Causal Locality Needs Spatial Input: The account rule and the edge-transfer premise

A debit and a credit conserve the combined balance of two books, yet their recording cubes may be arbitrarily far apart. This remains true when each post translates its window by one lattice edge: bounded motion and ordered entries leave the destination of a crossing unrestricted. A transfer represented by one spatial edge supplies the missing geometry. Its endpoint cubes coincide or are adjacent, and their arbitrary state corners have sharp separation bound five. For moving windows with crossing range r, a history of n double entries has no crossing when its initial separation exceeds r+2(n-1). We prove this bound sharp and show that it characterizes range for history classes closed under taking segments. A local rule for choosing updates then gives a sharp finite cone of dependence and, when the cone's joint input law is preserved, statistical no-signalling.