Quadratic In One Window

A research task requires exactly 48 tool calls. Every tool result is about 1,800 tokens and stays in the window that issued it. System prompt plus tool definitions is 3,000 tokens. Compare two shapes for the same 48 calls:

Fan-out. The lead writes a 2,400-token brief per subagent. Six subagents run 8 calls each. Each subagent's turn $t$ sends $3{,}000 + 2{,}400 + 1{,}800(t-1)$ input tokens. Lead overhead (writing briefs, synthesizing six summaries) is about 40,000 tokens.

Single agent. One agent runs all 48 calls in one window. Its turn $t$ sends $3{,}000 + 2{,}400 + 1{,}800(t-1)$ input tokens.

Which shape bills fewer input tokens, and why?

Fan-out costs about 6x more, because each of the six subagents repeats the 5,400-token preamble on every one of its turns.

Fan-out, roughly 0.6M against 2.3M, because accumulation inside one window is quadratic in the number of calls.

The single agent, roughly 0.26M, because it pays the 5,400-token preamble once instead of six times.

Roughly equal, since both shapes read the same 48 tool results exactly once.