Effectiveness of Hidden-State Probes for Localizing Reasoning Failures

Shubham Subhnil; Yannis Belkhiter; Erik Miehling; Adam Dahlgren Lindström; Elizabeth M. Daly

cs.LGOct 15, 2026

Effectiveness of Hidden-State Probes for Localizing Reasoning Failures

Authors: Shubham Subhnil, Yannis Belkhiter, Erik Miehling, Adam Dahlgren Lindström, Elizabeth M. Daly

Organizations: Trinity College Dublin, Ireland · IBM Research, Dublin, Ireland · Umeå University, Sweden

Abstract

Verifying the final answer of a long reasoning trace may show that the trace is wrong but it does not distinguish three cases. The problem may lie beyond the model's capability, another attempt would succeed, or one step in an otherwise recoverable trace made the trace fail. Standard hidden-state probes are trained on final-answer correctness, a label that conflates these cases. Monte Carlo rollouts distinguish the cases and locate the failing step at additional inference cost. We test the hidden states of 4B and 8B reasoning models for the failure type and the failing step and compare each probe with the strongest baseline that uses no hidden states. On incorrect 8B traces, the confidence probe, a calibrated correctness probe at a midpoint cutoff, misses 81% of cases in which a single step makes the trace unrecoverable and 47% of cases in which another attempt would succeed. A probe with a different training design over the same states reverses the ordering of these two kinds of failure, so either ordering can be read from those states. At both scales, no single tested probe locates the failing step more accurately than a rule based only on trace length, and at 4B a probe trained to predict that step fails to outperform this rule in any of 10 runs on identical traces. The confidence probe therefore indicates that an answer is wrong, but its failure-type ordering depends on its training design.

Figures & tables

Figure 1: The three failure types (Definition 1 ) on three Qwen3-4B traces from the gold collection. Each panel shows, for one trace, the rollout rates

\hat{p}_{t}

at the evaluated prefixes and the envelope, which is the best reachable value

V^{*}_{t}

at each step. For a capability gap, the envelope is at or below

\theta=0.1

from step 0. For a pit , the envelope first falls to

\theta

or below partway through the trace, at the pit step. For a near miss, the envelope stays above

\theta

at every step.

Figure 2: Failure types and pit steps in the broad collection at both model scales (Table 9 ). Left: Shares of capability gaps, pit s and near misses among incorrect traces. Center: The relative depth of the pit step, the pit step divided by the number of steps in the trace, with the medians marked. The late median depths of the pit step motivate a baseline that predicts a late step. Right: The best reachable value immediately before the pit step,

V^{*}_{t_{\textsc{pit}}-1}

, with the share above

0.25

marked.

V^{*}_{t_{\textsc{pit}}-1}

depends on later sampled states, so a probe that reads only the steps generated so far need not determine it (Section 4 ).

Figure 3: Confidence-probe predictions at the final step, on the broad collection at both model scales, with training and validation split by problem. (A) The miss rates for capability gaps, pit s, and near misses. (B) The miss rates of pit s and near misses within four bins of the best reachable value at step 0,

V^{*}_{0}

, at 4B (solid) and 8B (dashed); the star marks the miss rate of capability gaps at their mean

V^{*}_{0}

. The incorrect traces differ between scales, so no comparison between scales is paired. Bands and bars are 95% problem-clustered bootstrap intervals at both scales. (C) AUROC of pit s against near misses: the confidence probe ranks pit s above near misses and the combined probe ranks pit s below near misses, each by its final-step score on its own held-out traces. Table 11 gives the values, intervals and trace counts.

