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S≤2πkBRE/(ℏc)S\leq 2\pi k_BRE/(\hbar c)

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A second, independent ceiling bounds not the rate of erasure but the total information a bounded system can hold at all: the Bekenstein bound limits the entropy of a system of effective size R and energy E to S≤\leq 2π\pi kBk_BRE/(ℏ\hbar c) [ 8 ] . For a processor of size R=1\,cm\mathrm{cm} holding E=1\,J\mathrm J of available energy, this gives S≲\lesssim27\,J K−1\mathrm{J\,K^{-1}} , equivalent to roughly 2.9×\times10^{24} bits of maximum distinguishable entropy — a static capacity ceiling, not a rate. Erasing that many bits at 10\,mK\mathrm{mK} would cost at least Qmin⁡Q_{\min}=ST≈\approx0.27\,J\mathrm J , more than a quarter of the entire energy budget assumed available, entirely independent of any horizon.…

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SS

Symbol S

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π\pi

Symbol pi

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kBk_B

Symbol k_B

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RR

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EE

Symbol E

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cc

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S≤2πkBRE/(ℏc)S\leq 2\pi k_BRE/(\hbar c)

Equation 97 · Evolutionary Physics

A Horizon Is a Toll Booth, Not a Loophole

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

A second, independent ceiling bounds not the rate of erasure but the total information a bounded system can hold at all: the Bekenstein bound limits the entropy of a system of effective size R and energy E to S≤\leq 2π\pi kBk_BRE/(ℏ\hbar c) [ 8 ] . For a processor of size R=1\,cm\mathrm{cm} holding E=1\,J\mathrm J of available energy, this gives S≲\lesssim27\,J K−1\mathrm{J\,K^{-1}} , equivalent to roughly 2.9×\times10^{24} bits of maximum distinguishable entropy — a static capacity ceiling, not a rate. Erasing that many bits at 10\,mK\mathrm{mK} would cost at least Qmin⁡Q_{\min}=ST≈\approx0.27\,J\mathrm J , more than a quarter of the entire energy budget assumed available, entirely independent of any horizon.…

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