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Equation 37 · Part 9 · A Horizon Is a Toll Booth, Not a Loophole

Denominator: dτ

∣ddτln⁡ ⁣[⟨H(τ)⟩−E0(τ)]∣≪2[⟨H(τ)⟩−E0(τ)]πℏ.\left|\frac{d}{d\tau}\ln\!\left[\langle H(\tau)\rangle - E_0(\tau)\right]\right| \ll \frac{2\left[\langle H(\tau)\rangle-E_0(\tau)\right]}{\pi\hbar}.
dτd\tau

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

dτ occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

The dossier for this paper flags something the two equations above do not, by themselves, settle: the periodic-evolution bound above was derived for a Hamiltonian with a genuinely time-independent spectrum. A real computation drives H(τ\tau) through a sequence of different instantaneous Hamiltonians, one per gate, and nothing guarantees in advance that a chain of instantaneously applied bounds sums to a valid bound on the whole chain. The condition that licenses treating N⊥N_\perp[Γ\Gamma] as a legitimate partition into instantaneously static Levitin-Toffoli steps is a quasi-static one: over one local orthogonalization time τML(τ)\tau_{\rm ML}(\tau) = π\piℏ\hbar/[2(⟨\langle H(τ\tau)⟩\rangle-E0(τ)E_0(\tau))] ,…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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