Knowing Is Not Choosing: What Explicit Verification Adds Beyond Generative Preference

Yilong Li; Chengpo Yan; Aayan Arish; Suman Banerjee

cs.CLSep 27, 2026

Knowing Is Not Choosing: What Explicit Verification Adds Beyond Generative Preference

Authors: Yilong Li, Chengpo Yan, Aayan Arish, Suman Banerjee

Organizations: Department of Computer Science University of Wisconsin-Madison Madison, WI 53703, USA

Abstract

Generating a correct answer does not mean that a language model will select it. We separate factual recall into three steps: generating a correct candidate, ranking the available candidates, and selecting the final answer. Pre-generation readouts predict factual recall and which questions sampling will cover across three model families, but say little about whether an available correct answer will ultimately be selected. Explicit verification with $P(\mathrm{True})$ improves within-question ranking over mean log-likelihood in Gemma, Qwen3, and Llama, with AUROC gains of $0.08$ -- $0.12$ . In a prospectively defined Gemma cohort, verification raises plurality accuracy by about $5$ points, and still gains about $2$ points over chat-template likelihood, a stronger generative baseline. The advantage is strongest for relations with common-answer priors and depends on access to the entity; masking the entity removes the ranking advantage in larger Qwen models. Finally, the measured benefit depends on how correctness is defined: recall-oriented reference matching can credit option lists favored by likelihood and substantially understate the improvement seen under human semantic judgments. Prior work shows that models can carry latent factual knowledge and judge candidate answers; we show that these capabilities do not collapse into a single notion of ``knowing,'' and trace where information is gained, lost, or mismeasured between availability, ranking, and final choice.

Figures & tables

Figure 1: Same candidates, different rankings. Generation likelihood reflects “Would I say this?” ; $P(\mathrm{True})$ asks the same model whether the answer is correct (prompt in Appendix Y ) and takes its Yes probability, normalized over Yes and No . Box texts paraphrase the two scores; rankings are illustrative.

Panel of Figure 2 — Figure 2: What verification adds beyond generative baselines. Gemma plurality accuracy on 541 questions under semantic labels. (a) Generative baselines versus $P(\mathrm{True})$ . (b) Results split by common-answer prior. (c) Entity masking on high-prior questions. Brackets show paired differences with 95% intervals.

Model	$N$	$\Delta$ AUROC		Plurality accuracy (%)	Gain (points)
Gemma 2B	$541$	$+0.102$	$[0.056,0.149]$	$23.11{\to}28.10$	$+4.99$	$[2.7,7.3]$
Gemma 9B	$581$	$+0.032$	$[0.006,0.058]$	$43.55{\to}43.72$	$+0.17$	$[-0.86,1.20]$
Gemma 27B	$564$	$+0.030$	$[-0.033,0.091]$	$48.05{\to}48.67$	$+0.62$	$[-1.03,2.28]$
Llama 8B	$598$	$+0.119$	$[0.084,0.154]$	$43.48{\to}45.15$	$+1.67$	$[0.05,3.30]$
Qwen3.5 2B	$528$	$+0.008$	$[-0.053,0.069]$	$11.55{\to}9.94$	$-1.61$	$[-3.29,0.00]$
Qwen3.5 4B	$565$	$+0.139$	$[0.101,0.176]$	$19.47{\to}21.59$	$+2.12$	$[0.00,4.26]$

Table 1: Verification improves ranking more consistently than final selection. Stricter criterion,

K=16

; accuracy runs from mean log-likelihood to

P(\mathrm{True})

, and

\Delta

AUROC (pair-weighted) and gain are their differences with paired

95\%

intervals. Gemma 2B is the prospective cohort; the other rows reuse its questions, and the four newer Qwen models are post hoc (Appendix E ). Coverage and conversion: Appendices A , M , and U .

Panel of Figure 3 — Figure 3: Recall information beyond exposure. $\Delta L$ is the held-out log-loss reduction from adding the readout. Panel (a) shows type-macro gains with $95\%$ joint-refit entity-bootstrap intervals; (b) shows Gemma by type.

Panel of Figure 4 — Figure 4: Pre-generation readout across and within questions. (a) $K=16$ coverage by readout quintile under the stricter criterion ( $521$ Gemma questions after excluding entities used to fit the direction; $598$ Llama). (b) Pair-weighted AUROC on mixed questions ( $199$ Gemma, $302$ Llama, $168$ Qwen3). (c) Held-out log-loss improvement from adding the readout to $M_{0}$ , for coverage and for plurality success on covered questions ( $192$ , $355$ , and $244$ ). Intervals are $95\%$ entity-clustered.

Panel of Figure 5 — Figure 5: Verification across recent Qwen models. Stricter criterion, $16$ raw samples per question. (a) Coverage and plurality conversion across five models (likelihood amber, $P(\mathrm{True})$ teal). (b,c) Ranking and selection differences relative to chat-template likelihood for $P(\mathrm{True})$ and entity-masked $P(\mathrm{True})$ on Qwen3.6-27B and Qwen3.8-27B. Error bars show paired $95\%$ intervals.

