arXiv:2606.19111v2 Announce Type: replace-cross Abstract: Team science holds that leadership is contingent: it helps only under specific conditions. We ask the same question of multi-agent LLM teams: under what measurable conditions does process-level coordination control add value, and do those conditions match what team science predicts? This takes measurement, not just accuracy: we use behavioral signatures (majority lock-in, exploration, recovery from an incorrect round-0 consensus) and per-action ablations, clean because each controller is an explicit action set. We operationalize three classical leadership styles (transactional, transformational, situational) as controllers over a shared action vocabulary (explore, revise, accept, synthesize). A matched controller with the same actions but an arbitrary rule recovers no better than majority voting. Across four task regimes and three open-weight model families, no controller dominates by accuracy. Transactional control matches a shared round-0 vote on all 12 combinations to within 1.3pp, and accuracy gains appear only on the single llama-4-scout social combination, where the round-0 majority is unreliable. Against the stronger flat baseline, only situational still gains (+8pp). A recovery-advantage account, tested with four boundary probes, says a controller beats plain interaction only where the round-0 majority is unreliable, the task is recoverable, and undirected interaction does not already repair it. Because these conditions map onto contingency theory (leadership substitutes, path-goal redundancy, the readiness gap), a largely null accuracy result is what the theory predicts, not a failure of the controllers. We advance this readiness-gap boundary as an explanatory hypothesis rather than an established general principle. Its null side is corroborated across every other combination, including under held-out prediction, while its positive side is observed in a single combination.