Budgeted Act-or-Defer Multi-Agent LLM Deliberation with Local Reliability Bounds

Multi-agent deliberation among LLMs can improve reasoning, but deployment requires deciding when the current answer is reliable enough to act on and when it should be escalated to human review. We formulate this as budgeted act-or-defer decision making. At each round, the system maps the debate prefix to a low-dimensional state, computes a $k$-nearest-neighbor lower confidence bound on state-conditional correctness using calibration data, and acts only when the bound exceeds a user-specified reliability threshold. The certificate controls wrong actions through the decomposition $\beta = \delta + \alpha + \epsilon_{act}$, separating calibration failure, residual action risk, and representation gap. The guarantee is conditional, not distribution-free: it relies on a valid local bias envelope and an action-region representation-gap bound, and each assumption is paired with falsification-style diagnostics. Because the same absolute wrong-action budget has different meanings across tasks of different difficulty, we set budgets relative to each task's final-round error using training data only, and evaluate safety by normalized budget usage $\mathrm{WA}/\beta$. On six benchmarks against nine baselines, the method uses 9--12\% of the pre-declared budget on activated datasets, reaching up to 84\% automation and 96\% acted-on accuracy; on stress-test datasets, it defers rather than forcing unreliable automation. The goal is not to beat post-hoc tuned scalar thresholds at matched automation, which can match subset quality after per-task search. Rather, the contribution is a prospective, auditable procedure for converting a user-declared wrong-action budget into an act-or-defer operating point before deployment, under explicitly stated assumptions.