What does this equation mean?

Putting the pieces together gives a short, honest account of where a tool protocol’s overhead actually lives. A single request pays a fixed tool-use tax $\tau_0$ the moment any tool is attached, plus roughly $\bar{s}$ tokens for every one of the n tools visible in that request, repeated on every turn because the underlying inference API carries no memory between calls. Lazy discovery — Anthropic’s Tool Search Tool, or a disciplined use of MCP’s own paginated listing — replaces the n in that formula with a much smaller m , the tools actually retrieved, at the cost of a small fixed search overhead $\sigma$ . Separately, and not reducible to a token count at all, the choice the model makes over whatever shortlist it is shown carries its own error, decomposable into a retrieval gap that shrinks as the shortlist widens and a confusion gap that grows as it does, with namespacing removing one specific, avoidable source of confusion — a same-named collision — before the shortlisting step ever runs. Schema minimization shrinks $\bar{s}$ directly; it does not touch n or the selection error at all. Four separate levers, four separate places in the same formula, and none of them a substitute for any other.