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Equation 108 · The Bit Comes Back Before the Bearing

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tG(δ)t_G(\delta)

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tGt_G

Symbol t_G

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δ\delta

Symbol delta

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subscript

subscript

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​ is not a slower recovery — it is a request for better precision than the compass itself possessed, which no observer, inside or outside any horizon, could ever meet. Within that ceiling, tG(δ)t_G(\delta) grows toward the hole’s full evaporation lifetime as δ\delta is tightened, which is a specific, checkable form of the “information remnant” Nakata, Wakakuwa, and Koashi found directly in their symmetry-constrained analysis of the Hayden-Preskill protocol: a residue that a purely logical decoder never has to wait for, because it was never entitled to it in the first place [ 8 ] .

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