Equation 31 · The Economics and Physical Limits of Running AI Agents at Scale
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol q
q is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
Symbol i
i appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.
Symbol c_i
is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
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.
Denominator: 1-q
The complete quantity below the fraction bar; it must be nonzero for this division.
Starting index or lower bound: i
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.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction.
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.
Sources cited in the article section
These citations give research context. Read each source to check which claims it supports.
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