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Equation 11 · Part 5 · The Unit Economics of a Token

Symbol a

Tin(k)=∑j=1k(b+ja)=kb+a k(k+1)2T_{\mathrm{in}}(k) = \sum_{j=1}^{k} \left( b + j a \right) = k b + a \, \frac{k (k+1)}{2}
aa

What this part means

the number of tokens.

Its job in the formula

a is one of the signed contributions combined to compute the quantity on the left.

Where the article explains it

Suppose a loop runs for k steps, each step appends roughly a tokens of tool output and assistant text, and the context started at b tokens.

The passage around this formula

The nonlinearity that surprises people does not live in this equation, which is linear. It lives in agent loops. Suppose a loop runs for k steps, each step appends roughly a tokens of tool output and assistant text, and the context started at b tokens. The total input billed across the whole loop is then Tin(k)=∑j=1k(b+ja)=kb+a k(k+1)2T_{\mathrm{in}}(k) = \sum_{j=1}^{k} \left( b + j a \right) = k b + a \, \frac{k (k+1)}{2}. which is quadratic in the number of steps. Doubling the length of an agent trajectory roughly…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

Open the illustrated variables: a letter stands for a value guide →

See this notation across published equations →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.