Equation 4 · Part 11 · The Token Tax of Giving a Model More Tools

Ending index or upper bound: n

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

n

What this part means

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

Its job in the formula

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

Full expressionEnding index or upper bound: nArticle 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)…

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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.

Open the illustrated sums and products: repeat an operation over an index guide →

See this notation across published equations →

Sources cited in the article section

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