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Equation 6 · Part 7 · RAG in 2035: Four Scenarios, Their Signals, and What Would Falsify Them

=

cfull(M)=Θ(M),cretrieve(M)=O(log⁡M)+Θ(k),k≪M.c_{\text{full}}(M) = \Theta(M), \qquad c_{\text{retrieve}}(M) = O(\log M) + \Theta(k), \qquad k \ll M.
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What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

The two axes are related but not reducible to one another, and the coupling has an economic root worth making explicit. Autoregressive decoding is bounded by how much has to be read from memory to produce each token — the weights, plus whatever context has accumulated — and Pope and colleagues formalized exactly this partitioning of cost between arithmetic and memory movement in transformer serving [ 5 ] . Write n for the number of tokens a system must hold in an answer’s working set. Pasting a whole corpus of size M into context puts a floor under n near M itself; retrieving costs roughly the price of an index lookup plus the price of reasoning over the k ≪\ll M tokens actually returned:…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.