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Equation 22 · Part 13 · A Horizon Is a Toll Booth, Not a Loophole

Denominator: pihbar

N⊥[Γ]=2πℏ∫Γ[⟨H(τ)⟩−E0(τ)]dτ.N_\perp[\Gamma] = \frac{2}{\pi\hbar}\int_\Gamma\left[\langle H(\tau)\rangle - E_0(\tau)\right]d\tau.
πℏ\pi\hbar

What this part means

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

Its job in the formula

pihbar 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

Define the object this paper needs: for a worldline Γ\Gamma traced by a processor, with instantaneous mean energy above its own ground state ⟨\langle H(τ\tau)⟩\rangle - E0(τ)E_0(\tau) measured in the processor’s own comoving rest frame at proper time τ\tau , the cumulative orthogonalization count is N⊥[Γ]=2πℏ∫Γ[⟨H(τ)⟩−E0(τ)]dτN_\perp[\Gamma] = \frac{2}{\pi\hbar}\int_\Gamma\left[\langle H(\tau)\rangle - E_0(\tau)\right]d\tau. N⊥N_\perp[Γ\Gamma] is a real-valued, non-negative functional of a timelike worldline in a fixed background spacetime together with a stated local energy convention; because the integrand is an energy and dτ\tau is a time, the integral has action units and N⊥N_\perp is dimensionless. It is invariant under reparametrizing Γ\Gamma ’s own curve parameter, since only proper time enters, and…

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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 article section

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