Panel of Figure 6 — Figure 6: From coverage to final answers. Gemma, $541$ questions, $K=16$ . (a, b) Argmax selection under the stricter criterion; in (b), $161$ of the $204$ covered questions have at least one correct selection. (c) Plurality gain of $P(\mathrm{True})$ over mean log-likelihood under four labelings of the same selected outputs, with paired $95\%$ intervals; the bracket gives the difference between semantic labels and reference matching.

Appendix figures & tables51 assets

Supplementary material from the paper’s appendix.

Appendix

Single-answer decoding
Method	Pair-weighted AUROC	Argmax selection	Plurality selection	Conversion
Mean log-likelihood	$0.6798$	$21.44\%$	$23.11\%$	$61.27\%$
Summed log-likelihood	$0.6870$	$24.40\%$	$24.58\%$	$65.20\%$
$P(\mathrm{True})$	$\mathbf{0.7820}$	$\mathbf{26.80\%}$	$\mathbf{28.10\%}$	$\mathbf{74.51\%}$
Plurality	—	—	$21.15\%$	$56.08\%$
Oracle	—	$37.71\%$	—	$100\%$

Appendix

Table 2: Ranking and selection on fixed candidates. Stricter criterion,

541

questions,

T=1

K=16

. AUROC is pair-weighted over

199

mixed questions; conversion divides accuracy by coverage (

204/541

), using plurality accuracy for the scored rows. Unscored plurality breaks ties by draw order, averaged over

400

random orderings of each pool. The oracle and greedy rows return one answer per question, so their accuracy is listed under Argmax; the oracle returns a correct candidate whenever one exists, so its conversion is

100\%

by definition. Greedy is a single-answer baseline on the same questions.

$K$	Coverage $C_{K}$	Accuracy $A_{K}$	Conversion $R_{K}$
$1$	$0.1743$	$0.1743$	$1.0000$
$2$	$0.2389$	$0.1952$	$0.8171$
$4$	$0.2960$	$0.2088$	$0.7055$
$8$	$0.3482$	$0.2216$	$0.6364$
$16$	$0.3771$	$0.2311$	$0.6127$

Appendix

Table 3: Coverage grows faster than accuracy.

541

eligible questions from the fresh

600

-question cohort,

T=1

, stricter criterion, plurality with likelihood tie-breaking.

K

is a retained-pool budget, capped at the pool size.

R_{K}=A_{K}/C_{K}

measures conversion.

Figure 7: Little of the added coverage is converted. The

541

prospective Gemma questions under the stricter criterion. Black shows coverage; teal and dashed amber show plurality accuracy with

P(\mathrm{True})

and likelihood. At

K=16

, verification raises conversion from

61.3\%

74.5\%

, improving accuracy by

4.99

points while

25.5\%

of covered questions remain unconverted.

Selection outcome	Questions
Both select a correct answer	$100$
Only likelihood selects a correct answer	$16$
Only $P(\mathrm{True})$ selects a correct answer	$45$
Neither selects a correct answer	$380$
Oracle over the two selections	$161/541=29.8\%$
Candidate coverage	$204/541=37.7\%$

Appendix

Table 4: Likelihood and verification recover different correct answers. Argmax outcomes partition the same

541

questions under the stricter criterion. The oracle uses labels to choose between the two selectors’ outputs, so it measures their combined opportunity.

Criterion	Selection	Likelihood	$P(\mathrm{True})$	Difference	95% CI
Reference	Argmax	$26.62$	$27.36$	$+0.74$	$[-1.71,3.19]$
Stricter	Argmax	$21.44$	$26.80$	$\mathbf{+5.36}$	$[2.56,8.19]$
Reference	Plurality	$26.99$	$28.28$	$+1.29$	$[-0.8,3.4]$
Stricter	Plurality	$23.11$	$28.10$	$\mathbf{+4.99}$	$[2.7,7.3]$

Appendix

Table 5: Selection gains under the two automatic criteria. Same

541

questions, candidates, scores, and selected outputs; only correctness labels change. Accuracies are percentages; differences and paired

95\%

entity-clustered intervals (

20{,}000

draws) are percentage points.

Figure 8: Reference matching credits option lists. Two option lists that contain the reference answer: reference matching accepts them and the stricter criterion rejects them. Figure 6 c shows the resulting plurality gains under each criterion.

Method	Argmax selection	Plurality selection	Conversion
Mean log-likelihood, filtered	$25.32\%$	$25.69\%$	$68.14\%$

Appendix

Table 6: Likelihood after the text filter. Stricter criterion,

541

questions,

T=1

K=16

; conversion is plurality accuracy divided by coverage.

Model	Metric	$M_{0}$	$M_{1}$	Improvement	95% CI
Gemma 2 2B-IT	Log loss	$0.295885$	$0.285492$	$+0.010393$	$[0.006103,0.014701]$
	Brier	$0.088757$	$0.084993$	$+0.003764$	$[0.002253,0.005272]$
Qwen3-8B	Log loss	$0.328104$	$0.322458$	$+0.005646$	$[0.002218,0.008922]$
	Brier	$0.100192$	$0.098240$	$+0.001952$	$[0.000765,0.003144]$
Llama 3.1 8B-Instruct	Log loss	$0.436713$	$0.419618$	$+0.017095$	$[0.009969,0.024223]$
	Brier	$0.139928$	$0.133543$	$+0.006385$	$[0.003911,0.008907]$

Appendix

Table 7: Internal recall information extends across model families. Each model is evaluated on

3{,}840

entities, including a

1{,}280

-entity confirmation role. Gemma and Llama use matched cohorts and layer-

16

and layer-

15

class-mean readouts; Qwen3 uses a separate cohort and fixed layer-

23

feature under its amended protocol (Appendix G ). Losses and improvements are equal-weighted type macros; intervals use

10{,}000

joint-refit entity-bootstrap draws.

