Organizations: Enkira.ai, USA · School of Computing, Queen’s University, Kingston, Ontario, Canada · IBM T.J. Watson Research Center, USA · Department of Computer Science, University of Chicago, USA · Tensormesh Inc., USA
Test-time compute can substantially improve Large Language Model (LLM) reasoning performance, yet how and when additional compute helps remains poorly understood. We study Divergent-Convergent Reasoning (DCR), a simple two-phase primitive consisting of an exploration phase that generates multiple candidate solutions followed by a convergent reconciliation phase. We present three core results. First, we show that even a single reconciliation step can reliably amplify correct minority reports: across datasets, DCR often recovers the correct answer when correct exploration outputs are in the minority, a regime where majority voting fails. Second, we introduce recursive DCR, an autoregressive reconciliation system that iteratively analyzes disagreements and allocates additional test-time compute. Recursive DCR achieves higher accuracy than fixed-compute baselines-reaching 93.3% on AIME 2024 and 92.0% on AIME 2025-while using roughly 27% less compute on average, demonstrating that attentive resource allocation is superior to uniform scaling. Third, we analyze disagreement among exploration outputs via a simple, training-free dispersion metric. Dispersion reveals a structured relationship between disagreement and test-time gains: in regimes where DCR is effective, higher disagreement among exploration outputs is associated with larger accuracy improvements from reconciliation. Together, these results show that disagreement, often viewed as noise, can be systematically exploited to improve test-time reasoning and reveal emerging scaling laws for agentic LLM systems.
Figures & tables
Figure 1: Single-round DCR (linear): generate diverse proposals, then run one reconciliation round with fixed reviewer width K (default: K=25 ).
Figure 2: Recursive DCR: starting from an initial proposal pool, the system runs Convergent reconciliation in rounds and stops when the K reviewer calls in a round reach unanimous agreement , or when a maximum round/call budget is exhausted. In our implementation, the per-round reviewer batch size (reviewer width K ) is fixed (default: K=8 ); the depth (number of rounds) is therefore task-dependent and self-adjusting. If budget is exhausted without consensus, the system returns a majority answer along with a minority report for downstream review.
Dataset
Model
Trial acc. (%) ≈ Best-of-N
Consistency (%) ≈ Maj. Vote
Samp.
DCR(S)
DCR(M)
Samp.
DCR(S)
DCR(M)
MATH500
GPT-OSS
83.9
84.6
85.2
85.0
85.8
85.4
MATH500
Llama-4
36.4
73.7
79.1
35.8
74.0
79.0
MATH500
Llama-3.3
20.8
51.4
72.9
18.8
52.4
72.8
MATH500
Granite-4
26.2
53.3
73.6
26.0
54.6
74.6
AIME 2024
GPT-OSS
74.3
88.1
84.0
76.7
90.0
83.3
Table 1: Result 1: Single-round DCR ( K=25 ) amplifies correct minority reports. We compare Sampling (exploration baseline) against DCR (reconciliation). Trial Acc. is the average per-attempt accuracy (approximating Best-of-N ); Consistency measures reliability (percentage of problems where the method is correct >50% of the time, approximating Majority Vote ). DCR (Single) reviews the model’s own proposals; DCR (Mix) reviews a shared pool of diverse proposals. DCR consistently improves upon Sampling, often turning fragile minority correct answers into stable majority outcomes.
Dataset
Method
Acc. (%)
Avg. Calls
AIME 2024
Single-round DCR
88.1
50.0
Recursive DCR
93.3
36.4
AIME 2025
Single-round DCR
87.2
50.0
Recursive DCR
92.0
36.1
Table 2: Result 2: Recursive DCR is more accurate and efficient. compared to single-round DCR (budget=50). Recursive DCR stops early on easy problems (unanimity), reducing average compute by ∼ 27%, while achieving higher accuracy.
Figure 3: Accuracy versus test-time compute on (A) AIME 2024 and (B) AIME 2025. Each dot corresponds to a single problem (points are jittered slightly to avoid overlap). The y-axis reports per-problem trial accuracy (fraction correct over repeated stochastic trials for that problem, in %). The x-axis reports total API calls (exploration + reconciliation) as a proxy for budget spending. Colors indicate the aggregation method: Sampling (blue triangles, 25 calls), single-round DCR (green squares, 50 calls; reviewer width K=25 ), and recursive DCR (orange circles, variable calls; reviewer width K=8 per round). Dashed arrows highlight the progression on a representative problem. Notably, the S-DCR → R-DCR transition (bold orange annotation) shows that recursive DCR achieves both higher accuracy and lower budget than single-round DCR—a “win-win” enabled by early stopping on easy problems, while preserving the ability to allocate more compute to persistent disagreements.
Figure 4: Dispersion identifies the “Sweet Spot” for test-time compute. Top row (A–C): Trial accuracy for Sampling vs. DCR on the subset of tasks with dispersion ≥d . Low dispersion predicts high baseline accuracy; high dispersion signals difficulty. Bottom row (D–F): The accuracy gain ( ΔDCR-Samp ) on the same subsets. The green arrow marks the “Recoverable Uncertainty” regime, where filtering for higher dispersion isolates tasks where reconciliation is most effective. The red arrow marks the “Capability Collapse” regime ( d≳0.3 ), where extreme dispersion indicates the model is overwhelmed, and gains diminish.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Round
Reviewer 1
Reviewer 2
Reviewer 3
Reviewer 4
Consensus State
1
1024
13806
1024
122
Diverged (All Incorrect)
2
902
1024
902
1024
Emergence (Correct Minority)
3
902
902
902
902
Unanimity (Stopped)
Appendix
Table 3: Evolution of consensus over 3 rounds. Columns show the answer produced by each of the 4 reviewers in each round. The correct answer ( 902 ) is highlighted in bold. Starting from a completely incorrect state in Round 1, the correct answer emerges in Round 2 (discovered independently by Reviewers 1 and 3) and persuades the remaining reviewers to reach unanimity in Round 3.
Category
Model
N
Trial acc. (%)
Consistency (%)
Samp.
DCR(S)
DCR(M)
Samp.
DCR(S)
DCR(M)
biology
GPT-OSS
717
89.9
90.8
91.1
90.7
90.8
91.8
biology
Llama-4
717
85.4
88.4
90.2
85.6
88.3
90.5
biology
Llama-3.3
717
85.2
86.7
89.8
85.8
86.6
90.0
biology
Granite-4
717
74.1
74.8
87.1
74.1
74.5
87.3
business
GPT-OSS
789
84.5
86.3
85.1
85.7
86.7
87.1
Appendix
Table 4: MMLU-PRO per-category trial accuracy and consistency (%), part 1 of 3.
Category
Model
N
Trial acc. (%)
Consistency (%)
Samp.
DCR(S)
DCR(M)
Samp.
DCR(S)
DCR(M)
physics
GPT-OSS
1299
87.4
88.9
86.8
88.1
88.6
88.2
physics
Llama-4
1299
58.7
80.6
85.1
58.5
80.5
87.1
physics
Llama-3.3
1299
68.6
76.3
83.3
69.4
76.5
83.3
physics
Granite-4
1299
36.0
38.9
73.1
34.8
38.5
73.7
psychology
GPT-OSS
798
79.0
80.2
80.6
79.6
80.8
81.0
Appendix
Table 5: MMLU-PRO per-category trial accuracy and consistency (%), part 2 of 3.