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Equation 19 · How Fast Can a Horizon Learn Which Path You Took?

What does this equation mean?

Tdec(δ)=inf⁡{T:Idec(T)≥1−δ}.T_{\rm dec}(\delta)= \inf\{T:I_{\rm dec}(T)\geq1-\delta\}.

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Inputs and operationsinfT:I_rm dec(T) ≥ 1-delta
Result or conditionT_rm dec(delta)
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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.

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TdecT_{\rm dec}

Symbol T_rm dec

TrT_rm dec is part of the quantity the equation computes from the expression on the right.

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δ\delta

Symbol delta

the define a confidence deficit.

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TT

Symbol T

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

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IdecI_{\rm dec}

Symbol I_rm dec

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

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=

=

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

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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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How to interpret it

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What the article says around this equation

and a minimum-error decision time is Tdec(δ)=inf⁡{T:Idec(T)≥1−δ}T_{\rm dec}(\delta)= \inf\{T:I_{\rm dec}(T)\geq1-\delta\}. Both are deterministic transformations of the assumed overlap law. Neither is a new field-theory result. Numerically inverting the two binary-entropy equations gives:

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