On the Tip of the Tongue: Why LLMs Hallucinate Answers They Can Decode

Valeria Ruscio; Keiran Thompson

cs.AIMar 14, 2026

On the Tip of the Tongue: Why LLMs Hallucinate Answers They Can Decode

Authors: Valeria Ruscio, Keiran Thompson

Organizations: Intuition Machines Inc.

Abstract

A language model can give the wrong answer even when the correct answer is decodable from its intermediate states. To study this gap between decodability and selection, we distinguish read from write at the first answer token. Read asks whether the gold token can be decoded from intermediate residual states under same-relation decoy controls. Write asks whether the final readout ranks that token first among content tokens. Under three different readers, with a randomized-label control, a substantial fraction of failures remain readable while another content token is selected. We explain this through the selection margin at the final readout, the difference between the answer logit and the logit of its strongest alternative, which is answer support minus alternative support, and can also be split into a context-averaged baseline linked to token frequency and an item-specific term. Setting the answer support to the level typical of successful generations is sufficient to recover first-token selection for the majority of failures in most of the models we study; the original alternative remains ahead in most remaining failures under this edit, and this outcome follows directly from the readout geometry. Removing the frequency direction alone shifts selection but rarely recovers the answer. Prompt variants of the same fact that succeed supply support that transfers to failing variants through the residual stream and through late MLP outputs, with less consistent effects through late attention. First-token recovery leaves most full answers wrong, which limits the recovery achieved by these edits and separates three things that are easily conflated, decodability, recoverability, and generation.

Figures & tables

Model	$n_{F}$	Logit	Tuned	Probe	Shuffled	Consensus
Llama-3.1-8B	358	36.0%	42.5%	30.2%	12.6%	13.4%
Llama-3.2-1B	423	27.2%	38.5%	28.1%	13.9%	10.4%
Llama-3.2-3B	383	41.8%	41.5%	27.2%	13.3%	14.9%
Llama-3.2-3B-I	399	37.3%	39.3%	27.8%	11.8%	12.8%
Qwen2.5-3B	409	23.2%	33.3%	30.8%	11.2%	9.8%
Qwen2.5-3B-I	426	25.6%	37.8%	30.3%	10.6%	10.8%

Table 1: Hard-decoy readable fractions among content-selection failures under three readers and the randomized-label control. Consensus requires logit, tuned and trained-probe readability. I denotes an instruction-tuned checkpoint;

n_{F}

is the number of failures among 500 records. Intervals and overlaps are in Appendix C .

Model	$n_{F}$	A	B	A $\to$ B	Random
Llama-3.1-8B	358	84.9% [80.8%, 88.3%]	7.3%	90.5% [87.0%, 93.1%]	0.6%
Llama-3.2-1B	423	83.0% [79.1%, 86.3%]	4.3%	85.6% [81.9%, 88.6%]	0.5%
Llama-3.2-3B	383	83.0% [78.9%, 86.5%]	5.2%	90.9% [87.6%, 93.4%]	0.3%
Llama-3.2-3B-I	399	92.2% [89.2%, 94.5%]	3.5%	96.7% [94.5%, 98.1%]	0.0%
Qwen2.5-3B	409	90.7% [87.5%, 93.2%]	2.9%	94.6% [92.0%, 96.4%]	0.5%
Qwen2.5-3B-I	426	90.6% [87.5%, 93.0%]	2.6%	95.3% [92.9%, 96.9%]	0.0%

Table 2: First-token recovery under median-calibrated projection edits, on all failures. A sets the answer support to the median of successes, B sets the alternative support to the median alternative support of successes, and A

\to

B applies both. Random matches the norm of A. Brackets are 95% Wilson intervals.

Model	$n_{F}$	TP	FN	TN	Excess $>0$	Switches
Llama-3.1-8B	358	54	0	304	15.1%	0.0%
Llama-3.2-1B	423	68	4	351	16.1%	3.3%
Llama-3.2-3B	383	63	2	318	16.4%	1.0%
Llama-3.2-3B-I	399	29	2	368	7.3%	0.5%
Qwen2.5-3B	409	37	1	371	9.0%	0.5%
Qwen2.5-3B-I	426	39	1	386	9.2%	0.7%

Table 3: The alternative-excess criterion of equation 7 at the median target. A positive excess predicts that answer setting alone fails; TP, FP, FN and TN count agreement between the prediction and the offline outcome. Switches count a third token winning after the edit, whether or not the original alternative also remains above the gold token; they need not equal FN.

