Recovering Missing Support: Complementary Generation in Budgeted LLM Auditing

Collaborative auditing of large language models (LLMs) fundamentally relies on the availability of diagnostic actions. When no generation source proposes the action needed to expose an error, downstream selection cannot recover it: this is the missing support problem. Collaborative frameworks that focus on aggregating generated proposals can remain vulnerable to candidate anchoring, in which candidate-conditioned sources reproduce or reinforce latent errors. We introduce a two-stage collaborative auditing framework that separates action-support formation from budgeted allocation. First, candidate-conditioned generation is paired with a sealed task-only source that commits diagnostic actions before candidate materialization. Second, both routes feed a unified action pool, where shared triage and reconstruction evidence guide selection by calibrated proxy value under a fixed budget. Theoretically, finite repeated-call records leave irreducible uncertainty about the optimal support value of an allocator that knows task hit probabilities and assigns at most one additional call per task under an expected budget. Under iid tasks and conditionally binomial records, we characterize this uncertainty uniformly across budgets and establish matching minimax estimation rates over a specified call-depth range. In a controlled linear programming (LP) case study with 300 injected errors and 150 paired correct controls, task-only supplementation recovers 66 of 122 targets omitted by the candidate-conditioned route. All are purchased and repaired at $B=6$, raising repair from 59.00% to 81.00%. Three additional candidate-conditioned calls reach 69.00% at matched nomination slots, with unmatched computation. On identical augmented portfolios, the complete constraint-level verification procedure improves repair by 30.33 percentage points over whole-model verification (95% interval [25.33, 35.67]). All 150 controls remain correct. In this setting, the results support an architectural principle: budgeted auditing benefits from broadening diagnostic coverage before allocating resources.