Repeated emission and the loss-closed interface

Fix $K\ge 2$, $1\le c\le K$, and

The core is

Sector zero is the unique endpoint. Mode pair $r$ has energy ${\Omega }_r$. There is at most one quantum per retained mode pair.

Emission of $k\le \min (n,c)$ constituents occupies $I_{n,k}=\{n-k+1,\dots ,n\}$. Its energy is $E_n-E_{n-k}$.

Define the deduplicated occupation family

The loss-closed packet space is

with rail bits ordered by decreasing mode index and energy ${\sum }_{r\in S}{\Omega }_r$. A configuration is identified once, regardless of its hidden origin.

The label count and physical dimension are

For a span of length $d+1$, both endpoints are present; interior indices have two occupation choices or three physical choices. This proves the formulas.

The declared representation uses

The label is the surviving set; its actual rail bits are followed by zero padding. For stage A, $c=1$, $q_A=\lceil {\log }_2(K+1)\rceil +1$.