Equation 3 · How AI Models Learned to Use Tools: A Protocol History
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.
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Symbol C_bespoke
espoke is part of the quantity the equation computes from the expression on the right.
Symbol N
both large and still growing, the gap between M N and M + N dominates every other consideration.
=
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.
How to interpret it
Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
That parallel is precise enough to formalize, and doing so exposes the actual economic assumption both protocols are making rather than merely asserting it. Let M be the number of distinct hosts — editors, for LSP; agent frameworks and model-serving applications, for MCP — and N be the number of distinct targets — languages, for LSP; tools and data sources, for MCP. Wiring every host to every target independently, the way plugins and early framework wrappers did, requires . separate integrations, because nothing built for one pair transfers to any other. A protocol that both sides implement exactly once instead collapses this to
Sources cited in the article section
- [10] Introducing the Model Context Protocol ↗
- [11] Specification ↗
- [12] JSON-RPC 2.0 Specification ↗
- [13] Language Server Protocol ↗
These citations give research context. Read each source to check which claims it supports.
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