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Equation 19 · Part 2 · The Economics and Physical Limits of Running AI Agents at Scale

Symbol i

Cn=∑i=1nci=n pin(τ+δ)+pin σ n(n−1)2+n pout oˉC_n = \sum_{i=1}^{n} c_i = n\,p_{\text{in}}(\tau+\delta) + p_{\text{in}}\,\sigma\,\frac{n(n-1)}{2} + n\,p_{\text{out}}\,\bar{o}
ii

What this part means

the step.

Its job in the formula

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

Where the article explains it

The shape of the retransmission tax Because the API is stateless between calls, step i in the loop must include, as ordinary input, everything that made it into the transcript through step i-1 : the tool results, the intermediate reasoning, the earlier outputs.

The passage around this formula

Summed over an n -step trajectory, that gives Cn=∑i=1nci=n pin(τ+δ)+pin σ n(n−1)2+n pout oˉC_n = \sum_{i=1}^{n} c_i = n\,p_{\text{in}}(\tau+\delta) + p_{\text{in}}\,\sigma\,\frac{n(n-1)}{2} + n\,p_{\text{out}}\,\bar{o}. The middle term is the one that matters. It grows as n(n-1)/2 — quadratically in step count — while the other two terms grow only linearly. For a short loop, the quadratic term is negligible next to the fixed tax and the output cost. For a long one, it dominates completely, and no amount of shrinking τ\tau or oˉ\bar{o} changes that; the loop’s own length has become the largest line item, all by itself.

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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.

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Sources cited in the article section

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