Figure 4: Localization at 4B. Left: Hit@

\pm 2

on the reference pit steps for the five pointer probes trained without any problem that contributes a gold trace. Vertical lines mark the uniform-step baseline (dotted, no interval) and two fits of the length rule: the fit on broad-collection pit s outside the gold problems (solid, interval shaded), with which each pointer probe is paired, and the fit by five-fold cross-validation within the gold collection (dashed). Bars and the shaded band are 95% bootstrap intervals. Right: AUROC of the distance score for the pit step against the other steps of the trace, by layer, on 76 pit traces. Points are midpoints and bars the range over four distance measures, not confidence intervals. The horizontal line is the detection AUROC of the confidence probe, a different task, shown for reference.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Claim	Supporting evidence	Verdict	Nearest prior work	Contribution
Best reachable value partitions incorrect traces into capability gaps, pit s, and near misses	$0.56/0.39/0.05$ (4B) and $0.53/0.38/0.09$ (8B); recoverable $44.5$ and $46.8\%$ ; the shares change by less than two percentage points for $\theta$ between $0.05$ and $0.10$ , while $\theta{=}0.15$ raises the share of capability gaps to $0.68$ and $0.65$	Descriptive result under one decoding setting	[ Hwang et al., 2024 , Setlur et al., 2024 , Li, 2026 , Ahdritz et al., 2024 ]	Separates pit s from near misses among recoverable traces
The confidence probe predicts final correctness and remains calibrated	Detection AUROC $0.814$ and $0.809$ over 5 seeds; after controls $0.729$ and $0.722$ ; ECE $0.014$ / $0.024$ (Table 13 ); the text baseline reaches $0.7783$ at 4B and $0.7725$ at 8B	Supported for prediction and calibration; the probe exceeds the text baseline by less than the margin	[ Azaria and Mitchell, 2023 , Kadavath et al., 2022 , Yuan et al., 2025 , Yuan et al., 2026 ]	A step-level probe with a causal attention mask and surface-feature controls
Miss rates at the cutoff differ by failure type	Miss rates $0.717/0.714/0.490$ (4B) and $0.797/0.815/0.466$ (8B); difference between pit s and near misses after controls $+0.037$ and $+0.125$ (Table 11 )	Supported at 8B; partial at 4B , where the difference between pit s and near misses after controls is nonzero, its interval $[+0.012,+0.064]$ does not reach the minimum effect, and the confidence probe trained with the Brier loss gives $+0.052$	[ Yuan et al., 2026 ]	Breaks down misses by failure type
Probes with different training designs recover opposite failure-type orderings	AUROC of pit s against near misses: confidence probe $0.67/0.76$ , combined probe $0.36/0.40$ (4B/8B), each on its own held-out traces	Opposite orderings under partially matched training designs; the contribution of the target alone is open	[ Yu et al., 2024 , Lu et al., 2024 ]	Compares two training designs on the same hidden states at the same read site
A failure-type classifier on transcoder activations of one completed incorrect trace per problem separates capability-gap problems from pit problems	Difference from the strongest tested baseline that uses no hidden states $+0.055$ (8B) and $+0.049$ (4B), mean over six repetitions of five-fold cross-validation; every interval excludes zero and contains the margin; the same classifier on hidden states scores lower (Appendix B.1 )	Meets the margin at 8B, not at 4B; a difference between the scales is not established; applies after a trace has ended	[ Orgad et al., 2025 , Zhang et al., 2025b , Kantamneni et al., 2025 ]	Separates capability gaps from pit s; the confidence probe’s miss rates do not separate these two types
No single tested probe exceeds the length rule at locating the pit step; at 8B only the value probe was tested	Pointer probe $0.463$ versus length rule $0.469$ , above the length rule in $0$ of $10$ runs; pointer probe trained without gold problems $0.429$ versus $0.446$ , $0$ of $5$ ; ensemble $+0.026$ $[+0.002,+0.050]$ with inverse-frequency weights on the failure-type loss, interval containing zero with uniform weights; final-label comparison: the value probe and the final-label probe are below the length rule in every seed on the problems held out from the value probe’s training; pooled over all reference pit steps, which include training traces, the two probes score above the length rule in $0$ of $20$ runs (Table 12 ); 8B value probe with the two input sets $0.429$ and $0.405$ versus length rule $0.503$ , above the length rule in $0$ of $5$ seeds for each input set (Table 10 )	Negative result at 4B, paired on identical pit s, and at 8B, compared through seed means against the pit steps of the broad collection. Earlier configurations evaluated on $55$ to $134$ pit s are inconclusive	[ Zheng et al., 2025 , Alvarez and Baheri, 2026 , Zhao et al., 2026b , Cui et al., 2026 , Nadaf, 2026 ]	Uses a continuation-based pit -step target and refits the length rule for each evaluation set

Appendix

Table 1: Claims, supporting evidence, verdicts, nearest prior work, and contributions. Among the reviewed studies, none combines a three-way recovery taxonomy, probes on hidden states, and a continuation-based pit -step target.

