Equation 37 · Part 2 · The Economics and Physical Limits of Running AI Agents at Scale
Symbol σ
What this part means
the number of tokens.
Its job in the formula
σ is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
Full expression→Symbol σ→Article meaning
Where the article explains it
In the uncached model above, step i sends (i-1) tokens of retransmitted history as ordinary, fully metered input — every one of which counts against the per-minute ceiling, every single step, growing linearly with i within a single trajectory.
The passage around this formula
That exemption changes the shape of the rate-limit exposure the same way it changed the shape of the dollar cost. In the uncached model above, step i sends (i-1) tokens of retransmitted history as ordinary, fully metered input — every one of which counts against the per-minute ceiling, every single step, growing linearly with i within a single trajectory. A long, rapid, uncached loop can approach its own organization’s input-tokens-per-minute ceiling from its own growth alone, with no other caller involved: the trajectory throttles itself. With per-turn caching applied, the accumulated history is read from cache and, per the documentation above, does not count toward the limit at all;…
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.
Open the illustrated variables: a letter stands for a value guide →
See this notation across published equations →
Sources cited in the article section
- [3] Rate Limits ↗
- [8] Rate Limits ↗
- [11] Efficiently Scaling Transformer Inference ↗
- [4] Claude API Errors ↗
These citations provide research context; check each source for the exact claim it supports.