LLM judges are often evaluated with a single prompt and only a few repeated calls. When their verdicts vary, it remains unclear whether the variation comes from sampling noise within a prompt or systematic differences across prompts. We formalize this distinction using a second-order response law: the distribution of prompt-conditioned verdict distributions induced by a declared prompt policy. For a quadratic measure of prompt instability, we show that the usual plug-in estimator is biased upward at finite repeat budgets because it confounds within-prompt noise with between-prompt variation. We derive unbiased estimators for both sampled prompts and declared fixed prompt censuses from the difference between within- and across-prompt agreement. Under a crossed prompt-by-answer-order design, the same framework separates prompt, order, interaction, and residual call variation, while retaining invalid completed outputs as outcomes. Known-law simulations and a byte-identical live null recover the predicted finite-$R$ inflation. In a matched Qwen study, corrected low-repeat estimates are closer to an independently acquired $R=16$ reference than plug-in estimates, with the largest gains at small repeat budgets. A matched panel across four frozen judge configurations exhibits configuration-specific inflation magnitudes and component profiles. Prompt robustness can therefore be estimated separately from finite-call noise.

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