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Equation 2 · Part 9 · The Economics, Energy, and Physical Limits of Claude Code and Agentic Development Tools

Denominator: BW

ttoken  ≳  2 Nactive β+KV(c)BW,t_{\mathrm{token}} \;\gtrsim\; \frac{2\,N_{\mathrm{active}}\,\beta + \mathrm{KV}(c)}{\mathrm{BW}},
BW\mathrm{BW}

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

BW occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

Underneath all of the pricing mechanics sits a constraint pricing cannot remove: generating one token from a large language model is, for most of a request, bound by how fast bytes can move through accelerator memory rather than by how much arithmetic the accelerator can perform. Producing each token requires reading the model’s active weights and the accumulated key-value cache for that request from memory once; the arithmetic performed per byte read is low, so decoding is memory-bandwidth-bound while only the initial processing of a long prompt is compute-bound, a distinction Pope and colleagues formalized in their analysis of transformer inference efficiency [ 15 ] . Approximately,…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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