Isolating a Language Model's Judgment with a Label-Free Subspace

Chaemin Jang; Donggyu Lee; Jeesoo Kim; Dongman Lee; Jihee Kim

cs.LGOct 15, 2026

Isolating a Language Model's Judgment with a Label-Free Subspace

Authors: Chaemin Jang, Donggyu Lee, Jeesoo Kim, Dongman Lee, Jihee Kim

Organizations: Korea Advanced Institute of Science and Technology (KAIST)

Abstract

Language models are often asked to make judgments: whether a proposed answer is correct, whether one statement follows logically from another, or whether a response is safe. Yet changing the instructions that specify how the judgment should be expressed can make the final answer less reliable, even for the same input. We ask whether this reflects a change in internal judgment or a failure to express it. Comparing hidden-state vectors for the same input under different instructions, we find that their differences concentrate in a low-dimensional expression subspace. We identify this subspace without benchmark labels and remove its contribution from a linear predictor to obtain an isolated judgment. Across correctness, entailment, and safety, the isolated judgment remains reliable under instructions not used to identify the subspace, even when the model's final answer and the original predictor degrade. When the judgment question is negated, isolation raises the predictor's AUROC from 0.504 to 0.825 for entailment and from 0.480 to 0.891 for safety. We also find that judgment is readable in earlier layers, before the expression subspace begins to disrupt the predictor. Together, these results provide a way to separate judgment from expression and recover internal judgments even when the model's final answers are unreliable.

Figures & tables

Figure 1: Separating judgment from verdict. A: The same input under three instructions: the verdict changes, the judgment does not. B: Paired differences across instructions span a low-dimensional expression subspace

E

, estimated without labels; removing it from the readout

w

leaves the isolated judgment. C: On the four held-out conditions, the readout falls to 0.68 AUROC while the isolated judgment reaches 0.79, within 0.01 on average of a readout fitted on each condition.

Component	Setting
Judgments	Correctness, entailment, safety
Datasets	GSM8K, MMLU, ARC-Challenge, TriviaQA, ANLI, BeaverTails
Models	Qwen2.5-7B, Llama-3.1-8B, Mistral-7B, Qwen2.5-14B, Gemma-2-9B
Response formats	Yes/No, True/False, A/B, B/A, 1/0, Correct/Incorrect, sentence
Held-out conditions	No/Yes, False/True, 0/1, negated question

Table 1: Experimental setting. For each listed pair, the first response expresses a positive judgment. Held-out conditions are not used to identify the subspace and are introduced only when testing whether the judgment can be recovered under conditions held out from the estimate.

Position	AUROC
Before judgment question	0.741
After judgment question	0.812
Before verdict	0.818

Table 2: The readout is linked to the model’s judgment. The readout becomes stronger after the model is asked to judge the candidate answer, and intervening along its direction changes the model’s verdict.

Panel of Figure 2 — Figure 2: Across conditions, the readout can degrade while the isolated judgment remains close to a readout fitted directly on the target condition. Left: the seven response formats used to identify the expression subspace, averaged, with whiskers spanning the seven format means. Right: the four held-out conditions, with 95% intervals over the fourteen model–dataset cells. Gray ticks mark the condition-fitted readout. Means over fourteen cells and five instruction wordings; the sentence format has no native verdict margin.

Condition	Verdict	Readout	Isolated	Condition-fitted
No means correct	0.617	0.640	0.781	0.805
False means correct	0.593	0.665	0.787	0.803
0 means correct	0.769	0.787	0.809	0.810
Negated question	0.665	0.650	0.802	0.805

Table 3: Performance on the four held-out conditions. Values are AUROC against benchmark labels, averaged over fourteen model–dataset cells and five instruction wordings. The isolated judgment remains close to a readout fitted directly on each condition.

Panel of Figure 3 — Figure 3: Held-out conditions can substantially degrade the model’s verdict and the readout while leaving the isolated judgment comparatively stable. Panel (a) shows the mean AUROC drop across the four held-out conditions. Panel (b) shows the drop separately for each condition.

