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Equation 3 · Part 8 · Model Systems in 2035: Four Scenarios, Their Signals, and What Would Falsify Them

=

I∗(t)=F(t)B(t)=I0∗(gFgB)t,I^{*}(t) = \frac{F(t)}{B(t)} = I^{*}_0 \left(\frac{g_F}{g_B}\right)^{t},
=

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

Write F(t) for peak arithmetic throughput and B(t) for memory bandwidth. The quantity that matters for a serving system is their ratio, because it sets the arithmetic intensity — operations per byte moved — at which a machine becomes compute-bound rather than bandwidth-bound: I∗(t)=F(t)B(t)=I0∗(gFgB)tI^{*}(t) = \frac{F(t)}{B(t)} = I^{*}_0 \left(\frac{g_F}{g_B}\right)^{t}. with gFg_F and gBg_B the annual growth factors. When gFg_F > gBg_B , I∗I^{*} grows without bound, and the batch size required to keep the arithmetic units busy grows with it. Autoregressive decoding sits on the wrong side of this: generating one token requires streaming the weights and the accumulated key–value cache, so decode time is bounded below by

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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 article section

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