What does this equation mean?

Anthropic’s Tool Search Tool is a direct, documented implementation of that idea. The mechanism is described plainly: “You provide all your tool definitions to the API, but mark tools with defe$r_l$oading: true to make them discoverable on-demand,” with a lightweight search tool — built-in options include BM25-based and regex-based search — querying an index of names and descriptions, and the full definition of a matching tool injected into context only once the model actually requests it [ 5 ] . The search tool itself is not free, but its overhead is small and fixed rather than proportional to catalogue size: “Only the Tool Search Tool loaded upfront (~500 tokens)” [ 5 ] . Applied to the same 58-tool, roughly 55,000-token example described above, that reframes the earlier formula: instead of the full T(n) $\approx$ $\tau_0$ + n$\bar{s}$ , a session that only ever actually needs m of the n available tools pays approximately $\tau_0$ + $\sigma$ + m$\bar{s}$ , where $\sigma$ is the search tool’s own fixed overhead. The fraction of the original tax avoided is