	Gemma 2 2B-IT		Qwen3-8B		Llama 3.1 8B-Instruct
Type	$\Delta$ log loss	$\Delta$ Brier	$\Delta$ log loss	$\Delta$ Brier	$\Delta$ log loss	$\Delta$ Brier
Player	$+0.004327$	$+0.000986$	$-0.001800$	$-0.000648$	$+0.021228$	$+0.009470$
City	$+0.018755$	$+0.008791$	$+0.012650$	$+0.005363$	$+0.013850$	$+0.003681$
Movie	$+0.016274$	$+0.005028$	$+0.009230$	$+0.002342$	$+0.019916$	$+0.007289$
Song	$+0.002215$	$+0.000252$	$+0.002505$	$+0.000751$	$+0.013388$	$+0.005102$

Appendix

Table 8: Recall improvement by entity type. Type-specific point estimates for all three recall studies. The readout improves both scores in all four types for Gemma and Llama and in three for Qwen3.

Readout	Log-loss improvement (95% CI)	Brier improvement (95% CI)
Latent $7957$	$+0.007234\ [0.003492,0.011051]$	$+0.002566\ [0.001304,0.003877]$
$-\,$ latent $11898$	$+0.002124\ [0.000664,0.003566]$	$+0.000492\ [0.000017,0.000952]$

Appendix

Table 9: SAE latents also carry recall information. Improvements on the prospective Gemma recall cohort, with fitted models held fixed.

Comparison	Questions	Query-macro AUROC	95% CI
Greedy answer matches: matching vs. non-matching samples	$155$	$0.7569$	$[0.7323,0.7829]$
Greedy answer misses: matching vs. non-matching samples	$90$	$0.6073$	$[0.5685,0.6475]$

Appendix

Table 10: Within-question ranking persists when the greedy answer is wrong. In the earlier same-question sample, likelihood ranks sampled answers that match the reference against sampled answers that do not. Intervals are entity-clustered.

Temperature	Accuracy	Mixed questions	Query-macro AUROC	Pooled AUROC
$0.3$	$26.7\%$	$38$	$0.6312$	$0.7729$
$0.7$	$24.4\%$	$88$	$0.6692$	$0.7646$
$1.0$	$19.7\%$	$109$	$0.7055$	$0.7429$
$1.3$	$15.2\%$	$106$	$0.7378$	$0.7427$

Appendix

Table 11: Pooled and within-question ranking respond differently to decoding. The development sampling cohort is sampled at four temperatures (Appendix I ). Statistics use retained responses under reference matching; the mixed-question population changes with temperature. Query-macro AUROC weights mixed questions equally.

Study	Model and population	Evaluation
Prospective recall	Gemma 2 2B-IT; $3{,}840$ new entities	Fit/calibration/confirmation roles of $1{,}280$ entities each
Recall replication	Qwen3-8B; $3{,}840$ new entities	Separate fixed feature, cohort, and protocol
Recall replication	Llama 3.1 8B-Instruct; $3{,}840$ entities	Gemma-matched cohort and roles; separate layer- $15$ class-mean direction
Earlier same-question sample	Gemma; $600$ questions	$16$ samples at $T=1$ , top- $p$ $0.95$ ; $245$ retained mixed questions
Development sampling cohort	Gemma; separate sample of $250$ questions	Stricter-criterion temperature cohorts: $120$ name-valued, $170$ answerable
$512$ -entity chat cohort	Gemma 2 2B-IT and 2B; $512$ entities	$2{,}176$ human-labeled chat responses; paired pretrained and instruction-tuned recall study

Appendix

Table 12: Study populations. Denominators are not pooled across studies.

Analysis	Status
Gemma recall confirmation	Protocol fixed before measurement; positive
Qwen3 independent-cohort recall	Positive under the amended permutation limit
Llama matched-cohort recall	Positive; pre-specified controls met
Gemma prospective selection	Protocol fixed before generation
Llama selection inputs	Same questions and fixed rules as Gemma
Qwen3 selection inputs	Same questions and fixed rules as Gemma

Appendix

Table 13: Status of the reported analyses.

Direction	SAE	Latent	MaxMin	Pile frequency
Known	layer_12/width_16k/average_l0_82	$7{,}957$	$0.64049$	$0.0150$
Unknown	layer_14/width_16k/average_l0_84	$11{,}898$	$0.27795$	$0.0076$

Appendix

Table 14: Discovery recovers both published latents. Scores are bfloat16 on CUDA; float32 scores on Apple MPS differ slightly.

Condition	Refusals
Unsteered	$128$
Zero direction	$128$
Unknown latent $11{,}898$	$374$
Known latent $7{,}957$	$55$
Strongest matched random latent, unknown side	$194$

Appendix

Table 15: Refusal steering on Gemma 2 2B-IT. Refusals among

400

unknown-entity prompts, counted by the automatic matcher of the reference implementation.