Model	$n$	R / U recovery	Difference [95% CI]	Edit-norm difference [95% CI]
Llama-3.1-8B	129	75.2% / 91.5%	$-16.3$ [ $-25.6$ , $-7.8$ ]	$-2.05$ [ $-2.62$ , $-1.48$ ]
Llama-3.2-1B	114	76.3% / 83.3%	$-7.0$ [ $-18.4$ , $3.5$ ]	$-0.79$ [ $-1.31$ , $-0.24$ ]
Llama-3.2-3B	161	80.1% / 87.0%	$-6.8$ [ $-14.9$ , $1.2$ ]	$-1.55$ [ $-1.90$ , $-1.22$ ]
Llama-3.2-3B-I	148	87.2% / 96.6%	$-9.5$ [ $-15.5$ , $-3.4$ ]	$-2.09$ [ $-2.54$ , $-1.65$ ]
Qwen2.5-3B	95	83.2% / 91.6%	$-8.4$ [ $-16.8$ , $0.0$ ]	$-0.95$ [ $-1.40$ , $-0.47$ ]
Qwen2.5-3B-I	108	85.2% / 93.5%	$-8.3$ [ $-16.7$ , $-0.9$ ]	$-1.71$ [ $-2.59$ , $-0.94$ ]

Table 4: Readable (R) versus unreadable (U) failures under the logit reader, matched one-to-one on the initial answer–alternative margin, at the median answer-setting edit. Differences are R minus U with bootstrap 95% intervals; the norm difference is the difference in distance to the calibration target, not the minimum edit needed for recovery.

Model	Negative baseline	Mean margin	$r_{\mathrm{freq}}$	Freq. recovery	Random
Llama-3.1-8B	67.0%	$-1.242$ [ $-1.526$ , $-0.937$ ]	0.567	2.5% [1.3%, 4.7%]	0.8%
Llama-3.2-1B	77.8%	$-1.679$ [ $-1.923$ , $-1.421$ ]	0.568	1.9% [1.0%, 3.7%]	1.4%
Llama-3.2-3B	71.3%	$-1.335$ [ $-1.577$ , $-1.098$ ]	0.570	3.4% [2.0%, 5.7%]	2.1%
Llama-3.2-3B-I	69.9%	$-1.053$ [ $-1.293$ , $-0.810$ ]	0.554	3.5% [2.1%, 5.8%]	0.5%
Qwen2.5-3B	76.8%	$-1.828$ [ $-2.099$ , $-1.552$ ]	0.561	2.2% [1.2%, 4.1%]	2.0%
Qwen2.5-3B-I	67.6%	$-1.519$ [ $-1.864$ , $-1.170$ ]	0.503	1.9% [1.0%, 3.7%]	0.9%

Table 5: All-record baseline evaluated on failures, and frequency-direction ablation. Negative baseline is the fraction of failures whose baseline margin favors the selected alternative;

r_{\mathrm{freq}}

is the correlation between baseline logits and token log frequency; frequency recovery removes the frequency direction from the final state and random applies an equal-norm perturbation. Brackets are 95% intervals.

Model	$n$	Same fact	Different fact	Paired difference	$p_{\mathrm{Holm}}$
Llama-3.1-8B	45	88.9%	2.2%	86.4% [75.0%, 95.5%]	$<0.001$
Llama-3.2-1B	28	75.0%	0.0%	74.1% [55.6%, 88.9%]	$<0.001$
Llama-3.2-3B	31	83.9%	0.0%	83.9% [71.0%, 96.8%]	$<0.001$
Llama-3.2-3B-I	22	86.4%	0.0%	86.4% [72.7%, 100.0%]	$<0.001$
Qwen2.5-3B	18	55.6%	5.6%	50.0% [22.2%, 77.8%]	0.059
Qwen2.5-3B-I	25	80.0%	12.0%	68.0% [47.9%, 88.0%]	$<0.001$

Table 6: Residual transfer at the entrance to the late band, from a successful prompt variant of the same question (same fact) or of a different question (different fact), followed by a forward pass through the remaining layers.

n

counts record-level pairs for the arm rates. Paired tests use distinct questions: 44, 27, 31, 22, 18, 25, 20 and 33 in row order. Differences are percentage points with bootstrap 95% intervals and Holm-adjusted exact McNemar

p

values.