	broad collection		gold collection
	4B	8B	4B
evaluated steps	0, 1, 2, then every second step
rollouts per prefix in the scan	8	8	4
rollouts per prefix in the refinement	16	16	8
evaluated prefixes with the scan’s rollouts	$82.3\%$	$83.3\%$	$81.4\%$
evaluated prefixes with the refinement’s rollouts	$17.7\%$	$16.7\%$	$18.6\%$

Appendix

Table 2: The two collections of rollouts. A dash marks a quantity that was not computed for the gold collection. The refinement re-evaluates some prefixes with more rollouts than the scan.

model	target	training traces	mask	scales
confidence probe	final label $Y$ , broadcast to every step	all $144{,}000$ traces	causal	4B, 8B
value probe	rollout rate $\hat{p}_{t}$ at each step	recoverable traces	bidirectional	4B, 8B
combined probe	$Y$ on correct traces, $\hat{p}_{t}$ on incorrect ones	$143{,}979$ traces (21 incorrect traces without rollout rates dropped)	bidirectional	4B, 8B
pointer probe	pit step; further outputs, including the failure type	$22{,}877$ incorrect traces of the broad collection, of which $8{,}971$ pit s	bidirectional	4B
failure-type classifier	failure type of the problem, capability gap or pit	one scored trace per problem	—	4B, 8B

Appendix

Table 3: Training configuration of every probe and of the failure-type classifier. The four probes read hidden states at the read site (Appendix D ); the failure-type classifier reads transcoder activations at layers 19 to 24 (Appendix B.1 ). The pointer probe was trained at 4B only.

	4B	8B
pre-norm encoder layers	4
width	512
attention heads	8
maximum number of steps	48
input dimension per step	$5{,}120$	$8{,}192$
parameters	15.5M	17.1M

Appendix

Table 4: Architecture and training settings of the confidence probe. A value spanning both columns applies at both scales.

	4B	8B
detection AUROC	$0.8165$	$0.8165$
AUROC after controls	$0.7322$	$0.7267$
ECE	$0.027$	$0.019$
miss rate, capability gap	$0.717$	$0.795$
miss rate, pit	$0.742$	$0.823$
miss rate, near miss	$0.469$	$0.522$

Appendix

Table 5: The confidence probe under the trace split, the leakage comparison. Table 13 and Table 11 give the same quantities under the problem split.

labels	scale	problems	classifier	hidden states	strongest baseline	difference	repetitions $\geq 0.05$
stored	4B	1,722	0.667	0.580	0.618	$+0.0488\pm 0.0071$	3 of 6
stored	8B	6,372	0.721	0.646	0.666	$+0.0549\pm 0.0026$	6 of 6
defects removed	4B	1,294	0.643	0.581	0.578	$+0.0648\pm 0.0074$	6 of 6
defects removed	8B	5,078	0.700	0.638	0.617	$+0.0830\pm 0.0049$	6 of 6

Appendix

Table 6: Failure-type classifier on one completed incorrect trace per problem, capability-gap problems against pit problems. AUROCs are means over six repetitions of five-fold cross-validation; the difference is the classifier’s AUROC minus that of the strongest baseline that uses no hidden states, mean

\pm

standard deviation over repetitions. The hidden-state column is the same classifier on hidden states at the same layers. The last two rows remove the problems whose scored trace carries a grading defect and, at 4B, those whose scored trace has a different failure type from its label.

repetition	4B	8B
1	$+0.0364$ [ $+0.0040$ , $+0.0690$ ]	$+0.0581$ [ $+0.0468$ , $+0.0698$ ]
2	$+0.0558$ [ $+0.0278$ , $+0.0845$ ]	$+0.0550$ [ $+0.0426$ , $+0.0667$ ]
3	$+0.0505$ [ $+0.0218$ , $+0.0814$ ]	$+0.0537$ [ $+0.0413$ , $+0.0671$ ]
4	$+0.0491$ [ $+0.0189$ , $+0.0794$ ]	$+0.0553$ [ $+0.0425$ , $+0.0675$ ]
5	$+0.0463$ [ $+0.0158$ , $+0.0797$ ]	$+0.0506$ [ $+0.0377$ , $+0.0637$ ]
6	$+0.0547$ [ $+0.0222$ , $+0.0879$ ]	$+0.0568$ [ $+0.0442$ , $+0.0686$ ]

Appendix

Table 7: Difference between the failure-type classifier and the strongest baseline that uses no hidden states, per repetition of five-fold cross-validation, with stored labels. Each cell is the AUROC difference with its 95% problem-clustered bootstrap interval.