Predictor	AUROC	95% interval
Latent $11{,}898$	$0.6598$	$[0.6292,0.6880]$
Random ordering	$0.4959$	$[0.4577,0.5344]$
Baseline: exposure, missingness, type, relation	$0.8804$	$[0.8589,0.9002]$
Baseline and latent $11{,}898$	$0.8819$	$[0.8609,0.9013]$

Appendix

Table 16: Ranking wrong against correct attempted answers.

512

-entity chat cohort; AUROCs compare responses within each evaluation fold, with

2{,}000

type- and fold-stratified entity-bootstrap draws.

Policy	Retention	Correct	Wrong	Undeterminable	Risk bounds
Answer all	$100\%$	$476$	$1{,}204$	$84$	$[68.25,73.02]\%$
Random	$78.57\%$	$372$	$948$	$65$	$[68.45,73.14]\%$
Latent $11{,}898$	$80.30\%$	$441$	$908$	$64$	$[64.26,68.79]\%$
Baseline	$79.64\%$	$469$	$869$	$72$	$[61.63,66.74]\%$
Baseline and latent	$80.00\%$	$470$	$874$	$74$	$[61.64,66.85]\%$

Appendix

Table 17: Selective answering at about

80\%

retention. Thresholds are selected on calibration folds; counts use the human labels, and the risk bounds let every released undeterminable response be either correct or wrong.

Bare format	Chat correct	Chat wrong	Chat refusal	Undeterminable
Hit	$413$	$189$	$32$	$33$
Miss	$63$	$1{,}015$	$380$	$51$

Appendix

Table 18: Chat outcomes by bare-format outcome.

512

-entity chat cohort; bare-format hits and misses use reference matching and chat outcomes the human labels.

Criterion	$\tau$	Likelihood	$P(\mathrm{True})$	Difference	95% CI
Stricter	$0.05$	$21.63$	$27.54$	$+5.91$	$[3.33,8.58]$
	$0.1$	$22.00$	$27.54$	$+5.55$	$[2.99,8.16]$
	$0.3$	$22.74$	$27.36$	$+4.62$	$[2.18,7.16]$
	$1$	$23.11$	$27.54$	$+4.44$	$[2.19,6.80]$
Reference	$0.05$	$26.80$	$27.91$	$+1.11$	$[-1.13,3.42]$
	$0.1$	$26.99$	$27.91$	$+0.92$	$[-1.30,3.19]$

Appendix

Table 19: Verification improves confidence-weighted selection. Accuracies are percentages; differences and paired

95\%

intervals are percentage points. Scores, candidates, and aggregation temperatures are identical between criteria. Intervals are pointwise.

Automatic outcome	Responses	Annotator A	Annotator B
Both criteria accept	$91$	$79/2/10$	$69/3/19$
Reference matching only	$46$	$17/27/2$	$15/30/1$
Neither criterion	$263$	$1/180/82$	$3/219/41$

Appendix

Table 20: Annotator labels by automatic outcome. Correct / incorrect / ambiguous counts over the enriched audit set; these are not population rates.

Selector	Annotator A	Annotator B	Stricter criterion	Reference matching
Likelihood	$25$	$24$	$22$	$28$
$P(\mathrm{True})$	$28$	$23$	$30$	$31$

Appendix

Table 21: Correct argmax selections on the

100

uniformly sampled questions.

Criterion	Rule	Likelihood	$P(\mathrm{True})$	Difference	95% CI
Stricter	Argmax	$43.48$	$44.82$	$+1.34$	$[-0.48,3.16]$
	Plurality	$43.48$	$45.15$	$+1.67$	$[0.05,3.30]$
Reference	Argmax	$44.98$	$46.15$	$+1.17$	$[-0.61,2.96]$
	Plurality	$44.98$	$46.49$	$+1.51$	$[-0.16,3.18]$

Appendix

Table 22: Llama selection on matched input questions. Original verification prompt,

598

eligible questions. Accuracies are percentages; differences and paired entity-clustered

95\%

intervals are percentage points.

Figure 9: Factual competence profiles. Recall improvement, the readout’s AUROC for

K=16

coverage (Section 4.2 ; Gemma on the

521

questions used there),

K=16

coverage on each model’s eligible questions (

541

598

, and

600

), pair-weighted AUROC, and plurality accuracy. Arrows compare mean log-likelihood with

P(\mathrm{True})

Panel of Figure 10 — Figure 10: Rank transitions. Stricter criterion. Mixed questions counted by the rank of the highest-scoring correct candidate under likelihood (rows) and verification (columns), under direct score ordering.

Model	Candidates	Mixed questions	Likelihood	$P(\mathrm{True})$	Query-macro gain [95% CI]
Gemma	$5{,}649$	$184$	$0.722$	$0.776$	$+0.0667$ $[0.0257,0.1072]$
Llama	$8{,}242$	$287$	$0.633$	$0.754$	$+0.1234$ $[0.0942,0.1521]$

Appendix

Table 23: Factual ranking after the text filter. Stricter criterion. Pair-weighted AUROC uses mixed questions; the gain and interval average question-level differences with entity-clustered resampling.

