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Equation 43 · How Fast Can a Horizon Learn Which Path You Took?

What does this equation mean?

Iirr(tc)=1−h2 ⁣(1−Dirr(tc)2).I_{\rm irr}(t_c)= 1-h_2\!\left(\frac{1-D_{\rm irr}(t_c)}{2}\right).

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start with1-D_rm irr(t_c)
Divide by2
This relates toI_rm irr(t_c)
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

Read it piece by piece

IirrI_{\rm irr}

Symbol I_rm irr

IrI_rm irr is part of the quantity the equation computes from the expression on the right.

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tct_c

Symbol t_c

the become irrevocable at an intermediate cutoff.

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h2h_2

Symbol h_2

h2h_2 is one of the signed contributions combined to compute the quantity on the left.

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DirrD_{\rm irr}

Symbol D_rm irr

monotone.

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=

=

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

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fraction

fraction

Divide the expression above the line by the one below it.

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subscript

subscript

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.

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1−Dirr(tc)1-D_{\rm irr}(t_c)

Numerator: 1-D_rm irr(t_c)

The complete quantity above the fraction bar.

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22

Denominator: 2

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

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

The infimum asks how little path distinguishability remains after the best allowed attempt to erase it. For equal priors, an associated unavoidable decision information is Iirr(tc)=1−h2 ⁣(1−Dirr(tc)2)I_{\rm irr}(t_c)= 1-h_2\!\left(\frac{1-D_{\rm irr}(t_c)}{2}\right). Finally, the proposed horizon receipt time is

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Sources cited in the article section

These citations give research context. Read each source to check which claims it supports.

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