	yes	8B
qwen_small_nothink	Qwen3-8B (non-thinking)	Qwen3	no (same weights)	8B
qwen_large	Qwen3-32B (thinking)	Qwen3	yes	32B
qwen_large_nothink	Qwen3-32B (non-thinking)	Qwen3	no (same weights)	32B
olmo_small	Olmo-3-Think-7B	Olmo-3	yes	7B
olmo_large	Olmo-3-Think-32B	Olmo-3	yes	32B

Appendix

Table 2: The eight models. The two Qwen3 pairs share parameters, tokenizer, pre-training corpus and post-training recipe exactly within each pair; only whether a reasoning chain is generated differs. Short names in the first column are used in the tables below.

dataset	problems	answer format	split
MATH-500	500	open-ended numeric	in-domain
AIME	90	open-ended numeric	out-of-domain
AMC	83	open-ended numeric	out-of-domain
GPQA-Diamond	198	4-option multiple choice	out-of-domain
MMLU-Pro	300	10-option multiple choice	out-of-domain

Appendix

Table 3: The five benchmarks. Domain and answer format vary independently: GPQA-Diamond and MMLU-Pro are both out-of-domain and multiple choice, which would confound the two were it not for AIME and AMC, which are out-of-domain and open-ended.

dataset	pair	problems (on / off / common)			collision on / off	$\Delta$
MATH-500	8B	25	117	21	0.457 / 0.374	$+0.084$
MATH-500	32B	26	103	19	0.453 / 0.297	$+0.156$
AIME	8B	28	79	28	0.263 / 0.130	$+0.134$
AIME	32B	28	77	28	0.193 / 0.099	$+0.095$
AMC	8B	13	39	12	0.274 / 0.139	$+0.135$
GPQA-Diamond	8B	100	135	94	0.771 / 0.642	$+0.129$

Appendix

Table 4: Collision on the matched subset. Nine of nine favor reasoning (exact sign test

p=0.00195

), median

\Delta=+0.124

against

+0.133

unmatched. AMC at 32B is dropped: only seven problems meet the criterion in both arms, below the pre-set minimum of ten.

dataset	pair	$\Delta$	95% CI	$\Delta$	95% CI	$n$ matched
		unmatched (Table 1 )		matched (Table 4 )
MATH-500	8B	$+0.129$	$[-0.037,\ +0.294]$	$+0.084$	$[-0.075,\ +0.238]$	21
MATH-500	32B	$+0.190$	$[+0.034,\ +0.345]$	$+0.156$	$[-0.021,\ +0.311]$	19
AIME	8B	$+0.122$	$[-0.003,\ +0.259]$	$+0.134$	$[-0.011,\ +0.285]$	28
AIME	32B	$+0.097$	$[-0.012,\ +0.220]$	$+0.095$	$[-0.023,\ +0.221]$	28
AMC	8B	$+0.035$	$[-0.135,\ +0.218]$	$+0.135$	$[-0.026,\ +0.308]$	12

Appendix

Table 5: Cluster-bootstrap intervals for the collision difference, 20,000 resamples, paired over problems. Six of the ten unmatched intervals and four of the nine matched intervals exclude zero; all nineteen point estimates are positive. The index is low-powered per cell by construction : it is defined only where an arm produces at least two errors, and the matched version requires that of both arms, which is rare precisely because the reasoning arm is accurate. Every multiple-choice cell, where 47 or more problems qualify, excludes zero in both analyses except MMLU-Pro at 32B matched. The inference in Section 4 is the consistency of sign across cells and the two independent weight-pairs, not per-cell significance ; that is what a sign test tests. The contrast with Table 8 , where every cell is individually significant, is the expected consequence of conditioning: the mechanism is measured on a small selected subset, its downstream consequence on every problem.