Model	Rule	Likelihood	$P(\mathrm{True})$	Gain [95% CI]
Gemma	Argmax	$25.37$	$27.78$	$+2.41$ $[0.22,5.12]$
	Plurality	$25.74$	$29.26$	$+3.52$ $[1.57,5.47]$
Llama	Argmax	$44.15$	$45.48$	$+1.34$ $[-0.43,3.10]$
	Plurality	$44.15$	$45.65$	$+1.51$ $[0.02,3.05]$

Appendix

Table 24: Selection after the text filter. Stricter criterion. Accuracies are percentages; gains and entity-clustered

95\%

intervals are percentage points.

Model	Relations	$n$	Mixed	AUROC	Likelihood	$P(\mathrm{True})$	Gain [95% CI]
Gemma	High-prior	$229$	$113$	$0.618\rightarrow 0.810$	$27.51$	$37.55$	$+10.04$ $[5.53,14.31]$
	Low-prior	$312$	$86$	$0.744\rightarrow 0.753$	$19.87$	$21.15$	$+1.28$ $[-0.98,3.59]$
	High $-$ low	—	—	—	—	—	$+8.76$ $[3.5,13.7]$
Llama	High-prior	$283$	$162$	$0.631\rightarrow 0.756$	$47.70$	$51.59$	$+3.89$ $[0.34,7.46]$
	Low-prior	$315$	$140$	$0.649\rightarrow 0.761$	$39.68$	$39.37$	$-0.32$ $[-3.73,3.08]$
	High $-$ low	—	—	—	—	—	$+4.20$ $[-0.75,9.15]$

Appendix

Table 25: Ranking and selection by relation group. High-prior relations are the four with at least

10\%

common-answer coverage;

n

counts questions. Stricter criterion; pair-weighted AUROC over mixed questions; plurality accuracy in percent; gains and entity-clustered

95\%

intervals in points. High

-

low rows give the difference between the two gains.

Model	Wording	Pair-weighted AUROC	Argmax	Plurality	Plurality gain [95% CI]
Gemma	Original	$0.7827$	$26.80\%$	$28.10\%$	$+4.99\ [2.7,7.3]$
	Short	$0.7806$	$26.80\%$	$27.91\%$	$+4.81\ [2.60,7.12]$
	True/False	$0.7697$	$25.69\%$	$27.17\%$	$+4.07\ [1.83,6.46]$
	No/Yes	$0.7479$	$24.40\%$	$26.43\%$	$+3.33\ [1.09,5.70]$
Llama	Original	$0.7586$	$44.82\%$	$45.15\%$	$+1.67\ [0.05,3.30]$
	Short	$0.7568$	$44.65\%$	$44.98\%$	$+1.51\ [-0.20,3.20]$

Appendix

Table 26: Verification under four prompt wordings. Stricter criterion; pair-weighted AUROC and selection accuracy for

P(\mathrm{True})

. The last column is its paired plurality gain over likelihood, in percentage points with a pointwise

95\%

entity-clustered interval. Gemma has

541

eligible questions; Llama has

598

$K$	Coverage	Likelihood accuracy	Verification accuracy	Gain	Likelihood cost	Verification cost
$1$	$36.75$	$36.75$	$36.75$	$0.00$	$52$	$121$
$2$	$44.55$	$38.98$	$40.91$	$+1.93$	$104$	$241$
$4$	$50.25$	$40.50$	$43.40$	$+2.90$	$208$	$482$
$8$	$55.04$	$41.94$	$44.65$	$+2.71$	$416$	$965$
$16$	$59.36$	$43.48$	$45.15$	$+1.67$	$812$	$1{,}885$

Appendix

Table 27: Selection and token-accounted cost. Llama,

598

questions, stricter criterion, plurality with the named tie-break. Accuracy and coverage are percentages; gain is verification minus likelihood accuracy, in points. Cost is mean accounted tokens per question.

Model	Budget	Uniform	Random	Risk	R+T	R+T+S	Difference [95% CI]
Gemma	$7.00$	$25.49$	$24.92$	$24.17$	$25.70$	$25.42$	$-0.07\ [-0.86,0.74]$
	$9.79$	$26.61$	$26.07$	$24.83$	$26.89$	$26.63$	$+0.02\ [-0.74,0.76]$
	$11.86$	$27.09$	$26.92$	$26.69$	$26.95$	$26.87$	$-0.23\ [-0.72,0.21]$
Llama	$7.00$	$44.52$	$43.73$	$43.69$	$44.15$	$44.27$	$-0.25\ [-1.03,0.54]$
	$9.99$	$44.88$	$44.14$	$43.80$	$44.48$	$44.73$	$-0.15\ [-0.95,0.65]$
	$12.88$	$45.04$	$44.54$	$44.06$	$44.59$	$45.09$	$+0.05\ [-0.56,0.67]$

Appendix

Table 28: Recall-guided allocation at matched candidate budgets. Percent accuracy with verification-based plurality. The final column is the readout allocator’s gain over uniform in points (

95\%

interval). R+T denotes relation/type; S adds the readout. Risk allocates extra candidates to the questions with the lowest readout scores first.

