Test-time scaling can improve LLM reasoning, but existing approaches typically adapt generation or apply answer selection after a predetermined generation schedule. We study within-instance allocation: under a hard per-instance prompt-plus-completion token cap, when should inference stop, generate another candidate, or spend residual tokens adjudicating candidates already observed? BOSS is a training-free policy for this three-way decision, with worst-case budget checks before every generation and judge call. Its key contribution is online generation-adjudication allocation under one shared token ledger. Across two Llama-3.1 models and four reasoning benchmarks, BOSS attains the highest accuracy in 22 of 24 model-task-budget cells under the common hard cap. At Llama-3.1-8B and B=4k, it reaches 60.5 macro-average accuracy versus 58.6 for Adaptive-Consistency, 59.1 for TALE-EP, and 58.9 for fixed-schedule League. On identical Stage-I candidate pools, randomized A/B judging with repeated voting improves plurality from 58.4 to 60.3, with the full BOSS selector reaching 60.5. Approximately matched-cost controls further characterize how the benefit of adjudication changes with available inference budget. Together, these results establish generation-adjudication allocation as a distinct design axis for reasoning under strict per-instance caps.

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