Figure 5: Read-site selection for Qwen3-4B and Qwen3-8B. (A) Detection AUROC of a probe at each layer, on hidden states and on transcoder activations (not used by any result in this paper; see the text of this appendix); on hidden states it forms a broad upper-middle-layer plateau and peaks at layer 23. (B) Detection AUROC of a probe on 2, 3, 5 and 8 adjacent layers centred on layer 23; the gain from a third layer over two is smaller than the interval width. Bands and bars are 95% bootstrap intervals that resample held-out traces. We select hidden states from

\mathcal{S}^{\star}=\{23,24\}

DeepMath topics
problem type	4B problems	4B share	8B problems	8B share
Algebra	4,136	0.563 [0.549, 0.577]	3,728	0.537 [0.522, 0.552]
Calculus	2,564	0.511 [0.494, 0.530]	2,256	0.477 [0.457, 0.497]
Other	928	0.542 [0.513, 0.572]	826	0.510 [0.478, 0.541]
Discrete Mathematics	890	0.603 [0.573, 0.632]	801	0.599 [0.564, 0.627]
Geometry	856	0.555 [0.524, 0.584]	769	0.535 [0.506, 0.569]

Appendix

Table 8: Capability-gap share of incorrect traces by problem type in the broad collection at both model scales, with the stored labels and

95\%

problem-clustered bootstrap intervals. The pooled shares are

0.5555

at 4B and

0.5321

at 8B. DeepMath topics and Hendrycks MATH subjects are listed separately because the two sources differ in difficulty scale.

set	population	used for	4B	8B
all traces	every sampled trace (35,854 problems)	training the confidence and combined probes	144,000	144,000
held-out traces	traces whose problems were held out by at least one of the five training seeds	detection and calibration of the confidence probe	119,736	119,756
broad collection	every incorrect trace	shares of the failure types (Figure 2 )	28,699	24,871
held-out incorrect traces	incorrect traces among the held-out traces	miss rates by failure type (Section 3 )	23,837	20,590
held-out pit s	pit s of problems outside the training split	pointer probe (4B); value probe, per seed (8B)	2,289	2,805–2,918
classifier problems	one incorrect trace per problem, capability gaps and pit s	failure-type classifier (Appendix B.1 )	1,722	6,372

Appendix

Table 9: Sizes of the trace sets and of the populations drawn from them, and the use of each population. The broad collection labels every incorrect trace at both model scales. The gold collection is a 4B collection with half as many rollouts per evaluated prefix, and the pit steps of its pit s are the reference pit steps. A later table reports a separate

n

when its evaluation population differs from the corresponding row here.

measurement	4B	8B	interpretation
Failure types and detection
detection AUROC of the read site a	0.7245 [0.7047, 0.7432]	0.7149 [0.6960, 0.7326]	read site chosen from the final label only
share of pit s whose pit step equals $\arg\min_{t}\hat{V}_{t}$	0.154	0.125	the $\arg\min$ step equals the pit step for a minority of pit s
combined probe $-$ confidence probe, detection AUROC on held-out problems	$-0.026$ [ $-0.029$ , $-0.022$ ]	$-0.025$ [ $-0.029$ , $-0.021$ ]	the confidence probe has the higher detection AUROC
confidence probe’s predicted correctness at the pit step minus other steps b	$-0.0206$ [ $-0.0272$ , $-0.0140$ ]	$-0.0351$ [ $-0.0372$ , $-0.0327$ ]	lower at the pit step
same difference against the position-matched steps b	$-0.0018$ [ $-0.0055$ , $+0.0022$ ]	$-0.0051$ [ $-0.0062$ , $-0.0040$ ]	8B: the decrease remains; 4B: no detected difference

Appendix

Table 10: Supporting measurements cited in the main text, at both model scales. Intervals are 95% bootstrap intervals that resample problems unless a note says otherwise, and

\pm

values are sample standard deviations over seeds (seed axes in Appendix B ).

