Multi-agent systems split a task across a tree of agents and justify the split with folklore: smaller contexts, cleaner separation, parallelism. We ask what the split does to how much of what the leaves discover reaches the root. Model a decomposition as a tree in which an agent handed
b items keeps any one with probability
r(b). If
r(b)=1/b, every tree delivers exactly one finding, for every task size and every shape; we verify this to
2.4×10−15 on 20,000 random irregular trees. If
r(b)=Cb−δ, a depth-
k tree over
N findings yields
CkN1−δ: task size and architecture separate, and architecture contributes only
C≤1 per level, so flat is optimal for yield and no arrangement of agents escapes the exponent
δ. On 600 production deep-research traces
δ=0.34 [0.30, 0.38], by three identifications that do not share a failure mode. At a hop where item boundaries come from the tool rather than a text heuristic, and where
b=1 occurs 550 times,
C=0.571 [0.527, 0.615] is observed rather than extrapolated, over 16,082 hops. A tier also costs alignment: on 1,012 annotated multi-agent traces one brief in sixteen goes off-target, giving
μ=0.939 and a per-tier penalty
Cμ=0.536. Depth is bought on two other axes. The root context is the only state that persists and the only one that cannot cheaply forget, and depth cuts its exposure from
N items to
N1/k. Depth is also cheaper: production flat agents bill as
N1.39, not the
N2 an append-only context predicts, and at equal spend two tiers overtake flat at 403 findings. Across every parameter we measured the model says 0.7% to 11.3% of production sessions are worth delegating, against 7.8% that do. A hazard model on 743,819 production tool calls finds that delegation does not respond to a filling context and is instead an opening move.