Selection, Recombination, or a Fresh Solve? A Candidate-Free Control for Single-Pass Test-Time Aggregation

Guiv Farmanfarmaian

cs.LGAug 18, 2026

Selection, Recombination, or a Fresh Solve? A Candidate-Free Control for Single-Pass Test-Time Aggregation

Authors: Guiv Farmanfarmaian

Organizations: Department of Computer Science, ETH Zurich Zurich, Switzerland

Abstract

An aggregation pass gives a language model two things: candidate solutions and another chance to answer. We ask whether the candidates improve that attempt, and whether aggregation improves on selecting an existing answer. Our candidate-free control asks the same model to solve the problem again from the problem alone, using the same decoding settings and maximum output-token allowance. On AIME-2025 and HMMT-2025, Qwen3-4B-Instruct-2507 reaches 15.5% accuracy when all four candidates are wrong, compared with 27.7% for a fresh solve. When at least two candidates are correct, the ordering reverses: 88.7% for aggregation versus 59.7% for a fresh solve. With exactly one correct candidate, the benefit remains unresolved under the specified test. Yet plurality voting reaches 96.0% in the multiple-correct regime without another model generation; its accuracy difference from aggregation is inconclusive (raw $p = 0.180$ ). The all-wrong accuracy loss and multiple-correct gain survive a statistical correction for adding the second benchmark, developed after data collection, and persist when fresh responses are scored by their final answers. These findings concern one 4B model, two mathematics benchmarks, and a single aggregation pass over candidates truncated at their first answer. All-wrong recovery can arise from solving the problem afresh. The fresh-solve control tests whether candidates improve accuracy. Comparing with selection asks whether the additional generation improves on an existing answer.

Figures & tables

Figure 1: Three ways to reach a correct answer. The examples are hypothetical. Selection requires a correct candidate; recombination can use steps from candidates with wrong answers; independent solving can succeed without using the displayed candidates. Here

c

counts correct candidates. Our control measures net candidate value. Establishing the process behind an individual answer requires additional evidence.

Figure 2: Five arms separate candidate information from another opportunity to generate. AGGREGATE reads solution fragments and answers; ANSWER-ONLY reads answers; NO-CANDIDATE reads the problem alone. All three generate a response with the same model and output limit. VOTING returns an existing answer. ORACLE scores 0 at c0 and 1 at c1 or c2+. Gold answers are used only by ORACLE, scoring, and stratum construction. Every panel uses the same illustrative candidate set.

Figure 3: Aggregation, a fresh solve, and voting by candidate correctness. Left: pooled problem-level accuracies. At c2+, voting reaches 96.0% versus 88.7% for aggregation and requires no generation after the pool exists; their paired accuracy comparison is inconclusive (raw sign-test

p=0.180

). Right: mean AGGREGATE minus NO-CANDIDATE, with Holm-adjusted

p

-values and decisions at family level 0.025. These sign tests assess directional consistency across problems; c1 remains unresolved despite its positive mean. Generating arms share a 16,384-token output limit. Each set contains four first-answer-truncated solution fragments.

Regime	AGG	Fresh	Vote	$\Delta_{\mathrm{cand}}$	Raw $p$	Holm $p$
c0	0.155	0.277	0.000	$-0.123$	0.0118	0.0236
c1	0.672	0.430	0.156	$+0.242$	0.0755	0.0755
c2+	0.887	0.597	0.960	$+0.290$	0.0044	0.0133

Table 1: Primary pooled results. AGG, Fresh, and Vote are problem-level accuracies.

\Delta_{\mathrm{cand}}

and both

p

-value columns compare AGG with Fresh. The exact two-sided sign tests are Holm-adjusted across three regimes at stage-2 family level 0.025: c0 and c2+ pass; c1 remains unresolved. The voting comparison is separate (Section 3.1 ). Values are shown as fractions and rounded independently.

Contrast	First-answer scoring	Final-answer scoring
c0: AGGREGATE $-$ NO-CANDIDATE	$-0.118$	$-0.159$
c2+: AGGREGATE $-$ NO-CANDIDATE	$+0.290$	$+0.242$
c2+: AGGREGATE $-$ ANSWER-ONLY	$+0.161$	$+0.008$

Table 2: Extraction audit on fresh generations from the same process. These are separate estimates from the original run in Table 1 . Values are accuracy differences. The c0 and c2+ candidate contrasts retain their signs and reported Holm decisions. The c2+ reasoning contrast loses its reported significance under final-answer scoring. Detailed truncation and extraction checks are in Appendix H .

