Equation 3 · Model Systems in 2035: Four Scenarios, Their Signals, and What Would Falsify Them
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.
Read it piece by piece
Symbol I^*
is part of the quantity the equation computes from the expression on the right.
Symbol t
t is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol F
F occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol B
B occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol I^*_0
is one factor in the product that computes the quantity on the left.
=
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.
superscript
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.
See an illustrated explanation →Denominator: B(t)
The complete quantity below the fraction bar; it must be nonzero for this division.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
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: . with and the annual growth factors. When > , 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
Sources cited in the article section
- [6] Compute Trends Across Three Eras of Machine Learning ↗
- [2] Key Trends and Figures in Machine Learning ↗
- [3] Scaling Laws for Neural Language Models ↗
- [4] Training Compute-Optimal Large Language Models ↗
- [5] Will We Run Out of Data? Limits of LLM Scaling Based on Human-Generated Data ↗
These citations give research context. Read each source to check which claims it supports.
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