Panel of Figure 4 — Figure 4: Judgment and expression emerge at different depths. At intermediate layers, the readout is nearly identical across Yes/No, No-means-correct, and the negated question. The conditions diverge in the later layers (a), while the isolated judgment remains aligned across depth (b). Full depth-wise results are in Appendix C.3 .

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

cell	Yes/No own	frozen $\to$ A/B	A/B fitted	frozen $\to$ B/A	B/A fitted	frozen $\to$ sentence	sentence fitted
Llama-3.1-8B ARC	0.737	0.703	0.695	0.714	0.702	0.730	0.731
Llama-3.1-8B GSM8K	0.966	0.971	0.970	0.972	0.972	0.952	0.961
Llama-3.1-8B MMLU	0.719	0.679	0.672	0.688	0.678	0.708	0.699
Llama-3.1-8B TriviaQA	0.842	0.846	0.843	0.845	0.842	0.844	0.846
Mistral-7B ARC	0.730	0.662	0.696	0.649	0.670	0.685	0.690
Mistral-7B GSM8K	0.826	0.814	0.822	0.812	0.821	0.829	0.827

Appendix

Table 4: Per-cell transfer of the frozen Yes/No readout (five wordings): its Yes/No AUROC, its AUROC on A/B, B/A, and sentence states, and the AUROC of a readout fitted on each of those formats.

cell	stem	response position
Llama-3.1-8B ARC	0.691	0.746
Llama-3.1-8B GSM8K	0.967	0.971
Llama-3.1-8B MMLU	0.683	0.710
Llama-3.1-8B TriviaQA	0.810	0.827
Mistral-7B ARC	0.694	0.717
Mistral-7B GSM8K	0.776	0.788

Appendix

Table 5: Response format specified first, per cell: Yes/No AUROC of the class mean fitted at the end of the judgment question (stem, before any response token) and at the response position. Five wordings. Mistral-7B follows these prompts on few rows and is descriptive. Qwen ARC cells are descriptive.

cell	rank	direction norm in subspace	readout, B/A	isolated, B/A	readout, negated	isolated, negated
Llama-3.1-8B ARC	7	0.524	0.714	0.729	0.718	0.732
Llama-3.1-8B GSM8K	9	0.473	0.972	0.972	0.944	0.965
Llama-3.1-8B MMLU	6	0.277	0.688	0.692	0.664	0.703
Llama-3.1-8B TriviaQA	8	0.504	0.845	0.846	0.726	0.841
Mistral-7B ARC	8	0.569	0.649	0.695	0.646	0.687
Mistral-7B GSM8K	6	0.444	0.812	0.823	0.744	0.817

Appendix

Table 6: The expression subspace per cell (paired differences among the five response formats plus entailment and safety negation pairs; rank by the split-half rule, median over folds and wordings): fraction of the Yes/No readout direction’s squared norm inside it (standardized coordinates), and the AUROC of the readout and of the isolated judgment on B/A states and on negated-question states. Five wordings.

condition	verdict	readout	pooled	isolated	condition-fitted
No means correct	0.617	0.640	0.789	0.781	0.805
False means correct	0.593	0.665	0.771	0.787	0.803
0 means correct	0.769	0.787	0.804	0.809	0.810
negated question	0.665	0.650	0.794	0.802	0.805

Appendix

Table 7: Held-out conditions. AUROC against benchmark labels, means over fourteen cells, five wordings. The expression subspace is estimated from the five response formats and the entailment and safety negation pairs; none of these conditions enters it. Verdict is the model’s own response margin; readout is the frozen Yes/No readout; pooled is a class mean fitted on labeled A/B and B/A states; condition-fitted is a class mean fitted on the condition itself. Cell-bootstrap intervals on the isolated-minus-condition-fitted gap: [

