When an LLM judge only has to assign a three-way support label to a candidate answer given a reference, does asking it to decompose the answer into atomic claims help, and at what cost? We compare four single-call designs that share the judge model, the inputs, and the level of instruction detail: candidate-side atomic decomposition, a matched holistic rubric, reference-side decomposition that checks whether each reference claim is covered, and a bidirectional combination. Support labels are constructed from TruthfulQA, ASQA, and QAMPARI references (200 questions and 400 rows per dataset). All four designs are run with Opus-4.6, GPT-4.1, and Gemini Flash Lite; Sonnet-4.6 is added for the candidate-side and holistic designs. Candidate-side decomposition is weak where the label depends on completeness: the holistic rubric is more accurate on ASQA and QAMPARI for every judge while using fewer tokens. Reference-side decomposition is 12.5-21.3 points more accurate than holistic on ASQA for 28-29% more tokens, and stays near the holistic ceiling on QAMPARI at 63-69% more. On TruthfulQA misconceptions, candidate-side decomposition is competitive and significantly better for two judges. On a 60-row single-author subset whose labels are looser than strict reference completeness, candidate-side decomposition edges ahead of holistic for every judge, reversing the construction-label order. What a judge decomposes should follow what the label measures.

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