Model	Base full	A first	A full	A $\to$ B first	A $\to$ B full
Llama-3.1-8B	28.0%	84.6%	38.0% [33.1%, 43.1%]	90.5%	36.9% [32.0%, 42.0%]
Llama-3.2-1B	14.2%	82.7%	30.5% [26.3%, 35.0%]	85.3%	30.5% [26.3%, 35.0%]
Llama-3.2-3B	22.6%	82.8%	33.7% [29.1%, 38.6%]	90.9%	36.8% [32.1%, 41.8%]
Llama-3.2-3B-I	19.2%	92.5%	35.1% [30.6%, 39.9%]	97.0%	36.3% [31.8%, 41.2%]
Qwen2.5-3B	17.8%	90.5%	34.2% [29.8%, 39.0%]	94.4%	35.5% [31.0%, 40.2%]
Qwen2.5-3B-I	13.8%	89.9%	31.2% [27.0%, 35.8%]	94.8%	34.0% [29.7%, 38.7%]

Table 7: Greedy full-answer accuracy on all 500 records, and recovery under the median edits applied during generation on failures. First-token recovery uses the canonical token; full-answer correctness accepts any alias. Brackets are 95% Wilson intervals.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Full accuracy	Canonical	Any alias	$n_{S}$	$n_{P}$	Flips
Llama-3.1-8B	28.0% [24.2%, 32.1%]	28.4%	33.2%	124	18	10
Llama-3.2-1B	14.2% [11.4%, 17.5%]	15.4%	18.2%	62	15	10
Llama-3.2-3B	22.6% [19.2%, 26.5%]	23.4%	27.8%	101	16	12
Llama-3.2-3B-I	19.2% [16.0%, 22.9%]	20.2%	23.8%	81	20	16
Qwen2.5-3B	17.8% [14.7%, 21.4%]	18.2%	22.8%	76	15	13
Qwen2.5-3B-I	13.8% [11.1%, 17.1%]	15.0%	18.8%	57	18	16

Appendix

Table 8: Evaluation and calibration populations. Full, canonical-first and any-alias-first accuracy use 500 records;

n_{S}

counts successes (full answer correct and canonical first token),

n_{P}

counts records whose first token is canonical but whose full answer is wrong, and flips counts records whose label differs between the submitted and the corrected matcher.

Model	Corrected top-1	Submitted top-1	Corrected corr.	Submitted corr.	Protocol gate [95% CI]
Llama-3.1-8B	1.000	0.720	1.000	0.970	99.3% [96.1%, 99.9%]
Llama-3.2-1B	0.960	0.780	1.000	0.986	98.6% [92.4%, 99.8%]
Llama-3.2-3B	0.980	0.840	1.000	0.988	100.0% [96.7%, 100.0%]
Llama-3.2-3B-I	0.960	0.880	1.000	0.988	97.9% [92.7%, 99.4%]
Qwen2.5-3B	0.980	0.160	1.000	0.956	100.0% [95.9%, 100.0%]
Qwen2.5-3B-I	0.980	0.560	1.000	0.954	98.6% [92.2%, 99.7%]

Appendix

Table 9: Logit reconstruction on 50 records: normalization once (corrected) versus twice (submitted), top-1 agreement with the native forward pass and mean logit correlation, and the protocol gate.

Model	$n_{F}$	Logit	Tuned	Probe	Consensus
Llama-3.1-8B	358	36.0% [31.2%, 41.1%]	42.5% [37.4%, 47.6%]	30.2% [25.6%, 35.1%]	13.4% [10.3%, 17.3%]
Llama-3.2-1B	423	27.2% [23.2%, 31.6%]	38.5% [34.0%, 43.3%]	28.1% [24.1%, 32.6%]	10.4% [7.8%, 13.7%]
Llama-3.2-3B	383	41.8% [36.9%, 46.8%]	41.5% [36.7%, 46.5%]	27.2% [22.9%, 31.8%]	14.9% [11.7%, 18.8%]
Llama-3.2-3B-I	399	37.3% [32.7%, 42.2%]	39.3% [34.7%, 44.2%]	27.8% [23.6%, 32.4%]	12.8% [9.9%, 16.4%]
Qwen2.5-3B	409	23.2% [19.4%, 27.6%]	33.3% [28.9%, 38.0%]	30.8% [26.5%, 35.4%]	9.8% [7.3%, 13.0%]
Qwen2.5-3B-I	426	25.6% [21.7%, 29.9%]	37.8% [33.3%, 42.5%]	30.3% [26.1%, 34.8%]	10.8% [8.2%, 14.1%]

Appendix

Table 10: Reader-specific hard-decoy readable fractions among failures, with 95% Wilson intervals.