-0.012

0.004

] (negated question), [

-0.043

-0.008

] (No means correct), [

-0.028

-0.007

] (False means correct), [

-0.005

0.003

] (0 means correct).

target	verdict	readout	pooled	tuned probe	isolated	condition-fitted
Yes/No	0.788	0.815	0.809	0.824	0.818	0.815
True/False	0.779	0.815	0.812	0.811	0.814	0.814
A/B	0.744	0.798	0.803	0.774	0.808	0.804
sentence	–	0.807	0.801	0.776	0.806	0.809
1/0	0.794	0.815	0.806	0.784	0.810	0.814
correct/incorrect	0.779	0.816	0.810	0.800	0.815	0.810

Appendix

Table 8: Every readout on every target, means over fourteen cells, five wordings. The sentence format has no native response margin. Tuned probe: an L2-regularised logistic readout fitted under Yes/No, with the regularization strength chosen on the source folds.

target	verdict	readout	pooled	tuned probe	isolated	condition-fitted
Yes/No	0.801	0.812	0.802	0.812	0.812	0.812
True/False	0.800	0.812	0.809	0.794	0.804	0.812
A/B	0.779	0.812	0.807	0.787	0.815	0.811
sentence	–	0.802	0.793	0.762	0.804	0.807
1/0	0.809	0.804	0.805	0.789	0.810	0.813
correct/incorrect	0.794	0.807	0.803	0.802	0.808	0.807

Appendix

Table 9: Gemma-2-9B-IT on all four correctness datasets, the prospective fifth model, with the expression subspace estimated from paired differences among Gemma’s five response formats and from its entailment and safety negation pairs; means over the four cells (ARC and GSM8K descriptive), five wordings. Predictions were written before any Gemma state existed. Tuned probe as in Table 8 .

depth	Yes/No		B/A		No means correct		negated question
	readout	isolated	readout	isolated	readout	isolated	readout	isolated
0.25	0.699	0.702	0.699	0.700	0.700	0.701	0.699	0.702
0.50	0.786	0.788	0.780	0.785	0.781	0.782	0.783	0.784
0.75	0.818	0.824	0.800	0.813	0.756	0.808	0.756	0.818
0.90	0.813	0.820	0.794	0.808	0.648	0.799	0.632	0.811
1.00	0.815	0.818	0.792	0.805	0.640	0.781	0.650	0.802

Appendix

Table 10: Readout by depth: the Yes/No readout direction fitted at each depth and applied frozen, and the isolated judgment with the expression subspace estimated at that depth, under Yes/No, B/A, No-means-correct, and the negated question. Means over the fourteen primary cells, five wordings.

condition	readout	five formats only	negation pairs only	both	condition-fitted
No means correct	0.640	0.722	0.777	0.781	0.805
False means correct	0.665	0.754	0.784	0.787	0.803
0 means correct	0.787	0.805	0.810	0.809	0.810
negated question	0.650	0.733	0.801	0.802	0.805

Appendix

Table 11: Which pairs each held-out condition needs: the isolated judgment with the subspace estimated from paired differences among the five response formats only, from entailment and safety negation pairs only, or from both, against the readout and the condition-fitted readout. Means over fourteen cells, five wordings.

condition	readout	rank 1	rank 3	rank 8	condition-fitted
Yes/No	0.820	0.819	0.820	0.822	0.820
True/False	0.819	0.819	0.820	0.821	0.819
A/B	0.805	0.812	0.811	0.812	0.815
sentence	0.819	0.808	0.817	0.817	0.816
1/0	0.809	0.811	0.814	0.812	0.815
correct/incorrect	0.821	0.813	0.818	0.820	0.816

Appendix

Table 12: How much of the subspace the repair needs: the Yes/No readout direction with the leading direction from the negation pairs (rank 1), the three directions from the negation pairs (rank 3), and the full subspace (rank 8) removed, against the readout and the condition-fitted readout. Medians over fourteen cells, five wordings.

