Admissibility of the coupled floor profile at every q

Fix $q\ge 3$ and $\tau$ with $3\tau\ge 2$, and set

Then for all sufficiently large $N$ one has $L>0$, $G>0$, $L+G=N$ and $341L<100G$.

Here $\lambda$ is the fraction of the $N$ tensor positions that Coppersmith--Winograd assign to the two "low" blocks $\langle 1,q,1\rangle$ of the coupled constituent, the remaining $G$ positions carrying the two coupled $\langle q,1,q\rangle$ blocks. The last conjunct is the margin the hashing step consumes: since $G/L\to q^{3\tau}/2$ and $q\ge3$, $3\tau\ge2$ give $q^{3\tau}\ge 9$, the limiting ratio is at least $4.5$, comfortably above $3.41$. The quantitative input is mme_CW_coupled_pruning_ratio, which supplies $341/100 < q^{3\tau}/2$ for exactly this range of $q$ and $\tau$.

This is the general-$q$ form of mme_CW_q6_coupled_exact_floor_pruning, additionally recording $0<G$, which the capacity estimate needs and which does not follow from $L>0$ and $L+G=N$ alone (truncated subtraction allows $G=0$ when $L=N$).