Equation 4 · Part 10 · The Token Tax of Giving a Model More Tools

Starting index or lower bound: i=1

T(n) = \tau_0 + \sum_{i=1}^{n} s_i

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

Call this fixed, tool-independent charge $\tau_0$ . On top of it sits the actual content of the catalogue: for n tools attached to a request, each with a schema costing $s_i$ tokens for its name, description, and JSON Schema of parameters, the tool-visibility component of a single request’s input is $T(n) = \tau_0 + \sum_{i=1}^{n} s_i$ . which, if schemas run at roughly a uniform average size $\bar{s}$ , simplifies to T(n) $\approx$ $\tau_0$ + n $\bar{s}$ . Nothing about this term depends on whether the model calls any tool at all; it is the price of the menu, not the price of the order. And critically, because a stateless inference API resends its full input on every request, T(n) is not paid once at the start of a…

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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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