Large language models increasingly support scientific and algorithmic discovery through inference-time search over evaluated candidates. Existing adaptive discovery controllers assign credit based only on score progress, even though prompt length, retries, and guidance calls cause search actions to incur different token costs. Under a fixed search budget, the controller must decide which frontier to develop and whether its gain justifies the realized cost before the budget is exhausted. We prove that, in the worst case, cost-blind control can forfeit all but a vanishing fraction of attainable quality as frontier count and cost heterogeneity grow. To address this limitation, we introduce \textbf{CostAda}, an adaptive controller built on \emph{cost-calibrated frontier utility}. By valuing progress relative to realized action cost and conditioning credit on the remaining budget, CostAda coordinates local exploration intensity, frontier allocation, and budgeted tactic intervention. Realized action cost and remaining budget therefore shape the search rather than serving only as accounting variables or a stopping rule. Across eight benchmarks with GLM-5 and GPT-5.4, CostAda reaches the strongest baseline's full-budget quality with at most half the budget on thirteen of sixteen benchmark--backbone pairs and achieves the strongest final quality on all sixteen pairs.

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