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Published equation contexts

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}

Why this formula appears here

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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pinp_{\text{in}}

Symbol p_in

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

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poutp_{\text{out}}

Symbol p_out

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

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i=1i=1

Starting index or lower bound: i=1

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.

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nn

Ending index or upper bound: n

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

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

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}

Equation 19 · AI Agents & Systems

The Economics and Physical Limits of Running AI Agents at Scale

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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.

Meanings in this article

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