Figure 2: 10 of 10 paired comparisons favor reasoning; exact sign test

p=0.00098

. Collision among wrong answers with reasoning on against off, paired within fixed weights: each line joins the two arms of one weight-pair on one benchmark, and every line points the same way. The rule separates the two regimes of Section 4 : above it the answer space is unbounded and reasoning also emits about half as many distinct keys; below it both arms hold identical key sets of four and ten, so what changes is only how mass is distributed over them. This is Table 1 drawn.

reasoning
model	MATH-500	GPQA-Diamond	MMLU-Pro
qwen_small	0.509	0.771	0.748
qwen_large	0.487	0.744	0.771
olmo_small	0.480	0.662	0.624
olmo_large	0.707	0.718	0.730
non-reasoning

Appendix

Table 6: Collision among wrong answers for all eight models, on the three benchmarks large enough to support the unpaired comparison. Values are the cleaned-answer-space index of Section 4 . Dominances counts the reasoning–non-reasoning model pairs, sixteen per dataset, in which the reasoning model collides more;

p

is the exact one-tailed Mann–Whitney probability at

n_{1}=n_{2}=4

. Bold marks the three cells where a Gemma-3 value exceeds an Olmo-3-Think value, breaking the expected ordering; they yield four exceptions because one bold value exceeds both Olmo-3-Think models. The 0.45 threshold Section 4 refers to is prediction 7 in Table 21 .

model	$k=1$	$k=8$	gain	headroom	fraction converted
reasoning
qwen_small	0.948	0.964	$+0.016$	0.052	0.308
qwen_large	0.958	0.970	$+0.012$	0.042	0.286
olmo_small	0.958	0.968	$+0.010$	0.042	0.238
olmo_large	0.964	0.970	$+0.006$	0.036	0.167
non-reasoning

Appendix

Table 7: Per-model accuracy at

k=1

and

k=8

on the in-domain benchmark, the headroom each model has left, and the fraction of it that voting captures. The summary rows are the ranges Section 5 quotes. The final column is the point: the two arms differ almost threefold in raw gain and not at all in the fraction of available headroom converted, and the per-model values interleave rather than separating, so the equality is not an artifact of averaging over groups.

dataset	pair	reasoning on	reasoning off	ratio	95% CI
MATH-500	8B	0.034	0.114	$3.35\times$	[2.21, 5.56]
MATH-500	32B	0.030	0.118	$3.93\times$	[2.50, 7.00]
AIME	8B	0.170	0.798	$4.69\times$	[3.14, 8.50]
AIME	32B	0.139	0.787	$5.65\times$	[3.71, 11.20]
AMC	8B	0.060	0.305	$5.06\times$	[2.62, 16.20]
AMC	32B	0.048	0.314	$6.52\times$	[3.12, 25.00]

Appendix

Table 8: The danger zone, the fraction of problems with fewer than half the usable samples correct. Ten of ten favor reasoning (exact sign test

p=0.00098

); mean 0.136 against 0.369,

2.72\times

smaller , from identical weights. Intervals are a paired cluster bootstrap over problems, 20,000 resamples. Every lower bound exceeds

1\times

, which is the claim; upper bounds are unstable where the reasoning arm’s zone is small, since a single danger-zone problem out of 83 on AMC produces a ratio of 25.

reasoning
model	MATH-500	AIME	AMC	GPQA-D	MMLU-Pro	weighted mean
qwen_small	5,251	16,058	10,376	9,458	4,896	7,066
qwen_large	4,376	14,535	7,888	7,219	3,186	5,581
olmo_small	6,105	17,983	11,530	15,083	8,111	9,435
olmo_large	5,155	15,914	9,299	12,415	5,490	7,589
non-reasoning

Appendix

Table 9: Mean output tokens per problem at

k=1

, by model and dataset, from which the ratios above are computed. The last column is weighted by the number of problems in each dataset. The two Qwen3 pairs are the identified comparison; the wide spread between Olmo-3-Think and Gemma-3, which share no weights with anything, is why the eight-model ratio exceeds the within-weights one.

