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Equation 7 · The Metabolism of Civilization: Energy to 2100

What does this equation mean?

LR=1−2−bLR = 1 - 2^{-b}

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Inputs and operations1 - 2^-b
Result or conditionLR
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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.

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LL

Symbol L

L is part of the quantity the equation computes from the expression on the right.

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RR

Symbol R

R is part of the quantity the equation computes from the expression on the right.

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bb

Symbol b

b is one of the signed contributions combined to compute the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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subtraction

subtraction

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

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superscript

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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How to interpret it

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What the article says around this equation

where the experience exponent b sets the learning rate LR = 1 - 2^{-b} , the fractional cost drop for every doubling of z . Way et al.'s working version of the same relationship is stated as a stochastic first-difference equation in log-cost against log-experience, with an estimated autocorrelation parameter of 0.19 held fixed across all fifty-plus technologies studied, chosen because fitting it separately for each technology’s comparatively short historical record degraded genuine out-of-sample forecasting accuracy [ 1 ] . The mathematics is unremarkable; what matters for this argument is what it implies about heritability and compounding. A technology with a high, stable learning rate…
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where the experience exponent b sets the learning rate LR = 1 - 2^{-b} , the fractional cost drop for every doubling of z . Way et al.'s working version of the same relationship is stated as a stochastic first-difference equation in log-cost against log-experience, with an estimated autocorrelation parameter of 0.19 held fixed across all fifty-plus technologies studied, chosen because fitting it separately for each technology’s comparatively short historical record degraded genuine out-of-sample forecasting accuracy [ 1 ] . The mathematics is unremarkable; what matters for this argument is what it implies about heritability and compounding. A technology with a high, stable learning rate converts every unit of deployment — every subsidy dollar, every early adopter, every demonstration project — into a durable, non-reversing reduction in future cost, the economic analogue of a trait that increases fitness in every generation it is expressed. A technology without that property, no matter how much capital or attention it receives, does not compound in the same way; its next generation starts from roughly the same cost position as the last one did.

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