Model	$n_{F}$	Trained only	Shuffled only	Difference [95% CI]	L/T	L/P	T/P
Llama-3.1-8B	358	91	28	17.6% [11.7%, 23.2%]	0.471	0.288	0.469
Llama-3.2-1B	423	99	39	14.2% [9.0%, 19.4%]	0.397	0.251	0.431
Llama-3.2-3B	383	86	33	13.8% [8.6%, 18.8%]	0.587	0.300	0.437
Llama-3.2-3B-I	399	97	33	16.0% [10.8%, 21.3%]	0.530	0.300	0.374
Qwen2.5-3B	409	109	29	19.6% [14.4%, 24.7%]	0.383	0.277	0.456
Qwen2.5-3B-I	426	111	27	19.7% [14.6%, 24.4%]	0.357	0.293	0.443

Appendix

Table 11: Trained versus randomized-label probe on failures, and Jaccard overlaps between readable sets. Differences are percentage points with bootstrap 95% intervals; all paired exact tests give

p<0.001

. L, T and P denote the logit, tuned and trained-probe readers.

Model	Target	A	B	A $\to$ B	Random
Llama-3.1-8B	p25	63.7% [58.6%, 68.5%]	10.1% [7.4%, 13.6%]	74.9% [70.1%, 79.1%]	0.6% [0.2%, 2.0%]
	p50	84.9% [80.8%, 88.3%]	7.3% [5.0%, 10.4%]	90.5% [87.0%, 93.1%]	0.6% [0.2%, 2.0%]
	p75	94.4% [91.5%, 96.4%]	3.4% [1.9%, 5.8%]	97.8% [95.7%, 98.9%]	0.3% [0.0%, 1.6%]
	mean	86.0% [82.1%, 89.2%]	7.3% [5.0%, 10.4%]	92.5% [89.2%, 94.8%]	0.6% [0.2%, 2.0%]
Llama-3.2-1B	p25	31.9% [27.7%, 36.5%]	7.3% [5.2%, 10.2%]	44.0% [39.3%, 48.7%]	0.2% [0.0%, 1.3%]
	p50	83.0% [79.1%, 86.3%]	4.3% [2.7%, 6.6%]	85.6% [81.9%, 88.6%]	0.5% [0.1%, 1.7%]

Appendix

Table 12: Offline first-token recovery across calibration targets. Brackets are 95% Wilson intervals.

Model	$n$	Paired difference [95% CI]	$p_{\mathrm{Holm}}$
Llama-3.1-8B	355	5.1% [1.7%, 8.7%]	0.013
Llama-3.2-1B	419	2.6% [ $-0.5$ %, 5.7%]	0.135
Llama-3.2-3B	380	7.6% [4.5%, 10.8%]	$<0.001$
Llama-3.2-3B-I	396	4.5% [2.5%, 6.8%]	$<0.001$
Qwen2.5-3B	406	3.9% [1.5%, 6.7%]	0.013
Qwen2.5-3B-I	423	4.7% [2.6%, 6.9%]	$<0.001$

Appendix

Table 13: Combined minus answer-only recovery at the median target, paired by question. Differences are percentage points with bootstrap 95% intervals; Holm correction covers the four targets within each model.

Model	$n_{L}$	Logit	$n_{T}$	Tuned	$n_{P}$	Probe	$n_{C}$	Consensus
Llama-3.1-8B	129	75.2% [67.1%, 81.8%]	151	77.5% [70.2%, 83.4%]	108	75.0% [66.1%, 82.2%]	47	70.2% [56.0%, 81.3%]
Llama-3.2-1B	114	76.3% [67.7%, 83.2%]	164	75.6% [68.5%, 81.5%]	119	73.9% [65.4%, 81.0%]	44	68.2% [53.4%, 80.0%]
Llama-3.2-3B	161	80.1% [73.3%, 85.6%]	159	77.4% [70.3%, 83.2%]	104	74.0% [64.9%, 81.5%]	58	77.6% [65.3%, 86.4%]
Llama-3.2-3B-I	148	87.2% [80.8%, 91.6%]	157	89.8% [84.1%, 93.6%]	112	90.2% [83.3%, 94.4%]	51	86.3% [74.3%, 93.2%]
Qwen2.5-3B	95	83.2% [74.4%, 89.4%]	136	85.3% [78.4%, 90.3%]	126	88.1% [81.3%, 92.7%]	40	82.5% [68.0%, 91.3%]
Qwen2.5-3B-I	108	85.2% [77.3%, 90.7%]	162	86.4% [80.3%, 90.9%]	130	90.0% [83.6%, 94.1%]	46	91.3% [79.7%, 96.6%]

Appendix

Table 14: Recovery under the median answer-setting edit on the readable subset of each reader, with subset sizes and 95% Wilson intervals.

