Correctness Is a Direction: Geometric Answer Selection in Language Models

Marcus Armstrong; Navid Ayoobi; Pradham Mummaleti; Alexander Chulzhanov; Arjun Mukherjee

cs.CLOct 3, 2026

Correctness Is a Direction: Geometric Answer Selection in Language Models

Authors: Marcus Armstrong, Navid Ayoobi, Pradham Mummaleti, Alexander Chulzhanov, Arjun Mukherjee

Organizations: Department of Computer Science University of Houston Houston, TX 77204

Abstract

Answer correctness is encoded as a recoverable geometric direction in the hidden states of language models. We show that the mean displacement from incorrect to correct answer representations, computed at approximately 70% of model depth from fifty labeled examples with no parameter updates, yields a scoring direction that outperforms zero-shot log-probability scoring by up to +32.0 percentage points on factual benchmarks (ARC-Challenge and MMLU) and by +38.1 to +51.8 percentage points on TruthfulQA, across five models spanning 1B to 8B parameters in three architecture families (Llama, Qwen, Gemma). The method requires one forward pass and one dot product per candidate; no generation is performed at inference. Applied as a hallucination detector on individual (question, answer) pairs, the recovered direction achieves 0.693~AUROC versus 0.578 for log-probability scoring. We additionally find that correctness directions for factual reasoning, domain knowledge, and calibrated truthfulness are near-orthogonal in representation space, revealing that language models allocate geometrically independent subspaces to qualitatively distinct notions of correct answer, with architecture-dependent variation in the degree of separation. This structure explains the observed transfer pattern---the direction calibrated on factual questions transfers within task type but not across it---and suggests that LLM calibration failures may reflect a routing problem: the model's internal representation contains more correctness signal than its output behaviour exploits.

Figures & tables

Model	Layer	Zero-shot	Standard $\hat{\delta}$	Contrastive $\hat{\delta}$	Std SNR	Con SNR
Llama-3.2-1B	0.60	35.0%	26.1% ( $-$ 8.9)	44.7% (+9.7)	43.4 $\times$	8.1 $\times$
Qwen-2.5-1.5B	0.65	41.0%	35.1% ( $-$ 5.9)	61.1% (+20.1)	49.4 $\times$	9.0 $\times$
Gemma-2-2B	0.70	44.4%	27.6% ( $-$ 16.8)	66.5% (+22.1)	27.2 $\times$	10.1 $\times$

Table 1: Standard vs. contrastive direction at the layer selected via held-out validation for each model. Zero-shot log-probability on identical held-out test examples (ARC-Challenge) provides the baseline. Layer selection uses a 20% validation split; all reported accuracies are on the remaining 80% test set. Accuracy changes relative to zero-shot in parentheses.

Figure 1: ARC-Challenge accuracy (%) for standard

\hat{\delta}

(orange, dashed) and contrastive

\hat{\delta}

(blue, solid) across five layer fractions, for all three model families. The zero-shot log-probability baseline is shown as a dotted horizontal line for each model. The faint vertical guide marks fraction 0.70. Contrastive extraction outperforms standard at all tested depths for Qwen and Gemma. For Llama, the directions converge at fraction 0.75 (layer 12 of 16), where the standard direction reaches its global maximum of 58.0%; at all other depths the contrastive direction is superior. Qwen and Gemma peak at fraction 0.70 despite differing architectures, consistent with a universal representational transition near this depth. Llama-3.2-1B points at fractions 0.75 and 0.80 are identical because both map to layer 12 in a 16-layer architecture.

		$N=50$		$N=100$
Model	Layer	SNR	Acc. (%)	SNR	Acc. (%)
Qwen-2.5-1.5B	0.70	8.4 $\times$	60.6	8.4 $\times$	60.6
Gemma-2-2B	0.75	10.0 $\times$	67.8	9.7 $\times$	65.6

Table 2: Effect of calibration set size on direction SNR and ARC-Challenge accuracy at the validation-selected optimal layer. SNR and accuracy do not improve with doubled calibration set size. Llama-3.2-1B results omitted from tabular comparison as the N=50 vs. N=100 systematic comparison was conducted at the previously identified 0.75 layer; the general pattern (no improvement with N=100) is consistent across all models and layer configurations tested.

