Equation 10 · Part 11 · The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol
Starting index or lower bound: i=1
What this part means
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.
Its job in the formula
i=1 appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.
Full expression→Starting index or lower bound: i=1→Article meaning
The passage around this formula
Model it simply. Suppose a session runs for k turns, and each turn adds on average tokens of new tool result to a transcript that is resent in full on every request, because nothing is evicting or compressing it. The tokens billed on turn i include roughly i tokens of accumulated result history, so the total billed across the whole session is . Doubling the length of a session does not double the bill; it roughly quadruples it, because the earlier results are being paid for again on every one of the later turns. This is not a hypothetical — it is the direct arithmetic consequence of the same design choice OpenAI documents for its own tool schemas: whatever is…
Learn the underlying idea
Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.
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Sources cited in the surrounding passage
These citations provide research context; check each source for the exact claim it supports.