condition	readout	after LEACE	after INLP	isolated
No means correct	0.690	0.689	0.694	0.772
False means correct	0.675	0.674	0.688	0.783
0 means correct	0.766	0.760	0.743	0.815
negated question	0.694	0.695	0.689	0.805

Appendix

Table 13: Concept erasure of condition identity against the isolation. LEACE and INLP fitted on the paired states of all conditions (condition identity one-hot), then the readout on the erased states; medians over fourteen cells, five wordings.

condition	readout	identical-verdict pairs	verdict sign regressed out	as in Table 7
No means correct	0.690	0.772	0.771	0.772
False means correct	0.675	0.763	0.783	0.783
0 means correct	0.766	0.820	0.818	0.815
negated question	0.694	0.784	0.782	0.805

Appendix

Table 14: The subspace without verdict information: estimated from pairs on which the model’s verdict is identical under both conditions (identical by expressed judgment, so Yes under “correct?” and No under “incorrect?” count as identical; both prompts must be answered with one of their two response tokens; 44–74% of negation pairs are kept per model, almost all drops being disagreements); from all pairs with the model’s verdict sign regressed out of the differences before the singular decomposition; and as in Table 7 . Isolated judgment, medians over fourteen cells, five wordings.

estimation $\to$ evaluation	condition	readout	other wordings	same wording	fitted
P01,P03,P05 $\to$ P07,P09	No means correct	0.680	0.745	0.760	0.804
P01,P03,P05 $\to$ P07,P09	False means correct	0.674	0.763	0.763	0.810
P01,P03,P05 $\to$ P07,P09	0 means correct	0.770	0.814	0.816	0.818
P01,P03,P05 $\to$ P07,P09	negated question	0.727	0.788	0.811	0.809
P07,P09 $\to$ P01,P03,P05	No means correct	0.704	0.760	0.780	0.801
P07,P09 $\to$ P01,P03,P05	False means correct	0.676	0.764	0.795	0.808

Appendix

Table 15: Held-out wordings: the subspace estimated on three instruction wordings and applied at the other two, and the reverse, against the subspace estimated at the evaluation wording; isolated judgment, medians over fourteen cells.

condition	readout	isolated judgment	fitted on the new family
Yes/No	0.810 (0.819)	0.813 (0.816)	0.814 (0.820)
True/False	0.810 (0.815)	0.807 (0.815)	0.812 (0.818)
A/B	0.795 (0.814)	0.802 (0.816)	0.800 (0.813)
B/A	0.778 (0.779)	0.795 (0.809)	0.797 (0.812)
sentence	0.806 (0.817)	0.804 (0.814)	0.804 (0.815)
negated question	0.652 (0.695)	0.792 (0.793)	0.804 (0.801)

Appendix

Table 16: A different instruction family: five new wordings unrelated to the original ten, with the readout and the expression subspace taken from the original family and applied unchanged. AUROC against benchmark labels for the original-family readout on the new family’s states, the original-family isolated judgment on the same states, and a readout fitted on the new family; means (medians) over fourteen cells.

	entailment				safety
condition	readout	isolated	fitted	verdict	readout	isolated	fitted	verdict
Yes/No	0.866	0.866	0.866	0.864	0.903	0.902	0.903	0.894
True/False	0.866	0.860	0.866	0.865	0.905	0.904	0.907	0.891
A/B	0.848	0.846	0.846	0.812	0.888	0.896	0.901	0.889
B/A	0.851	0.848	0.846	0.767	0.878	0.894	0.899	0.829
sentence	0.845	0.846	0.838	–	0.874	0.895	0.896	–

Appendix

Table 17: Entailment (ANLI) and safety (BeaverTails) as judgment tasks: AUROC against the task label for the readout (frozen Yes/No readout), the isolated judgment, a readout fitted on the condition, and the model’s verdict. Means over the four primary models, five wordings. Conditions below the rule are held out from the subspace.

model	Yes/No readout	frozen vs fitted	fall under negation	isolated vs fitted
Qwen2.5-32B-Instruct	0.804	0.016	0.380	0.007
Gemma-2-27B-IT	0.750	0.016	0.140	0.006

Appendix

Table 18: Larger models, means over the two cells (TriviaQA, MMLU); three wordings, all rows, rank fixed at 8.