Figure 2: Pairwise cosine similarity between contrastive correctness directions calibrated on ARC-Challenge, MMLU-Biology, and TruthfulQA for all five tested models (rows), across the three task-direction pairs (columns). Warm colours indicate positive alignment; white indicates orthogonality; values are annotated directly in each cell. The dashed line separates sub-3B models (above) from 7B–8B models (below). Near-orthogonality ( $|\cos\theta|<0.16$ ) is the dominant pattern: Llama-3.2-1B, Qwen-2.5-1.5B, and Llama-3.1-8B satisfy this across all three pairs, with Llama-8B showing the strongest separation of any tested model (max $|\cos\theta|=0.10$ ). Gemma-2-2B shows partial alignment between ARC and TQA ( $+0.302$ ); Qwen-2.5-7B shows moderate ARC $\leftrightarrow$ MMLU alignment ( $+0.311$ ) and partial ARC $\leftrightarrow$ TQA alignment ( $+0.166$ ), indicating family-specific geometric organisation that can increase with scale.

Category	Task	Qwen-1.5B	Gemma-2B	Llama-1B
Factual MC	ARC-C (source)	+24.3	+20.3	+6.3
	MMLU-Biology	+20.4	+19.7	+5.0
	MMLU-Chemistry	+4.9	+8.4	$-4.4$
	MMLU-World History	+13.5	+22.4	+6.8
Coherence	HellaSwag	$-11.0$	$-16.7$	$-26.7$
Calibration	TruthfulQA (ARC $\hat{\delta}$ )	+4.8	+12.9	+8.6

Table 3: Transfer of ARC-Challenge calibrated

\hat{\delta}

to target tasks (

\Delta

pp relative to zero-shot log-probability baseline on identical test examples). The bottom row shows TruthfulQA results when recalibrated on TruthfulQA examples (own

\hat{\delta}

), confirming that calibration source determines performance. Bold entries indicate positive transfer; negative entries indicate the ARC direction actively hurts relative to zero-shot.

		Llama-1B		Qwen-1.5B		Gemma-2B		Qwen-7B		Llama-8B
Task	ZS	GRM	$\Delta$	GRM	$\Delta$	GRM	$\Delta$	GRM	$\Delta$	GRM / $\Delta$
ARC-C (35 / 41 / 44 / 48 / 46%)	—	44.7	+9.7	61.1	+20.1	66.5	+22.1	80.0	+32.0	59.0 / +13.4
TruthfulQA (33 / 31 / 39 / 40 / 29%)	—	75.9	+43.4	72.5	+41.9	76.5	+38.0	81.2	+41.7	80.4 / +51.7
Probe gap (ARC)	—	$-1.0$		$+0.2$		$+0.0$		$+1.0$		$-1.0$
Probe gap (TQA)	—	$-2.1$		$-1.2$		$+0.7$		$-1.6$		$-0.5$

Table 4: Geometric scoring accuracy and improvement over zero-shot log-probability (same held-out test set; layer selected on a 20% validation split). Calibration: ARC

\hat{\delta}

for ARC-Challenge (50 ARC training examples), TQA

\hat{\delta}

for TruthfulQA (50 TQA examples). Probe gap rows show GRM minus a logistic regression probe trained on the identical 200 calibration hidden states (50 questions

\times

4 candidates); values within

\pm

3 pp confirm that the mean direction captures the full discriminative information available from fifty examples without optimisation. Improvements above

{\approx}8

pp are statistically significant at

\alpha{=}0.05

(two-proportion

z

-test).

Figure 3: Improvement in accuracy (

\Delta

pp) over the zero-shot log-probability baseline on ARC-Challenge (left) and TruthfulQA (right) for all five models. Error bars show 95% confidence intervals (two-proportion

z

-test). Annotated values above each bar show absolute GRM accuracy. Colours indicate architecture family: blues for Llama (light = 1B, dark = 8B), oranges for Qwen (light = 1.5B, dark = 7B), green for Gemma-2B. On ARC-Challenge, improvements scale with model size within each family: Qwen 1.5B

\to

7B gains an additional 18.9 pp; Llama 1B

\to

8B gains 14.3 pp. On TruthfulQA, where zero-shot baselines fall near or below chance (28.7–39.5%), geometric scoring achieves 72.5–81.2% across all five models, with the largest single improvement in the study coming from Llama-3.1-8B (+51.7 pp). MMLU and HellaSwag transfer results are reported in Table 3 .

