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Equation 15 · Part 6 · The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol

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nmax⁡=⌊C−Tsystem−Treservetˉtool⌋.n_{\max} = \left\lfloor \frac{C - T_{\mathrm{system}} - T_{\mathrm{reserve}}}{\bar{t}_{\mathrm{tool}}} \right\rfloor.
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What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

Put the three costs together and a deployment question comes into focus: given a fixed context window, how many tools can actually be offered before something has to give? A simple ceiling model makes the trade-off explicit. Let C be the model’s usable context window, TsystemT_{\mathrm{system}} the fixed cost of operator instructions, and TreserveT_{\mathrm{reserve}} a reserved allowance for the user’s message and the model’s own answer. If every tool schema costs on average tˉtool\bar{t}_{\mathrm{tool}} tokens and all of them are injected in full on every turn, the maximum number of tools a deployment can nominally offer before it must start pruning or retrieving is nmax⁡=⌊C−Tsystem−Treservetˉtool⌋n_{\max} = \left\lfloor \frac{C - T_{\mathrm{system}} - T_{\mathrm{reserve}}}{\bar{t}_{\mathrm{tool}}} \right\rfloor. The assumption doing the…

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An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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