# From Discovery to Decision: Finite-Budget Recoverability in LLM Voting

arXiv: https://arxiv.org/abs/2610.01014 · HTML: https://mlsanity.com/paper/2610.01014/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 31 of its 42 references that are in the map.
- First posted on arXiv 1 Oct 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 30 of 31)

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- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
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- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [MMLU-Pro: A More Robust and Challenging Multi-Task Language Understanding Benchmark](https://mlsanity.com/paper/2406.01574/) (2024)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [Gemma 4 Technical Report](https://mlsanity.com/paper/2607.02770/) (2026)
- [Deep Think with Confidence](https://mlsanity.com/paper/2508.15260/) (2025)
- [Let's Sample Step by Step: Adaptive-Consistency for Efficient Reasoning and Coding with LLMs](https://mlsanity.com/paper/2305.11860/) (2023)
- [Escape Sky-high Cost: Early-stopping Self-Consistency for Multi-step Reasoning](https://mlsanity.com/paper/2401.10480/) (2024)
- [The Limits of Inference Scaling Through Resampling](https://mlsanity.com/paper/2411.17501/) (2024)
- [How Do Large Language Monkeys Get Their Power (Laws)?](https://mlsanity.com/paper/2502.17578/) (2025)
- [Found in the Middle: Permutation Self-Consistency Improves Listwise Ranking in Large Language Models](https://mlsanity.com/paper/2310.07712/) (2023)
- [Single-Agent LLMs Outperform Multi-Agent Systems on Multi-Hop Reasoning Under Equal Thinking Token Budgets](https://mlsanity.com/paper/2604.02460/) (2026)
- [Provable Scaling Laws for the Test-Time Compute of Large Language Models](https://mlsanity.com/paper/2411.19477/) (2024)
- [On the Effect of Sampling Diversity in Scaling LLM Inference](https://mlsanity.com/paper/2502.11027/) (2025)
- [Sample Complexity and Representation Ability of Test-time Scaling Paradigms](https://mlsanity.com/paper/2506.05295/) (2025)
- [ROC-n-reroll: How verifier imperfection affects test-time scaling](https://mlsanity.com/paper/2507.12399/) (2025)
- [Optimal Self-Consistency for Efficient Reasoning with Large Language Models](https://mlsanity.com/paper/2511.12309/) (2025)
- [Optimal Bayesian Stopping for Efficient Inference of Consistent LLM Answers](https://mlsanity.com/paper/2602.05395/) (2026)
- [When More Sampling Hurts: The Modal Ceiling and Correlation Ceiling of Test-Time Scaling](https://mlsanity.com/paper/2606.28661/) (2026)
- [MARS: Margin-Adversarial Risk-controlled Stopping for Parallel LLM Test-time Scaling](https://mlsanity.com/paper/2606.12935/) (2026)
- [When Self-Consistency Backfires: Majority Vote Hurts the Majority of Hard Science Problems for Small LLMs](https://mlsanity.com/paper/2608.11403/) (2026)
- [PETS: A Principled Framework Towards Optimal Trajectory Allocation for Efficient Test-Time Self-Consistency](https://mlsanity.com/paper/2602.16745/) (2026)
- [EMS: Multi-Agent Voting via Efficient Majority-then-Stopping](https://mlsanity.com/paper/2604.02863/) (2026)
- [Oracle Gap and Signal Fidelity: A Fixed-Pool Diagnostic for Test-Time Collaboration](https://mlsanity.com/paper/2607.17531/) (2026)

## Abstract (arXiv)

Voting over multiple LLM responses is a common primitive in test-time scaling and ensemble inference. Collecting more responses can expand the candidate pool and increase the chance that a correct answer is discovered. Under a fixed call budget, a discovered answer still needs to accumulate enough support within the remaining calls to become the final plurality winner, creating a discovery-to-decision gap. In this work, we characterize this gap through the realized vote state and remaining call budget. We derive a sharp recoverability threshold and show that, as sampling proceeds, the observed candidate set can only expand while the set of reachable endpoint winners can only contract, inducing a candidate-level conversion window. Under a specified iid response law, the same state yields exact finite-horizon endpoint probabilities. We further show that merging wrong-answer identities preserves single-call correctness and cannot improve plurality accuracy, and that the effect of redistributing wrong-answer probability depends on the realized vote state. Singleton reachability yields a gold-free exact locking certificate. For a known answer universe, its first trigger is the earliest prefix at which all admissible continuations yield the same fixed-budget output. Empirically, most discovered-but-unselected correct answers lose reachability only after discovery. In a controlled Word16 study, input permutation improves raw-plurality accuracy by 21.1 points with essentially unchanged single-call correctness. Exact locking saves 28-30% of calls at a 16-call budget while preserving every fixed-budget output.