← All parts of this equation

Equation 44 · Part 5 · How Fast Can a Horizon Learn Which Path You Took?

=

Treceipt(δ)=inf⁡{tc:Iirr(tc)≥1−δ}.T_{\rm receipt}(\delta)= \inf\{t_c:I_{\rm irr}(t_c)\geq1-\delta\}.
=

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

Finally, the proposed horizon receipt time is Treceipt(δ)=inf⁡{tc:Iirr(tc)≥1−δ}T_{\rm receipt}(\delta)= \inf\{t_c:I_{\rm irr}(t_c)\geq1-\delta\}. These are proposed definitions, not results from the 2025 optimization paper. They deliberately make admissible control part of the quantity. If A\mathcal A grants arbitrary global unitaries on the horizon, the record can become fictitiously erasable. If it permits only “do nothing,” receipt time collapses into an ordinary decay threshold. A physically useful answer must live between those caricatures.

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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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Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.