Figure 3: What weighted voting actually does across the 280 combinations. It agrees with plain majority voting on 64,566 of 65,576 problem–method pairs (98.5%); where it differs it is right 56.3% of the time. The win rate in the differing cases fell monotonically as models were added to the study: 58.9%, then 56.6%, then 56.3%.

raw $p$	observed	expected under a uniform null
$[0,\ 0.01)$	0	2.8
$[0.01,\ 0.05)$	5	11.2
$[0.05,\ 0.10)$	2	14.0
$[0.10,\ 0.25)$	10	42.0
$[0.25,\ 0.50)$	34	70.0
$[0.50,\ 1.00]$	229	140.0

Appendix

Table 10: Distribution of the 280 raw

p

-values, against what a uniform null would produce. The median raw value is 1.000, and every Holm-adjusted value is 1.000.

pair	split	reasoning on	reasoning off	$\Delta$
8B	in-domain	0.918	0.952	$+0.034$
8B	out-of-domain	0.805	0.869	$+0.064$
32B	in-domain	0.939	0.956	$+0.017$
32B	out-of-domain	0.830	0.893	$+0.063$

Appendix

Table 11: Agreement AUROC pooled within each weight-pair and split.

\Delta

is reasoning off minus reasoning on.

Figure 4: Agreement AUROC with reasoning on against off, pooled within each weight-pair and split. This is Table 11 drawn.

model	dataset	acc	AUROC	95% CI	ceiling	margin
qwen_small	GPQA-D	0.629	0.742	[0.686, 0.796]	0.902	0.159
qwen_large	GPQA-D	0.695	0.773	[0.714, 0.828]	0.927	0.154
olmo_small	GPQA-D	0.550	0.745	[0.691, 0.794]	0.863	0.118
olmo_large	GPQA-D	0.613	0.800	[0.757, 0.850]	0.895	0.094
qwen_small_nothink	GPQA-D	0.508	0.735	[0.682, 0.789]	0.838	0.103
qwen_large_nothink	GPQA-D	0.574	0.772	[0.723, 0.822]	0.876	0.104

Appendix

Table 12: Agreement AUROC with its 95% cluster-bootstrap interval against the guessing ceiling, for all sixteen multiple-choice cells. Accuracy and chance rate are taken from the cleaned answer space. Margin is ceiling minus the point estimate, computed from unrounded values, so it can differ by 0.001 from the difference of the two rounded columns beside it.

model	dataset	$n$	answer	no_ans	unmap	trunc	acc
qwen_small	MATH-500	4000	0.9955	0.0045	0.0000	0.0055	0.9573
qwen_small	AIME	720	0.9222	0.0778	0.0000	0.0917	0.7222
qwen_small	AMC	664	0.9849	0.0151	0.0000	0.0226	0.8961
qwen_small	GPQA-Diamond	1584	0.9987	0.0000	0.0013	0.0019	0.6282
qwen_small	MMLU-Pro	2400	1.0000	0.0000	0.0000	0.0021	0.8333
qwen_large	MATH-500	4000	0.9968	0.0032	0.0000	0.0037	0.9593

Appendix

Table 13: Per-model answer, no-answer, unmappable, truncation and accuracy rates. Every cell answers at or above 0.9287. The four bold cells are the Gemma-3 multiple-choice rows discussed in Section 7 : the model returns a computed value rather than an option letter, and those responses are excluded from the answer space rather than counted as distinct answers. The acc column is over all samples, with unmappable responses scored incorrect; accuracy on the cleaned answer space is acc divided by answer , so gemma_small on GPQA-Diamond reads

0.3586/0.9287=0.386

, the figure the guessing-ceiling calculation uses.

dataset	pair	answer ON	answer OFF	$\Delta$	trunc ON	trunc OFF	verdict
MATH-500	8B	0.9955	0.9922	$+0.0033$	0.0055	0.0088	ok
MATH-500	32B	0.9968	0.9978	$-0.0010$	0.0037	0.0030	ok
AIME	8B	0.9222	0.9403	$-0.0181$	0.0917	0.0681	ok
AIME	32B	0.9403	0.9597	$-0.0194$	0.0792	0.0458	ok
AMC	8B	0.9849	0.9669	$+0.0181$	0.0226	0.0346	ok
AMC	32B	0.9880	0.9834	$+0.0045$	0.0166	0.0181	ok

Appendix

Table 14: Answer-rate parity within the paired comparison. Ten of ten cells fall within a 2% tolerance fixed before the check was run ; the largest divergence is

-0.0194

, on AIME at 32B.

model	reasoning	in-domain length AUROC
gemma_large	no	0.929
olmo_small	yes	0.897
gemma_small	no	0.893
qwen_small	yes	0.868
qwen_large	yes	0.865
olmo_large	yes	0.857