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Arm	Flip rate [95% CI]	Answer-level disagreement
AGGREGATE	0.045 [0.009, 0.091]	0.132
NO-CANDIDATE	0.077 [0.041, 0.118]	0.377
ANSWER-ONLY	0.105 [0.059, 0.155]	0.377

Appendix

Table H.1: Correctness-flip rates and extracted-answer disagreement between the scoring rules at c0, over all rows. Intervals use a problem-clustered bootstrap.

Quantity	First-answer	Final-answer	Conclusion
c0 $\Delta_{\mathrm{cand}}$	$-0.118$ $p=0.012$	$-0.159$ $p<0.0001$	Accuracy loss remains significant, including after excluding truncations.
c0 $\Delta_{\mathrm{reas}}$	$-0.041$ $p=0.263$	$-0.073$ $p=0.115$	Negative sign retained.
c2+ $\Delta_{\mathrm{cand}}$	$+0.290$ $p=0.0002$	$+0.242$ $p=0.0003$	Benefit remains significant.
c2+ $\Delta_{\mathrm{reas}}$	$+0.161$ $p=0.0013$	$\approx 0$ Holm $p=1.000$	Reasoning increment becomes unresolved.
c0 matching	0.885 / 0.417	0.874 / 0.475	Ordering retained, with at least 50% of the difference.

Appendix

Table H.2: Pooled results from the scoring rerun. Accuracy contrasts are problem-level differences. The p-values are raw exact sign-test values unless labelled Holm-adjusted; conclusions follow the within-family Holm procedure. Matching rates list AGGREGATE / NO-CANDIDATE.

Figure 5: Descriptive matching in the original c0 experiment. The figure shows the fraction of wrong extracted answers equal to a candidate answer displayed to AGGREGATE. These are output-level proportions for the original three generating arms; the placebo is not plotted. NO-CANDIDATE is compared with the same candidate answers although it never sees them. The original AIME paired matching test (33/4,

p<0.0001

) treats candidate sets as independent units. It is excluded from the primary inference, whose units are independent problems.

stratum	pooled $p$	Holm-adj. $p$	reject ( $\alpha_{\mathrm{fam}}{=}0.025$ )	reject ( $\alpha{=}0.05$ )
c2+	0.004425	0.0133	yes	yes
c0	0.011818	0.0236	yes	yes
c1	0.075519	0.0755	no	no

Appendix

Table 3: The adaptive two-benchmark extension, retrospectively corrected. Stage 1 (claim from the first benchmark alone) requires raw AIME c0

p_{1}\leq 0.025

; the observed

p_{1}=0.109375

instead entered the pre-committed extension band, so the pooled family is tested with Holm at family

\alpha=0.025

. FWER

\leq 0.05

by union bound, valid under any dependence and any partial-null configuration. The originally reported Holm correction at

\alpha=0.05

yields the same rejections with identical adjusted values.

Figure 6: Structured answer-field intervention on all-wrong (c0) candidate sets. All plotted values are in percentage points. Top: accuracy change from masking (MASKED

-

FULL),

-0.45

(95% CI [

-5.00,+3.64

]). Bottom: value-specific steering (SUBSTITUTED output matches an injected value minus FULL output matching the same values),

+6.82

(95% CI [

+2.27,+11.82

]). Problem-clustered percentile bootstrap, 55 problems and 220 sets. Causal conclusions are scoped to the structured intervention.

Endpoint	Effect [95% CI]	Raw $p$	Holm $p$
A: masking, accuracy	$-0.0045$ [ $-0.0500,+0.0364$ ]	1.000	1.000
B: substitution, matching	$+0.0682$ [ $+0.0227,+0.1182$ ]	0.0213	0.0425

Appendix

Table M.1: Primary intervention results under first-answer scoring. Effects are differences in probability, with 95% confidence intervals. The raw p-values are from sign tests; Holm correction covers the two endpoints.

