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Equation 15 · Part 10 · The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol

Numerator: C - T_system - T_reserve

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

What this part means

The complete quantity above the fraction bar.

Its job in the formula

C - TsT_system - TrT_reserve occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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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A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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