Model	Reader	$n$	R / U	Difference [95% CI]	Edit-norm difference [95% CI]	Gap
Llama-3.1-8B	logit	129	75.2% / 91.5%	$-16.3$ % [ $-25.6$ %, $-7.8$ %]	$-2.05$ [ $-2.62$ , $-1.48$ ]	0.553
	probe	108	75.0% / 88.0%	$-13.0$ % [ $-22.2$ %, $-3.7$ %]	$-1.34$ [ $-1.91$ , $-0.77$ ]	0.123
Llama-3.2-1B	logit	114	76.3% / 83.3%	$-7.0$ % [ $-18.4$ %, 3.5%]	$-0.79$ [ $-1.31$ , $-0.24$ ]	0.360
	probe	119	73.9% / 83.2%	$-9.2$ % [ $-20.2$ %, 1.7%]	$-0.77$ [ $-1.26$ , $-0.30$ ]	0.336
Llama-3.2-3B	logit	161	80.1% / 87.0%	$-6.8$ % [ $-14.9$ %, 1.2%]	$-1.55$ [ $-1.90$ , $-1.22$ ]	1.337
	probe	104	74.0% / 89.4%	$-15.4$ % [ $-25.0$ %, $-5.8$ %]	$-0.71$ [ $-1.00$ , $-0.42$ ]	0.048

Appendix

Table 15: Readable (R) versus unreadable (U) failures matched on the initial margin, for the logit and probe readers. Differences are R minus U with bootstrap 95% intervals; gap is the mean absolute margin mismatch of the pairs.

Model	$n$	SF	DF	Random	Attn	Attn DF	MLP	MLP DF	Middle	Early
Llama-3.1-8B	45	75.6%	6.7%	77.8%	24.4%	22.2%	64.4%	8.9%	62.2%	37.8%
Llama-3.2-1B	28	75.0%	3.6%	42.9%	46.4%	32.1%	50.0%	3.6%	53.6%	21.4%
Llama-3.2-3B	31	74.2%	6.5%	77.4%	35.5%	35.5%	61.3%	9.7%	74.2%	25.8%
Llama-3.2-3B-I	22	68.2%	0.0%	59.1%	31.8%	22.7%	59.1%	4.5%	72.7%	27.3%
Qwen2.5-3B	18	83.3%	5.6%	50.0%	55.6%	11.1%	55.6%	0.0%	38.9%	22.2%
Qwen2.5-3B-I	25	100.0%	0.0%	60.0%	64.0%	0.0%	84.0%	0.0%	48.0%	32.0%

Appendix

Table 16: First-token recovery under component replacement. SF and DF replace every late attention and MLP output with same-fact and different-fact donor outputs; Random replaces the same number of components sampled across all depths; Attn and MLP replace only the late attention or only the late MLP outputs, with same-fact and different-fact (DF) donors; Middle and Early replace same-size bands.

Model	Comparison	$n$	Difference [95% CI]	$p_{\mathrm{Holm}}$
Llama-3.1-8B	SF vs random	44	$-2.3$ % [ $-20.5$ %, 13.6%]	1.000
	Attn vs DF	44	2.3% [ $-13.6$ %, 18.2%]	1.000
	MLP vs DF	44	54.5% [38.6%, 70.5%]	$<0.001$
Llama-3.2-1B	SF vs random	27	33.3% [14.8%, 55.6%]	0.059
	Attn vs DF	27	14.8% [ $-7.4$ %, 37.0%]	1.000
	MLP vs DF	27	44.4% [22.2%, 66.7%]	0.011

Appendix

Table 17: Paired component-replacement comparisons paired by question string. Differences are percentage points with bootstrap 95% intervals; Holm correction covers the nine recovery comparisons per model.

Model	Control	$n$	Recovery difference [95% CI]	$p_{\mathrm{Holm}}$
Llama-3.1-8B	Different fact	44	84.1% [72.7%, 93.2%]	$<0.001$
	Random layer	44	0.0% [ $-6.8$ %, 6.8%]	1.000
Llama-3.2-1B	Different fact	27	85.2% [70.4%, 96.3%]	$<0.001$
	Random layer	27	22.2% [3.7%, 40.7%]	0.070
Llama-3.2-3B	Different fact	31	77.4% [58.1%, 93.5%]	$<0.001$
	Random layer	31	3.2% [ $-12.9$ %, 19.4%]	1.000

Appendix

Table 18: Residual patching at the layer where the logit lens reads the gold token most strongly, from a same-fact donor against a different-fact donor and against the same donor at a random other band layer. Differences are percentage points, paired by question string.

