Quantum circuit execution estimates output distributions by repeated measurements, yet developers commonly choose a fixed shot budget before execution. This static choice is brittle: low budgets can under-sample the distribution, while high budgets waste measurements. In this paper, we present StableShots, a black-box online stopping rule for static quantum circuits. The method executes a fixed circuit in small batches, monitors the total-variation distance between cumulative empirical distributions, and stops after repeated evidence of local stability. We evaluate StableShots on 180 QSimBench traces spanning six circuit families, six sizes from 4 to 14 qubits, and five noisy IBM simulated backends. With validation-only calibration and 100 repeated backend-holdout splits, the selected configuration reaches TVD <= 0.05 on all held-out test evaluations with median 7,650 shots, whereas fixed-shot baselines either fail more often or spend substantially more shots.

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