Endpoint	Analysis	Effect [95% CI]	Reported $p$
A: masking	Final-answer scoring	+0.0182	0.344
A: masking	Truncation excluded	$-0.0091$	1.0
B: substitution	Final-answer scoring	+0.0636 [ $+0.0182,+0.1136$ ]	0.0352
B: substitution	Truncation excluded	+0.0773 [ $+0.0273,+0.1303$ ]	0.0213

Appendix

Table M.2: Intervention sensitivities. Effects are differences in probability; 95% confidence intervals are shown where reported. These analyses use the same endpoint definitions as Table M.1 .

Group	Strategy	$n$	Median time (s)	p95 time (s)	Median input tokens	Median output tokens	Cap hits
c0	AGGREGATE-4	30	13.9	25.8	21,638.5	1,229.5	0
c0	FRESH-SOLVE	30	55.8	101.0	149.5	6,474.0	0
c1	AGGREGATE-4	30	13.0	31.8	21,320.5	1,243.0	0
c1	FRESH-SOLVE	30	59.4	90.2	149.5	6,856.5	1
c2+	AGGREGATE-4	30	14.3	23.5	17,576.0	1,437.5	0
c2+	FRESH-SOLVE	30	38.8	148.7	153.0	4,565.0	2

Appendix

Table 4: Time per call after the candidate pool exists. The frozen 30-set sample includes 10 sets per regime, with three unseeded repeats per arm (

n=30

per cell). Paired strategies run on the same A100 with batch size 1 and prompt-prefix caching disabled. Times use synchronized measurements; the table reports medians and empirical p95 (95th percentile) values. Token columns are medians; cap hits count outputs reaching 16,384 tokens. Candidate generation is excluded, so these measurements describe the additional call under these conditions. They do not estimate cached-serving or end-to-end cost. Table 5 reports mean token counts on a different sample.

Strategy	Additional generation calls	Mean input tokens	Mean output tokens
VOTE-4 (CPU plurality)	0	0	0
AGGREGATE-4	1	18,721.7	1,737.2
FRESH-SOLVE (after pool)	1	217.9	8,310.4

Appendix

Table 5: Additional generation after the candidate pool exists, measured on 220 c0 sets with the model’s chat template. Values are mean input and output token counts per request. They do not measure floating-point operations (FLOPs), GPU-seconds, monetary cost, or end-to-end cost. Candidate generation is also excluded. This sample differs from the 30-set timing sample in Table 4 . A complete four-candidate workflow also requires four candidate samples; their request count depends on batching.

Regime	AGG	Fresh	Ans-only	Vote	Oracle	$\Delta_{\mathrm{reas}}$ (raw $p$ )
c0	0.155	0.277	0.186	0.000	0.000	$-0.032$ (.359)
c1	0.672	0.430	0.539	0.156	1.000	$+0.133$ (.041)
c2+	0.887	0.597	0.710	0.960	1.000	$+0.177$ (.013)

Appendix

Table 6: Primary pooled problem-level accuracies for all five arms and the secondary contrast

\Delta_{\mathrm{reas}}=

AGGREGATE minus ANSWER-ONLY, with its raw paired sign-test

p

-value. The primary candidate contrast and corrected inference are in Table 1 . The apparent c2+ increment over ANSWER-ONLY does not persist under final-answer scoring of the process rerun (Table 2 ); it is descriptive.

bench	$c$	AGG	NoCand	AnsOnly	Vote	Oracle	$\Delta_{\mathrm{cand}}$	$\Delta_{\mathrm{reas}}$
AIME	c0	0.222	0.352	0.296	0.000	0.000	$-$ 0.130	$-$ 0.074
AIME	c1	0.697	0.500	0.618	0.237	1.000	$+$ 0.197	$+$ 0.079
AIME	c2+	0.893	0.583	0.762	0.976	1.000	$+$ 0.310	$+$ 0.131
HMMT	c0	0.089	0.205	0.080	0.000	0.000	$-$ 0.116	$+$ 0.009
HMMT	c1	0.635	0.327	0.423	0.038	1.000	$+$ 0.308	$+$ 0.212
HMMT	c2+	0.875	0.625	0.600	0.925	1.000	$+$ 0.250	$+$ 0.275

Appendix

Table 7: Per-benchmark accuracy by arm and

c

-bin (set-level point estimates). Both benchmarks show the three-regime sign pattern. The AIME

\leftrightarrow

HMMT homogeneity

p

(permutation on per-problem

\Delta_{\mathrm{cand}}

) is c0 0.91, c1 0.55, c2+ 0.70. Heterogeneity was not detected, and the benchmark-specific effects agree in sign. The pre-specified pooled estimate is reported alongside these per-benchmark results.