Classifier	AUROC	$\Delta$ vs Random
Random baseline	0.500	—
Zero-shot log-prob	0.578	+7.8 pp
GRM contrastive $\hat{\delta}$	0.693	+19.3 pp

Table 5: Hallucination detection AUROC on ARC-Challenge (Qwen-2.5-1.5B,

n{=}1{,}999

pairs). AUROC measures

P(\text{correct score}>\text{wrong score})

; 0.5 is random. GRM requires one forward pass and one dot product per pair; no training beyond the 50-example calibration step.

Figure 4: (a) Score distributions of the contrastive GRM projection for correct (blue, solid) and incorrect (orange, dashed) answer representations. Curves are Gaussian approximations fitted to the empirical means and standard deviations; the theoretical AUROC from these fits is 0.691, closely matching the empirical 0.693. The shaded region shows distribution overlap. Vertical dotted lines mark the class means (

\mu_{\mathrm{correct}}{=}{+}0.0255

\mu_{\mathrm{wrong}}{=}{-}0.0084

); the separation of 0.034 score units is modest but consistent. (b) ROC curves for GRM (AUROC 0.693, solid blue) and zero-shot log-probability (AUROC 0.578, dashed orange), derived analytically from Gaussian fits to empirical score distributions. The diagonal dotted line represents a random classifier.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

	Standard $\hat{\delta}$			Contrastive $\hat{\delta}$
Layer	SNR	ArgMax	ArgMin	SNR	ArgMax	ArgMin
0.60	43.4 $\times$	29.4	20.8	8.1 $\times$	44.2	11.2
0.70	38.5 $\times$	24.0	21.2	7.6 $\times$	40.2	14.2
0.75 †‡	36.4 $\times$	58.0	38.0	35.9 $\times$	53.2	54.0
0.80 †	36.4 $\times$	58.0	38.0	35.9 $\times$	53.2	54.0
0.90	35.7 $\times$	33.6	17.8	7.2 $\times$	38.4	16.2

Appendix

Table 6: Full layer sweep — Llama-3.2-1B-Instruct (16 layers). Fractions 0.75 and 0.80 map to the same absolute layer (layer 12); results are identical by construction.

Layer	SNR	ArgMax	ArgMin	SNR	ArgMax	ArgMin
	Standard $\hat{\delta}$			Contrastive $\hat{\delta}$
0.60	51.9 $\times$	36.8	15.2	8.5 $\times$	53.0	10.4
0.70	49.7 $\times$	39.0	15.0	8.4 $\times$	60.6	8.0
0.75	41.0 $\times$	37.4	18.2	7.7 $\times$	53.0	8.0
0.80	39.6 $\times$	35.4	17.0	7.5 $\times$	48.6	9.2
0.90	32.6 $\times$	31.2	19.0	6.5 $\times$	49.0	10.4

Appendix

Table 7: Full layer sweep — Qwen-2.5-1.5B-Instruct (28 layers).

Layer	SNR	ArgMax	ArgMin	SNR	ArgMax	ArgMin
	Standard $\hat{\delta}$			Contrastive $\hat{\delta}$
0.60	30.9 $\times$	33.2	21.2	11.0 $\times$	65.0	6.0
0.70	27.2 $\times$	28.4	24.6	10.1 $\times$	67.6	6.4
0.75	29.0 $\times$	33.2	20.8	10.0 $\times$	67.8	6.4
0.80	28.7 $\times$	35.4	18.4	9.7 $\times$	65.6	6.0
0.90	31.7 $\times$	36.4	16.8	9.9 $\times$	62.4	6.2

Appendix

Table 8: Full layer sweep — Gemma-2-2B-IT (26 layers).

Layer	Con SNR	Con ArgMax (val)
0.60	12.4 $\times$	62.8
0.65	13.5 $\times$	75.2
0.70	14.9 $\times$	82.5
0.75	15.4 $\times$	75.2
0.80	14.4 $\times$	74.8
0.85	13.7 $\times$	73.1

Appendix

Table 9: Layer sweep — Qwen-2.5-7B-Instruct (32 layers). Contrastive

\hat{\delta}

only; accuracy on 20% validation split used for layer selection. Zero-shot baseline: 48.0%.

