Equation 30 · The Economics and Physical Limits of Running AI Agents at Scale
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol q
q is part of the quantity the equation computes from the expression on the right.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
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What the article says around this equation
Holding = q constant across the trajectory — a simplification, since real failure rates are neither constant nor independent — the expected extra cost from retries is , and because increases with i , that extra cost is weighted toward the trajectory’s most expensive, latest steps rather than distributed evenly across it. A one-percent per-step retry rate is a rounding error at step five. Applied at step two hundred of an uncached or lightly cached loop, it is a rounding error on a much larger number.
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