Proposition: Exact forward decoding

Let $f$ be the map on canonical $B$-bit geometric words at the forward event that changes

and leaves the clock and particle packet fixed. Its outgoing reports are the canonical limiting reports at $s=0$. Then, on the forward event domain, decoding $f$ gives precisely the geometric part of (eq:Grule).

Proof. At $s=0$, all persistent finite edges have identical incoming and outgoing limiting lengths. Every new slot has limiting length zero. Thus the two canonical limiting report arrays agree. The mode change selects the outgoing germ rather than the incoming germ, and the permission field has the required decrement. All other decoded data agree with (eq:Grule).