Equation 1 · Part 12 · The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol

Starting index or lower bound: i=1

T_{\mathrm{turn}} = T_{\mathrm{system}} + T_{\mathrm{tools}} + \sum_{i=1}^{n} T_{\mathrm{result},i} + T_{\mathrm{exchange}}

i=1

What this part means

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

Its job in the formula

i=1 appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

Full expressionStarting index or lower bound: i=1Article meaning

The passage around this formula

A useful way to see the shape of the tax is to write out what actually occupies a turn’s context budget: $T_{\mathrm{turn}} = T_{\mathrm{system}} + T_{\mathrm{tools}} + \sum_{i=1}^{n} T_{\mathrm{result},i} + T_{\mathrm{exchange}}$ . $T_{\mathrm{system}}$ is the operator’s own instructions, $T_{\mathrm{tools}}$ is the serialised cost of every tool schema currently declared, $T_{\mathrm{exchange}}$ is the live user message and the model’s own reply, and the summation term is every prior tool result the session has decided to keep. The point of writing it this way is that $T_{\mathrm{tools}}$ is paid in full on the very first turn, before a single tool has been called and before it is known whether any tool will be needed at all. A deployment that registers thirty MCP servers, each exposing a handful of…

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Learn the underlying idea

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

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Sources cited in the article section

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