	TriviaQA				MMLU
model	Yes/No	frozen vs fitted	fall	isolated vs fitted	Yes/No	frozen vs fitted	fall	isolated vs fitted
Mistral-Nemo-12B-Instruct	0.776	0.007	0.151	0.002	0.756	0.007	0.033	0.002
Ministral-8B-Instruct	0.807	0.007	0.141	0.007	0.788	0.018	0.062	0.008
OLMo-2-7B-Instruct	0.764	0.006	0.014	0.005	0.737	0.012	0.030	0.002

Appendix

Table 19: Additional models by cell.

judge reads candidate	negatives	own-fitted	fitted here
Qwen2.5-7B reads Llama-3.1-8B, TriviaQA	115	0.801	0.800
Qwen2.5-7B reads Llama-3.1-8B, MMLU	128	0.810	0.810
Llama-3.1-8B reads Mistral-7B, TriviaQA	114	0.828	0.825
Llama-3.1-8B reads Mistral-7B, MMLU	160	0.763	0.771
Mistral-7B reads Qwen2.5-14B, TriviaQA	145	0.822	0.822
Mistral-7B reads Qwen2.5-14B, MMLU (descr.)	82	0.706	0.716

Appendix

Table 20: Another model’s answers: the judge’s own-answer readout applied to states where it judges another model’s answers, against a readout fitted on those states, under Yes/No. Descriptive: Mistral-7B reads Qwen2.5-14B MMLU (82 negatives).

condition	readout	isolated	fitted	verdict
Yes/No	0.885 (0.890)	0.880 (0.884)	0.885 (0.890)	0.881 (0.888)
True/False	0.881 (0.888)	0.868 (0.870)	0.881 (0.888)	0.876 (0.887)
A/B	0.865 (0.883)	0.863 (0.882)	0.862 (0.877)	0.793 (0.800)
B/A	0.848 (0.875)	0.855 (0.880)	0.854 (0.873)	0.753 (0.740)
sentence	0.867 (0.873)	0.865 (0.877)	0.875 (0.888)	–
negated question	0.518 (0.502)	0.827 (0.849)	0.854 (0.861)	0.574 (0.613)

Appendix

Table 21: Dataset-written candidates (MMLU and ARC, keyed option and one distractor, balanced). AUROC against the label, means (medians) over eight cells; readout and subspace fitted on this candidate set.

cell	verdict, correct?	verdict, incorrect?	readout, correct?	readout, incorrect?	isolated, incorrect?	fitted, incorrect?
Llama-3.1-8B ARC	0.698	0.528	0.737	0.718	0.732	0.727
Llama-3.1-8B GSM8K	0.852	0.821	0.966	0.944	0.965	0.968
Llama-3.1-8B MMLU	0.670	0.581	0.719	0.664	0.703	0.715
Llama-3.1-8B TriviaQA	0.832	0.798	0.842	0.726	0.841	0.839
Mistral-7B ARC	0.708	0.557	0.730	0.646	0.687	0.663
Mistral-7B GSM8K	0.792	0.761	0.826	0.744	0.817	0.822

Appendix

Table 22: The negated question per cell: AUROC of the model’s verdict under “correct?” and “incorrect?”; the readout under each; the isolated judgment under “incorrect?”; and a readout fitted on the negated states. Five wordings.