Panel of Figure 11 — Figure 11: Recall-guided allocation at matched candidate budgets. Gains are relative to uniform allocation. Shading shows the paired $95\%$ interval for the readout-augmented allocator.

Model	$K$	Coverage	Likelihood accuracy	Verification	Likelihood time	Verification time
Gemma	$4$	$24.80$	$18.90$	$21.03$	$0.849$	$0.969$
	$8$	$31.12$	$20.66$	$24.17$	$0.992$	$1.211$
	$16$	$36.45$	$21.97$	$26.31$	$1.289$	$1.717$
	$32$	$41.33$	$23.67$	$28.33$	$1.887$	$2.735$
Llama	$4$	$49.80$	$40.64$	$42.76$	$0.776$	$0.865$
	$8$	$54.42$	$41.83$	$43.74$	$0.876$	$1.029$

Appendix

Table 29: Coverage, accuracy, and measured time. Accuracies are percentages; costs are seconds per question in the implemented pipeline.

Figure 12: Higher temperature raises coverage and the verification gain. Gemma,

600

questions with new samples, raw budget

K=16

, stricter criterion, plurality selection. Lower brackets give the gain of

P(\mathrm{True})

at each temperature, and the top bracket the difference between the two gains, with paired

95\%

intervals.

Panel of Figure 13 — Figure 13: Accuracy at measured inference cost. Labels mark candidate budgets; bars show $95\%$ intervals. Cost is normalized to likelihood at $K=16$ .

Model	Quintile	Entities	Questions	Covered	Coverage [95% CI]
Gemma	1 (lowest)	$87$	$99$	$11$	$11.1$ $[5.3,17.8]$
	2	$87$	$104$	$17$	$16.3$ $[9.1,24.2]$
	3	$87$	$112$	$30$	$26.8$ $[18.6,35.9]$
	4	$87$	$102$	$68$	$66.7$ $[57.6,75.2]$
	5 (highest)	$87$	$104$	$66$	$63.5$ $[54.2,72.7]$
Llama	1 (lowest)	$98$	$121$	$19$	$15.7$ $[9.8,22.3]$

Appendix

Table 30: Coverage by readout quintile. Stricter criterion, full retained pools. Coverage is in percent, with

95\%

entity-clustered intervals.

Model	Target	$n$	$\Delta L$ [95% CI]	$\Delta$ Brier [95% CI]	AUROC
Gemma	Coverage	$521$	$+0.013$ $[0.005,0.021]$	$+0.005$ $[0.002,0.008]$	$0.750\rightarrow 0.790$
	Success, likelihood	$192$	$+0.0061$ $[-0.0004,0.0128]$	$+0.0022$ $[-0.0002,0.0046]$	$0.658\rightarrow 0.673$
	Success, $P(\mathrm{True})$	$192$	$+0.0030$ $[-0.0028,0.0089]$	$+0.0011$ $[-0.0010,0.0033]$	$0.667\rightarrow 0.675$
Llama	Coverage	$598$	$+0.017$ $[0.007,0.027]$	$+0.006$ $[0.002,0.010]$	$0.792\rightarrow 0.833$
	Success, likelihood	$355$	$+0.0037$ $[-0.0008,0.0082]$	$+0.0014$ $[-0.0003,0.0031]$	$0.625\rightarrow 0.636$
	Success, $P(\mathrm{True})$	$355$	$+0.0020$ $[-0.0020,0.0061]$	$+0.0007$ $[-0.0007,0.0023]$	$0.629\rightarrow 0.635$

Appendix

Table 31: Readout information beyond question difficulty. Held-out improvement from adding the readout to

M_{0}

(exposure, name, type, and relation features and a common-answer prior; no name likelihood for Qwen3), from out-of-fold predictions over five entity-disjoint folds, with

95\%

entity-clustered intervals. Coverage is at

K=16

; success is plurality success on covered questions under the named tie-break. The Qwen3 low-prior row uses the six low-prior relations. AUROC runs from

M_{0}

M_{1}

Measure	Likelihood	$P(\mathrm{True})$	Difference [95% CI]
Pair-weighted AUROC	$0.6206$	$0.6984$	$+0.0778$ $[0.038,0.118]$
Query-macro AUROC	$0.6269$	$0.6749$	$+0.0480$ $[0.0040,0.0921]$
Argmax selection	$26.17$	$27.33$	$+1.17$ $[-0.91,3.27]$
Plurality selection	$26.17$	$27.33$	$+1.17$ $[-0.75,3.10]$
Conversion (plurality)	$64.34$	$67.21$	—

Appendix

Table 32: Qwen3-8B ranking and selection. Stricter criterion,

600

questions,

K=16

;

244

questions (

40.67\%

) are covered. AUROC uses the

168

mixed questions (

5{,}707

correct–incorrect pairs). Accuracies and conversion are percentages; accuracy differences are points.

Measure	Likelihood	$P(\mathrm{True})$	Difference [95% CI]
Pair-weighted AUROC	$0.700$	$0.732$	$+0.032$ $[0.006,0.058]$
Argmax selection	$43.55$	$43.89$	$+0.34$ $[-0.91,1.59]$
Plurality selection	$43.55$	$43.72$	$+0.17$ $[-0.86,1.20]$
Conversion (plurality)	$81.88$	$82.20$	—
Option-list selections	$0$	$0$	—

Appendix

Table 33: Gemma 2 9B-IT ranking and selection. Stricter criterion,

581

eligible questions,

K=16

;

309

questions (

53.18\%

) are covered. Accuracies and conversion are percentages; accuracy differences are points.

