Equation 10 · The Economics, Energy, and Physical Limits of Tool Protocols and the Model Context Protocol
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 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.
Read it piece by piece
Symbol B
B is part of the quantity the equation computes from the expression on the right.
Symbol k
k occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol i
i appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →change
Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.
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.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
See an illustrated explanation →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.
Ending index or upper bound: k
This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.
Denominator: 2
The complete quantity below the fraction bar; it must be nonzero for this division.
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.
What the article says around this equation
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…
Read the full surrounding passage
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 placed in the context window is billed as input on every request that includes it, and a growing transcript is a context window that only gets larger [ 4 ] .
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.