Strategic voting is a canonical failure mode for collective choice: a voter may obtain a more preferred outcome by reporting a ballot that differs from its true preferences. This paper introduces an oracle-based benchmark for testing whether large language model (LLM) voters can discover and execute such manipulations. Each instance gives an LLM voter a true preference ranking, the other voters' ballots, a deterministic voting rule, and a prompt condition. An exact oracle enumerates every feasible report by the LLM voter, computes the sincere outcome, identifies all profitable reports, and records the best achievable outcome. The benchmark therefore supplies ground truth for strategic success without human labels or subjective grading of explanations. The benchmark covers plurality, Borda, approval, instant-runoff voting, and Copeland-style pairwise majority voting; prompt conditions separate sincere, strategic, civic, and expert framings. To keep the primary study defensible while preserving the main comparisons, the registered core design fixes a single electorate size, uses 600 balanced election instances, and produces 9,600 model--prompt responses when run with four model configurations and four prompt conditions. Because existing peer-reviewed work does not report manipulation discovery, optimal manipulation, false manipulation, near-miss, or invalid-ballot rates for this exact task, we do not impute LLM performance from unrelated studies. Instead, we report exact oracle-calibration baselines that bound and contextualize subsequent model results. By reducing strategic-voting behavior to exact counterfactual evaluation, the benchmark turns the question ``Do LLM voters vote sincerely or strategically?'' into a reproducible social-choice experiment.

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