condition	readout	isolated judgment
Yes/No	0.996 (0.98–1.00)	0.988 (0.94–1.00)
True/False	0.991 (0.94–1.00)	0.974 (0.89–1.00)
A/B	0.977 (0.87–1.00)	0.969 (0.86–0.99)
sentence	0.970 (0.86–0.99)	0.947 (0.84–0.99)
1/0	0.988 (0.93–1.00)	0.970 (0.86–1.00)
correct/incorrect	0.986 (0.90–1.00)	0.977 (0.89–1.00)

Appendix

Table 23: Readouts fitted to model verdicts: the class mean whose classes are the model’s own Yes/No verdicts, before and after removing the expression subspace, scored by AUROC against the model’s own Yes/No verdict on each condition’s held-out states. Medians over the cells with a defined verdict, five wordings; in parentheses, the range over cells.

cell	negated		No means correct		False means correct
	plain	isolated	plain	isolated	plain	isolated
Llama-3.1-8B ARC	0.427	0.762	0.757	0.860	0.829	0.883
Llama-3.1-8B GSM8K	0.425	0.935	0.845	0.922	0.817	0.910
Llama-3.1-8B MMLU	0.488	0.708	0.797	0.790	0.798	0.806
Llama-3.1-8B TriviaQA	0.226	0.905	0.610	0.930	0.817	0.920
Mistral-7B GSM8K	0.204	0.857	0.829	0.936	0.856	0.983

Appendix

Table 24: Readouts fitted to model verdicts per cell: AUROC against the model’s own Yes/No verdict, for the readout fitted to the model’s verdicts (plain) and the same with the expression subspace removed (isolated), under the negated question and two reversed conditions. Five wordings. Thirteen cells; Mistral-7B ARC has no defined verdict class.

cell	verdicts agree	vs Yes/No verdict	vs negated verdict	vs Yes/No, disagreeing rows
Llama-3.1-8B ARC	0.219	0.762	0.287	0.967
Llama-3.1-8B GSM8K	0.682	0.935	0.692	0.963
Llama-3.1-8B MMLU	0.251	0.708	0.394	0.855
Llama-3.1-8B TriviaQA	0.687	0.905	0.732	0.760
Mistral-7B GSM8K	0.801	0.857	0.783	0.877
Mistral-7B MMLU	0.476	0.820	0.403	0.817

Appendix

Table 25: Under the negated question, the isolated judgment fitted to the model’s own Yes/No verdicts (with the expression subspace removed, no benchmark label) scored against the model’s Yes/No verdict and against its negated-prompt verdict, on rows where both exist; “verdicts agree” is the fraction of rows on which the two verdicts express the same judgment; the last column restricts to rows on which they disagree. Means over five wordings per cell. –: fewer than twenty disagreeing rows.

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Cross-Model Disagreement as a Label-Free Correctness Signal

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Detecting when a language model is wrong without ground truth labels is a fundamental challenge for safe deployment. Existing approaches rely on a model's own uncertainty -- such as token entropy or confidence scores -- but these signals fail critically on the most dangerous failure mode: confident errors, where a model is wrong but certain. In this work we introduce cross-model disagreement as a correctness indicator -- a simple, training-free signal that can be dropped into existing production systems, routing pipelines, and deployment monitoring infrastructure without modification. Given a model's generated answer, cross-model disagreement computes how surprised or uncertain a second verifier model is when reading that answer via a single forward pass. No generation from the verifying model is required, and no correctness labels are needed. We instantiate this principle as Cross-Model Perplexity (CMP), which measures the verifying model's surprise at the generating model's answer tokens, and Cross-Model Entropy (CME), which measures the verifying model's uncertainty at those positions. Both CMP and CME outperform within-model uncertainty baselines across benchmarks spanning reasoning, retrieval, and mathematical problem solving (MMLU, TriviaQA, and GSM8K). On MMLU, CMP achieves a mean AUROC of 0.75 against a within-model entropy baseline of 0.59. These results establish cross-model disagreement as a practical, training-free approach to label-free correctness estimation, with direct applications in deployment monitoring, model routing, selective prediction, data filtering, and scalable oversight of production language model systems.

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