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Equation 106 · Part 4 · A Horizon Is a Toll Booth, Not a Loophole

Symbol E_infty

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

What this part means

EiE_infty is an input to the expression that computes the quantity on the left.

Its job in the formula

EiE_infty is an input to the expression that computes the quantity on the left.

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

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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