Language-model agents increasingly solve complex tasks through parallel reasoning and asynchronous information gathering. Controlled replay across two model families and three benchmarks shows that delivery timing redirects search decisions made before all results arrive, despite identical initial evidence. Acquisition disagreement reaches up to 24 percentage points above same-order repeat variation. This dependence persists after all initial evidence arrives and through a second search, turning temporary differences in what agents have seen into lasting differences in what they acquire. An acquisition-regret bound with a matching construction shows that invariance at a fixed early arrival count can limit the use of evidence already received. We introduce Confluent Evidence Acquisition (CEA), which instead releases local reasoning when each subquestion's evidence is complete and allocates shared search independently of completion order. For a fixed plan and stable conditional response law, we prove joint invariance of decisions, acquisitions, and answers. With question-only plans across three benchmarks, CEA brings acquisition disagreement to within 1.0 percentage point of repeat variation and achieves accuracy comparable to complete-input controls, improving over early commitment by up to 12.6 percentage points. Replacing CEA's fixed allocation with first-ready allocation under contention adds 21.6 percentage points of acquisition disagreement, while local query-target disagreement remains near repeat variation. Timed comparisons show that CEA reaches decisions earlier and responds faster than the same program waiting for all initial evidence.

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