Manuscript XXII-D@1 · XXII-D.tex · proposition · 17889d18

Proposition: Finite-stencil directional delay

Failure tests › F2: a fixed finite stencil does not cover every null direction · lines 1331–1377 · proof included or linked

XXII-D@1#prop:stencil, XXII-D@1#eq:stencil-bound

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In a chosen Lorentz coordinate frame, suppose every moving step has nonnegative duration and spatial velocity in a fixed finite set

S \subseteq {s \in R^{m - 1} : ∥ s ∥_{2} \leq 1}, 0 \in S .

For $m \geq 3$ , choose a unit vector $u \in / S$ and set

h_{S} (u) = s \in S max u \cdot s < 1.

If a protocol's endpoint has

∥ x_{n} - r u ∥_{2} \leq δ_{n}, r > 0,

and total step duration $τ_{n}$ , then

r - δ_{n} \leq h_{S} (u) τ_{n} .

If $h_{S} (u) = 0$ , sufficiently accurate endpoints are impossible. If $h_{S} (u) > 0$ and $δ_{n} \to 0$ , then

n lim inf τ_{n} \geq r / h_{S} (u) > r .

Thus this stencil cannot approximate the null target $(r, r u)$ with vanishing arrival delay.

Proof. For a unit $u$ , equality $u \cdot s = 1$ with $∥ s ∥_{2} \leq 1$ requires $s = u$ . Hence $h_{S} (u) < 1$ . Because $0 \in S$ , it is nonnegative.

Writing step durations as $Δ t_{j} \geq 0$ gives

u \cdot x_{n} = j \sum Δ t_{j} u \cdot s_{j} \leq h_{S} (u) j \sum Δ t_{j} = h_{S} (u) τ_{n} .

The endpoint estimate gives $u \cdot x_{n} \geq r - δ_{n}$ . The conclusions follow.

Displayed formulas in this passage

unlabeled · line 1335
$S \subseteq {s \in R^{m - 1} : ∥ s ∥_{2} \leq 1}, 0 \in S .$
unlabeled · line 1340
$h_{S} (u) = s \in S max u \cdot s < 1.$
unlabeled · line 1344
$∥ x_{n} - r u ∥_{2} \leq δ_{n}, r > 0,$
XXII-D@1#eq:stencil-bound · line 1348
$r - δ_{n} \leq h_{S} (u) τ_{n} .$
unlabeled · line 1354
$n lim inf τ_{n} \geq r / h_{S} (u) > r .$
unlabeled · line 1367
$u \cdot x_{n} = j \sum Δ t_{j} u \cdot s_{j} \leq h_{S} (u) j \sum Δ t_{j} = h_{S} (u) τ_{n} .$

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section: F2: a fixed finite stencil does not cover every null direction