Measure	Likelihood	$P(\mathrm{True})$	Difference [95% CI]
Pair-weighted AUROC	$0.643$	$0.673$	$+0.030$ $[-0.033,0.091]$
Plurality selection	$48.05$	$48.67$	$+0.62$ $[-1.03,2.28]$
Conversion (plurality)	$81.63$	$82.68$	—

Appendix

Table 34: Gemma 2 27B-IT ranking and selection. Stricter criterion,

564

eligible questions,

K=16

;

332

questions (

58.87\%

) are covered. Accuracies and conversion are percentages; the accuracy difference is in points. Argmax selection and option-list counts, reported for Gemma 2 9B-IT in Table 33 , are not available for this run.

Model	Eligible	Coverage	Likelihood argmax	Likelihood plurality	$P(\mathrm{True})$ plurality	Retention
			Conversion
Qwen3.5-2B	$528$	$19.70$	$59.62$	$58.65$	$50.48$	$65.85$
Qwen3.5-4B	$565$	$33.45$	$57.67$	$58.20$	$64.55$	$73.49$
Qwen3-8B	$600$	$40.67$	$64.34$	$64.34$	$67.21$	$98.36$
Qwen3.6-27B	$599$	$48.58$	$62.54$	$61.86$	$68.73$	$89.51$
Qwen3.8-27B	$594$	$41.25$	$60.41$	$60.00$	$68.23$	$81.49$

Appendix

Table 35: Coverage and conversion across Qwen models. Stricter criterion,

T=1

, no top-

k

truncation,

16

raw samples per question. Conversion is the share of covered questions that selection answers correctly, with likelihood or

P(\mathrm{True})

; retention is the share of the

9{,}600

samples kept by the format filter. All values except eligible counts are percentages.

Model	Pair-weighted AUROC	$\Delta$	Query-macro $\Delta$	Argmax accuracy (%)	Gain (points)
Qwen3.5-2B	$0.605{\to}0.613$	$+0.008$	$+0.012$	$11.74{\to}9.94$	$-1.80$
		$[-0.053,0.069]$	$[-0.049,0.073]$		$[-3.53,-0.10]$
Qwen3.5-4B	$0.617{\to}0.755$	$+0.139$	$+0.131$	$19.29{\to}21.59$	$+2.30$
		$[0.101,0.176]$	$[0.090,0.171]$		$[0.18,4.48]$
Qwen3.6-27B	$0.612{\to}0.792$	$+0.180$	$+0.198$	$30.38{\to}33.72$	$+3.34$
		$[0.149,0.213]$	$[0.162,0.235]$		$[0.87,5.87]$

Appendix

Table 36: Ranking and argmax selection in the newer Qwen models. Stricter criterion. Arrows run from mean log-likelihood to

P(\mathrm{True})

; differences are

P(\mathrm{True})

minus mean log-likelihood, with paired

95\%

entity-clustered intervals below. Pair-weighted and query-macro differences have separate intervals; plurality results are in Table 1 .

Labels	Coverage	Pair-weighted AUROC	Argmax gain	Plurality gain
Reference	$38.82\%$	$0.711\rightarrow 0.733$	$+0.74$ $[-1.71,3.19]$	$+1.29$ $[-0.8,3.4]$
Stricter	$37.71\%$	$0.680\rightarrow 0.782$	$+5.36$ $[2.56,8.19]$	$+4.99$ $[2.7,7.3]$
Semantic	$38.63\%$	$0.701\rightarrow 0.778$	$+4.07$ $[1.6,6.5]$	$+3.88$ $[1.6,6.1]$
Human	—	—	$+4.25$ $[1.63,6.87]$	$+3.70$ $[1.31,6.08]$
Semantic $-$ reference	—	—	$+3.33$ $[1.2,5.4]$	$+2.59$ $[0.9,4.3]$
Stricter $-$ semantic	—	—	$+1.29$ $[0.28,2.36]$	$+1.11$ $[0.21,2.05]$

Appendix

Table 37: Selection gains under four labelings. Gemma,

K=16

, all

541

questions (every reference passes the validity check). Pair-weighted AUROC is shown from likelihood to

P(\mathrm{True})

; gains are points with paired

95\%

entity-clustered intervals. Human labels cover every output on which the selectors differ, so the human gains apply to the whole cohort.

Selection	Likelihood	$P(\mathrm{True})$	Difference	Option lists
Argmax	$3.51$ $[1.95,5.15]$	$0.55$ $[0.05,1.30]$	$+2.96$ $[1.40,4.55]$	$18/19$ vs. $1/3$
Plurality	$2.22$ $[1.15,3.84]$	$0.37$ $[0.00,0.95]$	$+1.85$ $[0.58,3.16]$	$12/12$ vs. $0/2$

Appendix

Table 38: Reference-matching errors by selector. Errors are selected outputs whose label under reference matching differs from the semantic label, as percentages of the

541

questions, with paired

95\%

entity-clustered intervals. The last column counts the errors that are option lists.