Model	$n$	Read-direction shift	Random shift	Paired difference
Llama-3.1-8B	60	$-2.449$ [ $-2.847$ , $-2.087$ ]	0.002 [ $-0.031$ , 0.036]	$-2.452$ [ $-2.850$ , $-2.091$ ]
Llama-3.2-1B	60	$-3.157$ [ $-3.532$ , $-2.795$ ]	0.007 [ $-0.047$ , 0.058]	$-3.164$ [ $-3.549$ , $-2.801$ ]
Llama-3.2-3B	60	$-3.076$ [ $-3.611$ , $-2.554$ ]	$-0.033$ [ $-0.097$ , 0.018]	$-3.043$ [ $-3.560$ , $-2.536$ ]
Llama-3.2-3B-I	60	$-3.362$ [ $-4.002$ , $-2.774$ ]	0.024 [ $-0.026$ , 0.073]	$-3.387$ [ $-4.036$ , $-2.784$ ]
Qwen2.5-3B	60	$-1.598$ [ $-2.059$ , $-1.144$ ]	0.041 [ $-0.013$ , 0.099]	$-1.639$ [ $-2.114$ , $-1.188$ ]
Qwen2.5-3B-I	60	$-3.528$ [ $-4.552$ , $-2.484$ ]	$-0.051$ [ $-0.146$ , 0.041]	$-3.477$ [ $-4.522$ , $-2.415$ ]

Appendix

Table 19: Ablation of the gold read direction at the peak-read layer on 60 logit-readable failures per model. Values are changes in the final answer-versus-alternative margin, with paired bootstrap 95% intervals.

Model	Reference	Negative [95% CI]	Mean margin [95% CI]	$r_{\mathrm{freq}}$
Llama-3.1-8B	All records	67.0% [62.0%, 71.7%]	$-1.242$ [ $-1.526$ , $-0.937$ ]	0.567
	Failures	68.2% [63.2%, 72.8%]	$-1.272$ [ $-1.557$ , $-0.972$ ]	0.554
	Neutral text	65.4% [60.3%, 70.1%]	$-0.951$ [ $-1.262$ , $-0.620$ ]	0.508
Llama-3.2-1B	All records	77.8% [73.6%, 81.5%]	$-1.679$ [ $-1.923$ , $-1.421$ ]	0.568
	Failures	79.0% [74.8%, 82.6%]	$-1.804$ [ $-2.055$ , $-1.538$ ]	0.558
	Neutral text	73.0% [68.6%, 77.1%]	$-1.311$ [ $-1.576$ , $-1.038$ ]	0.530

Appendix

Table 20: Baseline sign and frequency association on failures under three reference means. Brackets are 95% intervals.

Model	Gate	Full acc	$n_{F}$	Recon new/old	Logit	Tuned	Probe	Shuffled	Probe $-$ shuffled	Neg. (all)	Neg. (neutral)	$r_{\mathrm{freq}}$	Freq. rec.	Random
Qwen3-0.6B	94.7%	11.4%	434	1.00 / 0.34	27.9%	33.6%	26.5%	12.0%	$+14.5$ [9.4, 19.4]	84.6%	77.2%	0.558	0.9%	1.6%
Qwen3-1.7B	100.0%	14.8%	411	0.98 / 0.12	29.0%	35.3%	25.1%	12.7%	$+12.4$ [7.5, 17.5]	85.2%	74.0%	0.566	1.0%	1.5%
Qwen3-4B	98.9%	17.4%	401	1.00 / 0.02	29.2%	39.2%	27.7%	11.5%	$+16.2$ [10.7, 21.7]	80.8%	72.3%	0.564	0.7%	2.7%
Qwen3-8B	98.1%	21.4%	388	1.00 / 0.00	26.8%	40.7%	26.3%	12.6%	$+13.7$ [8.5, 18.6]	76.8%	69.8%	0.568	0.8%	1.5%
Qwen3-14B	96.0%	25.2%	377	1.00 / 0.26	33.4%	38.7%	27.6%	13.5%	$+14.1$ [8.7, 19.4]	75.3%	70.8%	0.561	1.1%	2.7%

Appendix

Table 21: Qwen3-Base: protocol, readers and baseline on failures. Gate is the first-token protocol check among fully correct generations; reconstruction is top-1 agreement with the native logits for the corrected (new) and doubly normalized (old) extraction; reader columns are hard-decoy readable fractions among failures; probe

-

shuffled is the paired difference in percentage points with a bootstrap 95% interval (all exact

p<0.001

); negative baseline is reported under the all-record and neutral-text references; frequency recovery is first-token recovery after removing the frequency direction, with the random control.