quantity	4B	8B
miss rate, capability gap	$0.717\ [0.706,0.729]$	$0.797\ [0.785,0.808]$
miss rate, pit	$0.714\ [0.703,0.724]$	$0.815\ [0.805,0.824]$
miss rate, near miss	$0.490\ [0.460,0.521]$	$0.466\ [0.442,0.490]$
$n$ (incorrect traces)	23,837	20,590
$V^{*}_{0}\leq 0.25$ , pit s	$0.707[0.690,0.723]$ (3,160)	$0.787[0.770,0.804]$ (2,548)
near misses	$0.436[0.329,0.547]$ (78)	$0.449[0.357,0.549]$ (118)

Appendix

Table 11: Values behind Figure 3 : confidence-probe predictions at the final step, on the broad collection at both model scales, on held-out problems of the problem split. An incorrect trace is missed when its predicted correctness is at least the cutoff,

0.5

. The binned rows give the miss rates of pit s and near misses within bins of the best reachable value at step 0,

V^{*}_{0}

, with the number of traces in each bin in parentheses. The AUROC rows score each probe by its final-step score on its own held-out traces. The last block is the difference between pit s and near misses after controls, from a logistic model of trace misses on failure type, trace length, answer position, difficulty and

V^{*}_{0}

(Appendix B ). Intervals are 95% problem-clustered bootstraps.

comparison	pit s	probe	length rule	difference	runs above rule
pointer probe, held-out pit s of the broad collection a	2,289	$0.4632\pm 0.0134$	0.4692	$-0.0060$ [ $-0.0260$ , $+0.0141$ ]	0 of 10
pointer probe trained without gold problems	818	$0.4293\pm 0.0120$	0.4462	$-0.0169$ [ $-0.0473$ , $+0.0131$ ]	0 of 5
final-label comparison, held-out: value probe b	205–242	0.3640; 0.3402	0.4812	$-0.166$ to $-0.084$ ; $-0.182$ to $-0.113$	0 of 10
final-label comparison, held-out: final-label probe b	205–242	0.3041; 0.3423	0.4812	$-0.216$ to $-0.124$ ; $-0.170$ to $-0.116$	0 of 10
final-label comparison, pooled, training traces included c	818	0.3186–0.3795	0.4462	$-0.128$ to $-0.067$	0 of 20

Appendix

Table 12: Paired comparisons with the length rule, at 4B throughout. Each comparison scores the probe and the length rule on the same pit s; intervals are 95% problem-clustered bootstraps of the per-trace difference where given.

scale	loss	AUROC	AUROC after controls	ECE	ECE after scaling
4B	BCE (main)	$0.8136\pm 0.0049$	$0.7291\pm 0.0042$	$0.0143\pm 0.0040$	$0.0103$
8B	BCE (main)	$0.8088\pm 0.0084$	$0.7221\pm 0.0149$	$0.0243\pm 0.0111$	$0.0092$
4B	Brier (robustness)	$0.8107\pm 0.0046$	$0.7297\pm 0.0036$	$0.0273\pm 0.0103$	$0.0215$
8B	Brier (robustness)	$0.8085\pm 0.0047$	$0.7345\pm 0.0029$	$0.0229\pm 0.0061$	$0.0107$

Appendix

Table 13: Detection by the confidence probe at the final step and comparison with the text baseline, on held-out problems of the problem split at both model scales. The upper block reports the mean

\pm

standard deviation over five seeds. AUROC after controls scores the same five-seed predictions after the five control variables of Appendix B are removed from them. Expected calibration error uses ten equal-width bins on held-out scores, before and after temperature scaling. The lower block gives the held-out detection AUROC of the text baseline and of a logistic regression on the same features, the paired difference between the confidence probe and the text baseline, and the gain from adding the confidence probe’s score to the text baseline’s features; the lower block’s intervals are 95% problem-clustered bootstraps. Appendix B reports the trace split as a leakage comparison.

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Hidden-state probing reads model activations to study what they represent and whether those signals support monitoring or control.

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When post-trained language models fail on reasoning problems, the common test-time-scaling response is to spend more compute on additional attempts, and the failed traces play no further role. We argue this discards a crucial signal; some failures come from unlucky sampling, where more rollouts help, while others are structural and resist resampling regardless of budget. We propose that failed traces encode recoverability structure: the inference-time signature of which test-time interventions can rescue a given failure. Three problem-level trajectory features, derived from the structure of available interventions, recover this structure from the distributional signature of failed rollouts, not their text. They cluster failures into stable regimes, characterize the failure topography of different post-training methods (

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Abstract

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