Model	Target	$n$	AUROC	$\Delta L$ [95% CI]	$\Delta$ Brier [95% CI]
Gemma	Coverage	$521$	$0.779$	$+0.0120$ $[0.0040,0.0202]$	$+0.0044$ $[0.0013,0.0075]$
	Success, likelihood	$197$	—	$+0.0055$ $[-0.0010,0.0121]$	$+0.0020$ $[-0.0003,0.0043]$
	Success, $P(\mathrm{True})$	$197$	—	$+0.0028$ $[-0.0031,0.0087]$	$+0.0010$ $[-0.0010,0.0031]$
Llama	Coverage	$598$	$0.812$	$+0.0155$ $[0.0060,0.0252]$	$+0.0056$ $[0.0020,0.0093]$
	Success, likelihood	$359$	—	$+0.0033$ $[-0.0011,0.0079]$	$+0.0012$ $[-0.0004,0.0029]$
	Success, $P(\mathrm{True})$	$359$	—	$+0.0018$ $[-0.0022,0.0060]$	$+0.0007$ $[-0.0008,0.0022]$

Appendix

Table 39: Readout information under semantic labels. Held-out improvement from adding the readout to

M_{0}

, with

95\%

entity-clustered intervals; AUROC is the readout’s raw AUROC for coverage.

Panel of Figure 14 — Figure 14: Figure 2 under the stricter criterion. Same questions, scores, and layout. Brackets give paired differences in points with $95\%$ intervals.

Score	High-prior	Low-prior	All
Mean log-likelihood	$109$ ( $66.06\%$ )	$48$ ( $60.76\%$ )	$157$ ( $64.34\%$ )
Chat-template likelihood	$109$ ( $66.06\%$ )	$41$ ( $51.90\%$ )	$150$ ( $61.48\%$ )
PMI	$104$ ( $63.03\%$ )	$37$ ( $46.84\%$ )	$141$ ( $57.79\%$ )
$P(\mathrm{True})$	$115$ ( $69.70\%$ )	$49$ ( $62.03\%$ )	$164$ ( $67.21\%$ )

Appendix

Table 40: Qwen3-8B plurality selection by relation group. Stricter criterion. Correct selections, with conversion among covered questions (

165

high-prior,

79

low-prior,

244

in total).

Score	Qwen3.6-27B	Qwen3.8-27B
Mean log-likelihood	$30.05$ / $0.612$	$24.75$ / $0.625$
Summed log-likelihood	$29.38$ / $0.604$	$24.58$ / $0.620$
Chat-template likelihood	$31.22$ / $0.624$	$25.25$ / $0.643$
PMI	$30.38$ / $0.645$	$24.24$ / $0.659$
$P(\mathrm{True})$	$33.39$ / $0.792$	$28.14$ / $0.773$
Masked $P(\mathrm{True})$	$30.88$ / $0.612$	$24.47$ / $0.629$

Appendix

Table 41: Scores on the same candidates in two 27B Qwen models. Stricter criterion,

T=1

, no top-

k

truncation,

16

raw samples per question;

599

and

594

eligible questions. Each cell gives plurality accuracy (%) and pair-weighted AUROC. Masking differences in the text are paired against unmasked

P(\mathrm{True})

rescored in the same run, whose plurality accuracy is

33.44\%

and

27.98\%

Model	Measure	$P(\mathrm{True})-$ chat	Masked $\,-\,$ chat	Unmasked $\,-\,$ masked
Qwen3.6-27B	Accuracy	$+2.17$	$-0.33$	$+2.56$
		$[-0.16,4.58]$	$[-2.84,2.14]$	$[0.22,5.01]$
	Pair-weighted	$+0.1687$	$-0.0112$	$+0.1810$
		$[0.1368,0.2009]$	$[-0.0413,0.0192]$	$[0.1497,0.2125]$
	Query-macro	$+0.1828$	$+0.0085$	$+0.1748$
		$[0.1482,0.2178]$	$[-0.0284,0.0462]$	$[0.1427,0.2078]$

Appendix

Table 42: Paired differences in two 27B Qwen models. Plurality accuracy in points and pair-weighted and query-macro AUROC, with paired

95\%

intervals below. In the last column, unmasked

P(\mathrm{True})

is rescored in the same run as masked

P(\mathrm{True})

Source	Measurement	Result
Residual state	Last prompt token, $27$ layers	$+0.0015$
Subject attention	$26$ layers, $8$ heads	$-0.0212$
Output probability	Greedy chat answer, raw AUROC	$0.4813$
Binding	Subject-by-relation contrast	$-0.0051$

Appendix

Table 43: Readout diagnostics against relation identity. Reference-matching labels. The first, second, and fourth rows are AUROC improvements over relation identity; the third is a raw AUROC across questions.

Model	Yes	No	True	False
Gemma 2 2B-IT	$3553$	$1294$	$5036$	$8393$
Llama 3.1 8B-Instruct	$9642$	$2822$	$2575$	$4139$
Qwen3-8B	$9454$	$2753$	—	—

Appendix

Table 44: Label token IDs. True and False are used only by the true/false wording, which is evaluated for Gemma and Llama.

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