Model	A	B $\downarrow$	A $\to$ B $\downarrow$	Random	Combined $-$ A	Excess $>0$	FP / FN	A gen. first	A full	A $\to$ B $\downarrow$ full	Mask only full	Offline $=$ gen.
Qwen3-0.6B	47.0%	1.2%	67.3%	0.0%	$+20.3$ [16.6, 24.2]	50.7%	0 / 10	46.8%	20.7%	25.6%	3.7%	0.970
Qwen3-1.7B	56.9%	1.9%	67.6%	0.5%	$+10.7$ [7.8, 13.9]	42.8%	0 / 1	56.2%	22.1%	26.3%	1.5%	0.981
Qwen3-4B	75.1%	2.7%	83.8%	0.0%	$+8.7$ [6.0, 11.7]	24.4%	0 / 2	75.3%	34.4%	36.2%	2.2%	0.993
Qwen3-8B	73.2%	5.9%	77.6%	0.8%	$+4.4$ [2.3, 6.4]	26.3%	0 / 2	72.9%	29.6%	33.2%	3.6%	0.992
Qwen3-14B	85.4%	5.0%	91.5%	0.5%	$+6.1$ [3.7, 8.8]	14.3%	0 / 1	85.4%	37.1%	40.3%	5.0%	0.989

Appendix

Table 22: Qwen3-Base: median-calibrated edits on all failures. A sets the answer support, B

\downarrow

lowers the alternative support to at most the median of successes, A

\to

\downarrow

applies both, and Random matches the norm of A; combined

-

A is the paired difference in percentage points (bootstrap 95% interval, Holm-adjusted exact

p<0.001

in every model); excess

>0

is the share of failures the criterion of equation 7 predicts answer setting cannot repair, with its false positives and misses; the last four columns are generated outcomes, with the mask-only control (first-step content mask, no edit).

Model	$n$	R / U recovery	Difference	Edit-norm diff.	$\pi_{a}$ diff.	$\pi_{b}$ diff.	Alt-limited R / U
Qwen3-0.6B	121	31.4% / 43.0%	$-11.6$ [ $-23.1$ , 0.0]	$-0.93$ [ $-1.33$ , $-0.48$ ]	$+1.43$ [0.87, 1.96]	$-0.07$ [ $-0.74$ , 0.54]	0.67 / 0.55
Qwen3-1.7B	119	31.9% / 52.1%	$-20.2$ [ $-31.9$ , $-7.6$ ]	$-0.60$ [ $-0.89$ , $-0.30$ ]	$+1.02$ [0.65, 1.40]	$+0.19$ [ $-0.19$ , 0.58]	0.67 / 0.48
Qwen3-4B	117	49.6% / 80.3%	$-30.8$ [ $-41.0$ , $-20.5$ ]	$-1.48$ [ $-1.90$ , $-1.04$ ]	$+1.69$ [1.22, 2.15]	$+0.00$ [ $-0.59$ , 0.56]	0.50 / 0.20
Qwen3-8B	104	49.0% / 78.8%	$-29.8$ [ $-41.3$ , $-18.3$ ]	$-0.63$ [ $-1.06$ , $-0.20$ ]	$+1.03$ [0.53, 1.52]	$-0.09$ [ $-0.68$ , 0.48]	0.50 / 0.21
Qwen3-14B	126	77.8% / 85.7%	$-7.9$ [ $-18.3$ , 2.4]	$-0.60$ [ $-1.03$ , $-0.17$ ]	$+0.74$ [0.23, 1.24]	$-0.16$ [ $-0.66$ , 0.31]	0.22 / 0.13

Appendix

Table 23: Qwen3-Base: readable (R) versus unreadable (U) failures under the logit reader, matched on the initial margin, at the median answer-setting edit. Differences are R minus U with bootstrap 95% intervals;

\pi_{a}

and

\pi_{b}

are the projections on the answer and alternative directions before any edit; alternative-limited is the share of each subset with positive excess.

