Manuscript VII-J@1 · VII-J.tex · proposition · d8c01b04

Proposition: Robust cone margin

The fixed cone and VII-F's slicing event › Localization and timing uncertainty · lines 1526–1553 · proof included or linked

VII-J@1#prop:conemargin, VII-J@1#eq:robustcone

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A nominal dependency has slicing-time gap $h$ , nominal endpoint distance at most $D$ , spatial localization radii $r_{-}, r_{+}$ , slicing-time errors at most $ϵ_{s}$ at each endpoint, and additional absolute spacetime-time errors at most $ϵ_{T}$ at each endpoint.

It remains causally admissible if

h - (1 + λ_{0}) D \geq (1 + λ_{0}) (r_{-} + r_{+}) + 2 ϵ_{s} + 2 ϵ_{T} .

Strict inequality is a positive robustness margin.

Proof. The actual spatial distance is at most $D + r_{-} + r_{+}$ , and the actual slicing gap is at least $h - 2 ϵ_{s}$ . The time-difference estimate of Theorem thm:conegates then loses at most another $2 ϵ_{T}$ . Rearrangement gives (eq:robustcone).

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VII-J@1#eq:robustcone · line 1536
$h - (1 + λ_{0}) D \geq (1 + λ_{0}) (r_{-} + r_{+}) + 2 ϵ_{s} + 2 ϵ_{T} .$

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section: Localization and timing uncertainty