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Equation 37 · Part 8 · A Horizon Is a Toll Booth, Not a Loophole

subscript

∣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}.
subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

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 subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the surrounding passage

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