$K$	raw AGG c0	$\Delta_{\mathrm{cand}}$ c0	$\Delta_{\mathrm{cand}}$ c1	$\Delta_{\mathrm{cand}}$ c2+
2	0.250	$-$ 0.170 (.092)	$+$ 0.250 (.118)	$+$ 0.333 (.007)
4	0.222	$-$ 0.130 (.109)	$+$ 0.197 (.302)	$+$ 0.310 (.022)
8	0.037	$-$ 0.188 (.031)	$+$ 0.071 (1.00)	$+$ 0.274 ( $<.001$ )

Appendix

Table 8: Varying

K

(AIME-2025 only; problem-level). Raw AGGREGATE accuracy at c0 decreases as

K

grows, and

\Delta_{\mathrm{cand}}

remains negative for every

K

$c$	sets	prob	AGG	NoCand	AnsOnly	Vote	Oracle	$\Delta_{\mathrm{cand}}$ (p)	$\Delta_{\mathrm{reas}}$ (p)
c0	24	3	0.083	0.125	0.208	0.000	0.000	$-$ 0.042 (1.00)	$-$ 0.125 (1.00)
c1	16	2	0.375	0.438	0.500	0.000	1.000	$-$ 0.062 (1.00)	$-$ 0.125 (1.00)
c2+	200	25	0.960	0.840	0.915	0.980	1.000	$+$ 0.120 (.012)	$+$ 0.045 (.125)

Appendix

Table 9: RLVR (Qwen3-4B-Thinking) by

c

-bin. 200 of 240 sets fall in c2+ (25 of 30 problems); c0/c1 contain 3/2 problems, leaving insufficient power to assess model-class differences. CGT (correct given a valid terminated answer): instruct

244/644{=}0.379

vs RLVR

395/422{=}0.936

quantity	value
AGG (c0)	0.767
NoCand (c0)	0.000
AnsOnly (c0)	0.000
validation margin: AGG $-$ max(NoCand, AnsOnly)	0.767
criterion ( $\geq 0.30$ )	PASS

Appendix

Table 10: Validation experiment (constructed additive recombination, all c0). The pre-specified success criterion requires AGG to exceed both comparison arms by

\geq 0.30

run	arm	$n$	trunc_noans
instruct-AIME	AGG/NoCand/AnsOnly	268	0
instruct-HMMT	AGG/NoCand/AnsOnly	204	0
RLVR	AGGREGATE	240	0
RLVR	NO-CANDIDATE	240	37
RLVR	ANSWER-ONLY	240	13

Appendix

Table 11: Generation-completion checks by arm. trunc_noans counts outputs that hit the token cap without an extractable answer; cand_trunc counts candidate fragments additionally shortened at the word limit before aggregation. Both counts are 0 in the instruction-tuned runs. The RLVR counts occur entirely in NO-CANDIDATE (37) and ANSWER-ONLY (13 of 240). These unanswered outputs score wrong, lowering the affected arms’ accuracy and shifting AGG

-

NoCand and AGG

-

AnsOnly upward. The RLVR c2+

\Delta_{\mathrm{cand}}

\Delta_{\mathrm{reas}}

therefore require testing under other output allowances before broader interpretation. The primary instruction-tuned results have no such unanswered truncations.

bench	stratum	prob	sets	AGG	NoCand	AnsOnly	$\Delta_{\mathrm{cand}}$
AIME	pass@24 $=$ 0	9	36	0.000	0.000	0.000	$+$ 0.00
AIME	pass@24 $\geq$ 1	18	72	0.333	0.528	0.444	$-$ 0.194
HMMT	pass@24 $=$ 0	15	60	0.100	0.067	0.000	$+$ 0.033
HMMT	pass@24 $\geq$ 1	13	52	0.077	0.365	0.173	$-$ 0.288

Appendix

Table 12: c0 by pool difficulty (pass@24). The c0 accuracy loss relative to a fresh solve is concentrated on problems with at least one correct pool sample. A zero pass@24 observation does not establish that a problem is unsolvable; rates on that subset remain at or near zero.

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