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Equation 19 · Part 11 · The Economics and Physical Limits of Running AI Agents at Scale

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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}
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What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

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

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

Sources cited in the article section

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