Equation 1 · Part 11 · AI Accelerator Architecture in 2035: Scenarios, Signals, and Falsifiable Predictions

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\frac{F(t)}{B(t)} = \frac{F_0}{B_0}\left(\frac{r_F}{r_B}\right)^{t/2}

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What this part means

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.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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The passage around this formula

The structural fact behind every accelerator roadmap for a decade has been that peak arithmetic throughput has scaled roughly three times every two years while DRAM bandwidth has scaled roughly 1.6 times and interconnect bandwidth roughly 1.4 times over the same interval [ 1 ] . Analysis. Extending those three exponents unchanged from a 2024 baseline to 2035 — eleven years, or 5.5 doubling periods at the two-year cadence the source measures — gives a compact way to see how much the imbalance would compound if nothing structural intervenes: $\frac{F(t)}{B(t)} = \frac{F_0}{B_0}\left(\frac{r_F}{r_B}\right)^{t/2}$ . With $r_F$ $\approx$ 3.0 and $r_B$ $\approx$ 1.6 for DRAM bandwidth, ( $r_F$ / $r_B$ )^{5.5} = (1.875)^{5.5} $\approx$ 32 : the ratio of peak…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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