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

The corrected diamond–disk comparison

Directional paths and finite-time reachability › The corrected diamond–disk comparison · lines 936–978

VII-J@1#eq:diamond-disk

For cardinal hops of length $a$ , one per model interval $τ$ , or a wait,

K_{Sch} = {v : ∣ v_{1} ∣ + ∣ v_{2} ∣ \leq a / τ} .

For an isotropic physical speed $c = a / τ$ ,

K_{Phys} = {v : ∥ v ∥_{2} \leq c} .

Then

K_{Sch} \subseteq K_{Phys}, K_{Phys} \neq \subseteq K_{Sch} .

Indeed,

h_{K_{Sch}} (n) = c ∥ n ∥_{\infty} \leq c ∥ n ∥_{2} = h_{K_{Phys}} (n),

whereas the disk velocity $(c / 2, c / 2)$ lies outside the diamond.

This corrects an orientation ambiguity in the proposed plan: the diamond is safe relative to the equal-speed disk; it does not cover every physically possible direction. VII-E's directional example concerns that failed reverse coverage CheungVIIE (prop:direction).

Thus

implementation influence \subseteq physical possibility

is causal safety, while

target dependency \subseteq available numerical stencil

is an information-coverage requirement. Neither proves simulation accuracy.

Displayed formulas in this passage

VII-J@1#eq:diamond-disk · line 948
$K_{Sch} \subseteq K_{Phys}, K_{Phys} \neq \subseteq K_{Sch} .$
unlabeled · line 940
$K_{Sch} = {v : ∣ v_{1} ∣ + ∣ v_{2} ∣ \leq a / τ} .$
unlabeled · line 944
$K_{Phys} = {v : ∥ v ∥_{2} \leq c} .$
unlabeled · line 954
$h_{K_{Sch}} (n) = c ∥ n ∥_{\infty} \leq c ∥ n ∥_{2} = h_{K_{Phys}} (n),$
unlabeled · line 968
$implementation influence \subseteq physical possibility$
unlabeled · line 972
$target dependency \subseteq available numerical stencil$