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Equation 106 · Part 6 · A Horizon Is a Toll Booth, Not a Loophole

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Eloc dτ=E∞ dtE_{\rm loc}\,d\tau=E_\infty\,dt
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What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

Collect the ledger. N⊥N_\perp[Γ\Gamma] , the cumulative orthogonalization count, is an exact and dimensionless consequence of the Levitin-Toffoli bound integrated over proper time, valid without approximation in the quasi-static driving regime and bounded above by the sharper Deffner-Lutz result outside it — that piece is DERIVED. The identity ElocE_{\rm loc}\,dτ\tau=E∞E_\infty\,dt is an exact consequence of the definition of Killing energy in a static spacetime, proven here for the specific case of N⊥N_\perp[Γ\Gamma] rather than asserted by analogy — also DERIVED. That no energy-convention bookkeeping can extract a net gain from this identity is a direct corollary, not an assumption.

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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