$$\epsilon_L(t)$$

Equation 28 · Evolutionary Physics

The Bit Comes Back Before the Bearing

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

$\mathcal N_t$ is the effective channel from A to R\!$\cup$\!$C_t$ induced by the hole’s own scrambling dynamics up to time t ; $\epsilon_L(t)$ is a dimensionless number in [0,1] , and $t_L$ has units of time. This is a yes-or-no-shaped promise sharpened into a number: either a decoder within $\epsilon$ of perfect exists on the stated resources, or it does not, and $\epsilon_L(t)$ falls, in the regime this article works in, toward zero as t grows past a threshold.

Meanings in this article

• $\epsilon_L$: a dimensionless number in [0,1] , and $t_L$ has units of time.

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Equation 32 · Evolutionary Physics

The Bit Comes Back Before the Bearing

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

$\mathcal N_t$ is the effective channel from A to R\!$\cup$\!$C_t$ induced by the hole’s own scrambling dynamics up to time t ; $\epsilon_L(t)$ is a dimensionless number in [0,1] , and $t_L$ has units of time. This is a yes-or-no-shaped promise sharpened into a number: either a decoder within $\epsilon$ of perfect exists on the stated resources, or it does not, and $\epsilon_L(t)$ falls, in the regime this article works in, toward zero as t grows past a threshold.

Meanings in this article

• $\epsilon_L$: a dimensionless number in [0,1] , and $t_L$ has units of time.

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Equation 45 · Evolutionary Physics

The Bit Comes Back Before the Bearing

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

The shape difference between these two promises is the entire point of separating them. $\epsilon_L(t)$ is bounded in [0,1] and can, in principle, hit its floor abruptly once decoupling sets in. $\operatorname{Var}$$\hat g$ has no upper bound and shrinks only as fast as $F_Q$ grows, which — as the next two sections show — need not be abrupt at all. A reader who asks “has the black hole given it back?” is asking two different kinds of question depending on which object they mean, and conflating them is precisely the failure mode this construction is built to prevent.

Meanings in this article

• $\epsilon_L$: a dimensionless number in [0,1] , and $t_L$ has units of time.

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Equation 134 · Evolutionary Physics

The Bit Comes Back Before the Bearing

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

A third, more uncomfortable check is what the toy model of the previous section actually implies at the numbers already computed: $t_L$ $\approx$ 3.5\,$\mathrm{ms}$ , or $t_L$/$\beta$ $\approx$ 28.2 , for the solar-mass hole above. Taking $f_1$ = 1 as an illustrative, explicitly assumed per-quantum information content, a modest tolerance $\delta$ = 0.1\,$\mathrm{rad}$ (about six degrees) gives $t_G$/$\beta$ = 100 , so $t_G$ $\approx$ 12.4\,$\mathrm{ms}$ and $\Delta$ $t_{LG}$ $\approx$ +8.9\,$\mathrm{ms}$ : the compass is the slower debt, as the title claims. But a coarser tolerance, $\delta$ = 0.5\,$\mathrm{rad}$ (about twenty-nine degrees), gives $t_G$/$\beta$ = 4 , so $t_G$ $\approx$ 0.50\,$\mathrm{ms}$ and $\Delta$ $t_{LG}$ $\approx$…

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