Equation 3 · The Frontier Model Landscape in 2035: Four Scenarios, Their Signals, and What Would Falsify Them

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C_{\text{train}}(t) \approx C_0\, g^{\,t-t_0},

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This equation gives an approximation: it relates the quantities while allowing an approximation. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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C_{\text{train}}

Symbol C_train

$C_t$ rain is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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t

Symbol t

t is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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C_0

Symbol C_0

$C_0$ is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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g^{\,t-t_0}

Symbol g^t-t_0

$g^t$ - $t_0$ is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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≈

Approximately equal to; the equality is not exact.

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subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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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.

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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.

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Its accuracy depends on the assumptions and range of use described in the article.

What the article says around this equation

Write $C_{\text{train}}(t)$ for the cost of the largest training run in year t . Epoch’s estimate is well approximated by simple exponential growth, $C_{\text{train}}(t) \approx C_0\, g^{\,t-t_0}$ . with g between roughly two-point-four and three-point-five depending on method and window [ 5 , 6 ] . The form matters more than the exact rate: exponential cost growth does not by itself predict consolidation. It predicts a shrinking count of organisations able to clear an ever-rising bar only if the pool of adequately capitalised entrants grows more slowly than $g^{t}$ . Whether that pool keeps pace is a question about capital markets and state investment, not something the cost curve alone answers — which is exactly why axis…

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Write $C_{\text{train}}(t)$ for the cost of the largest training run in year t . Epoch’s estimate is well approximated by simple exponential growth, $C_{\text{train}}(t) \approx C_0\, g^{\,t-t_0}$ . with g between roughly two-point-four and three-point-five depending on method and window [ 5 , 6 ] . The form matters more than the exact rate: exponential cost growth does not by itself predict consolidation. It predicts a shrinking count of organisations able to clear an ever-rising bar only if the pool of adequately capitalised entrants grows more slowly than $g^{t}$ . Whether that pool keeps pace is a question about capital markets and state investment, not something the cost curve alone answers — which is exactly why axis A stays open below rather than settled by this one fact.

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