Appendix

Table 15: In-domain length AUROC on MATH-500, ordered by value. Bold marks the two models with no reasoning post-training of any kind; the Qwen3 non-thinking arms are also marked no but share weights with reasoning models.

model	agreement	probe	length	logprob
qwen_small	$+0.122$	$+0.039$	$+0.129$	$-0.141$
qwen_large	$+0.113$	$+0.042$	$+0.111$	$-0.146$
olmo_small	$+0.100$	$-0.084$	$+0.161$	$+0.000$
olmo_large	$-0.007$	$-0.152$	$+0.079$	$-0.027$
qwen_small_nothink	$+0.114$	$+0.053$	$+0.111$	$+0.013$
qwen_large_nothink	$+0.094$	$+0.069$	$+0.127$	$-0.010$

Appendix

Table 16: In-domain minus pooled out-of-domain AUROC, by signal and model. A positive value is degradation.

Figure 5: In-domain minus out-of-domain AUROC, by signal and model. Length is the only signal that degrades in all eight models. This is Table 16 drawn.

model	family	macro AUROC
gemma_small	Gemma-3	0.453
gemma_large	Gemma-3	0.458
qwen_large_nothink	Qwen3	0.517
qwen_large	Qwen3	0.521
qwen_small	Qwen3	0.542
qwen_small_nothink	Qwen3	0.580

Appendix

Table 17: Mean answer-token entropy as a confidence signal, macro-averaged across all five datasets. Bold marks the two models that remain below chance.

	ECE			AUROC
fitted on $\downarrow$	Gemma-3	Olmo-3	Qwen3	Gemma-3	Olmo-3	Qwen3
Gemma-3	0.148	0.343	0.142	0.819	0.837	0.728
Olmo-3	0.264	0.097	0.152	0.813	0.870	0.843
Qwen3	0.189	0.097	0.099	0.806	0.864	0.843

Appendix

Table 18: Cross-family transfer. Left: out-of-domain expected calibration error. Right: out-of-domain AUROC. Rows are the family the selector was fitted on, columns the family it was evaluated on. Ranking mostly transfers: five of six off-diagonal AUROCs lie between 0.806 and 0.864 against diagonals of 0.819 to 0.870, the exception being Gemma-3 fitted and applied to Qwen3 at 0.728. Calibration does not: Gemma-3 fitted, Olmo-3 evaluated gives ECE 0.343 against 0.097 for Olmo’s own, a 3.5-fold degradation.

reasoning
model	probe	entropy	logprob	length	agreement	hedging
qwen_small	$+0.776$	$-0.028$	$\mathbf{-0.176}$	$-0.015$	$+1.018$	$-0.029$
qwen_large	$+0.601$	$+0.047$	$\mathbf{-0.379}$	$+0.218$	$+1.149$	$+0.160$
olmo_small	$+0.492$	$-0.047$	$\mathbf{-0.101}$	$+0.236$	$+1.068$	$+0.019$
olmo_large	$+0.648$	$+0.037$	$\mathbf{-0.249}$	$+0.172$	$+0.830$	$+0.037$
non-reasoning

Appendix

Table 19: Learned selector coefficients: the six signals the selector is fitted on, one from each family, standardized so magnitudes are comparable within a row. Three patterns beyond the sign reversal discussed in Section 8 . Self-consistency dominates every model , which is why a fitted combination barely beats agreement alone (Table 20 ); its weight is also lower in every reasoning model (0.830–1.149) than in every non-reasoning model (1.311–1.970), and roughly doubles within each weight-pair when reasoning is switched off. Both answer entropy and hedging sit near zero throughout : given trace length in the same model, the selector declines to use the hedging count, which is the independent version of Section 7 ’s finding that hedge counts are length in disguise. These are descriptive observations from a single fit per model, not pre-registered tests: neither pattern appears in the scorecard of Table 21 , and both were found after the analysis was fixed. The fit produces point estimates only, so no intervals are reported.

