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Equation 8 · Part 9 · What a Rule Can Bind When the System Changes Weekly

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Ceq(t)=C∗⋅2 t/τ,τ≈8 months.C_{\mathrm{eq}}(t) = C^{*} \cdot 2^{\,t/\tau}, \qquad \tau \approx 8\ \mathrm{months}.
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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.

The passage around this formula

Second, the meaning of a fixed number drifts, and quickly . Ho and colleagues, analysing more than two hundred language-model evaluations from 2012 to 2023, estimated that the compute required to reach a set performance threshold has halved roughly every eight months, with a 95% confidence interval of about five to fourteen months [ 19 ] . If a threshold is fixed at C∗C^{*} and efficiency improves with a halving time τ\tau , the capability reachable below the threshold grows as though the budget were expanding: Ceq(t)=C∗⋅2 t/τ,τ≈8 monthsC_{\mathrm{eq}}(t) = C^{*} \cdot 2^{\,t/\tau}, \qquad \tau \approx 8\ \mathrm{months}. The analysis that follows is mine rather than the authors’. Taking the central estimate at face value, a threshold left unamended for two years admits models with…

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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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