Model	Band input	$p$	Late MLP	$p$	Late attn	$p$	Middle	Early	Late-all SF / DF / random	Peak-layer	Ablation
Qwen3-0.6B	90.0 / 23.3	$<0.001$	75.0 / 31.7	$<0.001$	36.7 / 21.7	0.216	70.0	41.7	83.3 / 23.3 / 66.7	$+70.0$ [56.7, 81.7]	$-2.56$ [ $-3.04$ , $-2.10$ ]
Qwen3-1.7B	83.3 / 6.7	$<0.001$	58.3 / 11.7	$<0.001$	33.3 / 21.7	0.568	76.7	50.0	71.7 / 5.0 / 65.0	$+68.3$ [53.3, 81.7]	$-1.98$ [ $-2.51$ , $-1.48$ ]
Qwen3-4B	80.0 / 3.3	$<0.001$	65.0 / 6.7	$<0.001$	45.0 / 15.0	0.004	53.3	36.7	86.7 / 1.7 / 63.3	$+68.3$ [55.0, 80.0]	$-1.62$ [ $-2.13$ , $-1.16$ ]
Qwen3-8B	73.3 / 1.7	$<0.001$	56.7 / 1.7	$<0.001$	31.7 / 23.3	1.000	51.7	28.3	58.3 / 3.3 / 51.7	$+56.7$ [43.3, 70.0]	$-2.20$ [ $-2.87$ , $-1.55$ ]
Qwen3-14B	66.7 / 0.0	$<0.001$	58.3 / 0.0	$<0.001$	56.7 / 15.0	$<0.001$	51.7	21.7	83.3 / 0.0 / 51.7	$+56.7$ [41.7, 70.0]	$-3.80$ [ $-4.49$ , $-3.13$ ]

Appendix

Table 24: Qwen3-Base: within-fact transfers on 60 pairs per model, first-token recovery for the same-fact donor / the different-fact donor, with Holm-adjusted exact

p

for the paired difference; middle and early replace same-size bands; late-all replaces every late block (same fact / different fact / random components across depth); peak-layer is the paired recovery difference of same-fact over different-fact residual patching at the peak-read layer; ablation is the change in the final margin from removing the gold read direction minus that of a random direction.

Hallucination in Language Models

Hallucination in language models studies how systems produce unsupported or false claims and how to detect, explain, and reduce them.

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A decodability criterion predicts when hidden-state selection beats majority voting in large language models
Aug 17, 2026Zhixiang wang, Ziliang Hong, Ulas BagciLanguage Model Decoding LLM Evaluation
Northwestern University, Chicago, United States
On the Tip of the Tongue: First-Token Cueing as a Behavioral Probe of Recall Failures in LLMs
Oct 16, 2026Jaewon Sok, Jewon Yeom, Seonghyeon Park +3Factual Knowledge in Language Models
Department of Rural Systems Engineering, Seoul National University · Graduate School of Data Science, Seoul National University · Department of Aerospace Engineering, Seoul National University
Hallucination as Commitment Failure: Larger LLMs Misfire Despite Knowing the Answer
May 21, 2026Jewon Yeom, Jaewon Sok, Heejun Kim +3LLM Interpretability Hallucination in Language Models
Graduate School of Data Science, Seoul National University · Department of Rural Systems Engineering, Seoul National University · Electrical Engineering and Computer Science, Gwangju Institute of Science and Technology +1

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A decodability criterion predicts when hidden-state selection beats majority voting in large language models

Zhixiang wang, Ziliang Hong, Ulas Bagci

Northwestern University, Chicago, United States

Language Model Decoding LLM Evaluation Confidence Estimation in Language Models

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Combining the answers a large language model (LLM) samples for a question into one decision is a test-time information fusion problem, usually solved by majority voting. Voting is unreliable on difficult questions, where the sampled answers share correlated errors, so the wrong answer can win and drawing more samples makes the decision worse. Selecting a candidate by reading a correctness signal from the model's hidden states is a promising alternative, but its accuracy varies across models and tasks, and no measure indicates when it can be trusted. In this paper, we propose CASE (Correctness-Axis SElection), a dynamic selection combiner that trains a linear gate on the answer-token hidden state and selects the highest-scoring candidate. Its main contribution is decodability, a leakage-free measure of how well the gate ranks a question's correct candidates above its incorrect ones, which predicts whether hidden-state selection will outperform voting. A conventional probe appears accurate only because of question-identity leakage, which vanishes under question-grouped evaluation. On held-out data, decodability predicts the accuracy gain of selection over voting with a Pearson correlation r=0.75 and a decision threshold near AUC=0.60. Across general and medical LLMs, CASE improves over voting by up to 19 points on medium-difficulty questions and 16.8 points on hard questions. Decodability depends on the aligned knowledge a model must recall, not on its scale, and its prediction transfers to an unseen scientific domain within 3.8 points. It thus provides a practical criterion, measurable in advance for a given model and task, for choosing between learned selection and majority voting.

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Hallucination in Language Models

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A decodability criterion predicts when hidden-state selection beats majority voting in large language models

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