Theorem: Exact minimal reached dimension

The smallest subspace containing the prepared stored code and invariant under $N_{\mathrm{tot}},G_{\mathrm{hop}}$ is

Thus

Restricted propagation is exact on the code. For P8, $4\le \dim {\mathcal{K}}_{\mathrm{ext}}\le 40$.

Proof. Number-sector projections are polynomials in $N_{\mathrm{tot}}$ on this finite space. Any invariant subspace containing the initial code contains $B_{m,0}{\mathcal{V}}_m$. The recurrence successively generates every $k$, and proves invariance of the displayed sum. Orthogonality proves the dimension formula.

For P8 the endpoint ranks are at most one and interior ranks at most two, giving $1+6+10+14+9=40$. Also $2=\mathrm{rank}V\le {\sum }_m{r}_m$. If $r_0=1$, a further rank has weight at least three. If $r_0=0$, the total weight is at least six. Hence the unconditional lower bound four.