Layer	Con SNR	Con ArgMax (val)
0.60	11.2 $\times$	55.6
0.65	11.1 $\times$	56.0
0.70	10.8 $\times$	53.4
0.75	10.5 $\times$	49.6
0.80	10.4 $\times$	50.9
0.85	10.1 $\times$	52.6

Appendix

Table 10: Layer sweep — Llama-3.1-8B-Instruct (32 layers). Contrastive

\hat{\delta}

only; accuracy on 20% validation split used for layer selection. Zero-shot baseline: 45.6%.

Standard direction ( $n=200$ )
		$N=50$		$N=100$
Model	Layer	SNR	Acc. (%)	SNR	Acc. (%)
Llama	0.90	35.7 $\times$	32.0	35.6 $\times$	26.5
Llama	0.75	36.3 $\times$	28.0	35.5 $\times$	31.5
Qwen	0.90	32.6 $\times$	28.5	30.9 $\times$	30.0
Qwen	0.75	41.0 $\times$	39.0	38.5 $\times$	34.5

Appendix

Table 11: Effect of calibration size on SNR and ARC-Challenge accuracy. Upper section: standard direction at two layer fractions per model (

n{=}200

test). Lower section: contrastive direction at optimal layer per model (

n{=}500

test). SNR consistently decreases as

N

increases; accuracy shows no systematic benefit beyond

N{=}50

Subject	Model	ZS	Acc.	$\Delta$	Acc.	$\Delta$
			ARC $\hat{\delta}$		Own $\hat{\delta}$
Biology	Llama	37.7	42.7	+5.0	35.0	$-2.7$
	Qwen	40.3	60.7	+20.4	40.3	$+0.0$
	Gemma	41.0	60.7	+19.7	48.7	$+7.7$
Chemistry	Llama	31.0	26.6	$-4.4$	—	—
	Qwen	32.0	36.9	+4.9	—	—

Appendix

Table 12: ARC-calibrated contrastive direction on MMLU subjects (

n{=}300

test examples per subject). “Own” columns show accuracy when the direction is calibrated on the same subject’s validation split (

N{\leq}50

, typically 10–30 examples available). Dashes indicate subject calibration was not separately tested.

Threshold $t$	TPR (wrong flagged)	FPR (correct flagged)	Precision
$-0.04$	30.7%	10.5%	89.7%
$-0.02$	40.4%	17.1%	87.6%
$\phantom{-}0.00$	56.9%	29.7%	85.2%
$+0.02$	64.5%	40.0%	82.8%
$+0.04$	72.0%	50.1%	81.1%

Appendix

Table 13: GRM detector operating points at selected score thresholds. TPR: fraction of wrong answers correctly flagged (high GRM score

<

threshold flags answer as suspect — note: we flag low-scoring answers, so high TPR requires a high threshold

t

). FPR: fraction of correct answers incorrectly flagged.

Model	Params	Layers	$d$	Opt. frac	Opt. layer
Llama-3.2-1B-Instruct	1.24B	16	2,048	0.60	9
Qwen-2.5-1.5B-Instruct	1.54B	28	1,536	0.65	18
Gemma-2-2B-IT	2.61B	26	2,304	0.70	18
Qwen-2.5-7B-Instruct	7.61B	32	3,584	0.70	22
Llama-3.1-8B-Instruct	8.03B	32	4,096	0.65	20

Appendix

Table 14: Model architectures and optimal extraction layers.

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Locating Answer-Correctness Signals in Frozen Large Language Models

Yuansen Liu, Yixuan Tang, Anthony Kum Hoe Tung

School of Computing, National University of Singapore

LLM Answer Verification Language Model Error Detection Language Model Probing

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Language models expose internal signals that predict whether an answer is correct, readable from a single forward pass of a frozen model without additional generations. Yet existing probes often commit to one signal family or layer and can be brittle under distribution shift; in retrieval-augmented settings, many specialized detectors instead target passage faithfulness, which can diverge from correctness when retrieved evidence is unhelpful or conflicting. We therefore ask where answer correctness is readable, which internal signal families carry it, and how they should be combined. We search over hidden states, token probabilities, residual-stream features, attention, and their fusion, treating the selected readouts as a predictive measurement rather than a mechanistic localization. We run this analysis separately in closed-book and with-context settings, since context can change which readouts are informative. A consistent anatomy emerges: correctness concentrates in the answer span, recovered from the answer tokens even under retrieval, and the families carry it complementarily, so fusing them helps most out of distribution, where a single signal is weakest. The protocol is effective across two backbones and gates a retrieval controller as one downstream use.

Abstract

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