model	selector	agreement	$\Delta$ (all)	$\Delta$ (out-of-domain only)
qwen_small	0.897	0.869	$+0.028$	$+0.028$
qwen_large	0.888	0.887	$+0.000$	$-0.017$
olmo_small	0.899	0.884	$+0.015$	$+0.014$
olmo_large	0.912	0.890	$+0.021$	$+0.030$
qwen_small_nothink	0.892	0.878	$+0.014$	$-0.003$
qwen_large_nothink	0.898	0.898	$-0.001$	$-0.014$

Appendix

Table 20: A fitted six-signal selector against raw agreement, the strongest single signal. The distinction between the last two columns is the result: pooled over all five datasets (which includes the benchmark the selector was fitted on) the selector gains

+0.0118

and improves six of eight models; restricted to the four benchmarks it was not fitted on it gains

-0.0003

, nothing to four decimal places, and loses on five of eight. Both

\Delta

columns are computed from unrounded AUROCs, so a

\Delta

can differ by 0.001 from the difference of the two rounded columns beside it; qwen_large gains by under 0.0005 and is counted among the six.

#	prediction	result
1	Guessing ceiling bounds agreement AUROC	PASS , 16/16, no violations
2	Non-thinking 8B GPQA agreement $\leq 0.834$	PASS , 0.741
3	…and $\approx 0.07$ below the thinking arm’s 0.743	FAIL , moved 0.002
4	MMLU-Pro ceilings near-equal, so any change is real signal	PASS as a test, within 0.006
5	The reasoning arm shows stronger length coupling	FAIL , see Section 7
6	Length replicates on all large models, $>0.75$ on mathematics	PASS , 9/9

Appendix

Table 21: Sixteen pre-registered predictions and their verdicts. Eleven passed; four failed, and one was partial.

Correlated Errors in Language Models

See all 8 papers

How Uncertainty Estimation Scales with Sampling in Reasoning Models
Mar 19, 2026Maksym Del, Markus Kängsepp, Marharyta Domnich +4LLM Uncertainty Estimation Uncertainty Quantification
Institute of Computer Science, University of Tartu
Pause and Reflect: Conformal Aggregation for Chain-of-Thought Reasoning
May 13, 2026Yu Gu, Zijun Yu, Vahid Partovi Nia +1Selective Prediction Self-Consistency Decoding
Department of Mathematics and Statistics McGill University Montréal, QC, Canada · Department of Mathematics and Industrial Engineering Polytechnique de Montréal Montréal, QC, Canada
Reliable Chain-of-Thought via Prefix Consistency
May 8, 2026Naoto Iwase, Yuki Ichihara, Mohammad Atif Quamar +1Self-Consistency Decoding Reasoning Consistency in Language Models
Nagoya University · Nara Institute of Science and Technology · Mohamed bin Zayed University of Artificial Intelligence +1

Mar 19, 2026J/K move · Enter open · S save

How Uncertainty Estimation Scales with Sampling in Reasoning Models

Maksym Del, Markus Kängsepp, Marharyta Domnich, Ardi Tampuu+3

Institute of Computer Science, University of Tartu

LLM Uncertainty Estimation Uncertainty Quantification Confidence Estimation in Language Models Confidence Estimation Large Reasoning Models Verbalized Confidence Estimation+1

Open paper AskPDF arXiv

Uncertainty estimation is critical for deploying reasoning language models, yet remains poorly understood under extended chain-of-thought reasoning. We study parallel sampling as a fully black-box approach using verbalized confidence and self-consistency. Across three reasoning models and 17 tasks spanning mathematics, STEM, and humanities, we characterize how these signals scale. Both self-consistency and verbalized confidence scale in reasoning models, but self-consistency exhibits lower initial discrimination and lags behind verbalized confidence under moderate sampling. Most uncertainty gains, however, arise from signal combination: with just two samples, a hybrid estimator improves AUROC by up to

+12

on average and already outperforms either signal alone even when scaled to much larger budgets, after which returns diminish. These effects are domain-dependent: in mathematics, the native domain of RLVR-style post-training, reasoning models achieve higher uncertainty quality and exhibit both stronger complementarity and faster scaling than in STEM or humanities.

Abstract

Correlated Errors in Language Models

Explore similar work

How Uncertainty Estimation Scales with Sampling in Reasoning Models

Pause and Reflect: Conformal Aggregation for Chain-of-Thought Reasoning

Reliable Chain-of-Thought via Prefix Consistency