Sub-base cyclic value of the coupled constituent at every q

Every base strictly below $4q^{3\tau}(q^{3\tau}+2)$ is a $\tau$-value of the cyclic symmetrisation of the coupled Coppersmith--Winograd constituent, for every $q\ge3$.

Formally: for $q\ge3$, $3\tau\ge2$ and $0\le V<4q^{3\tau}(q^{3\tau}+2)$,

This is obtained from the even-power extractions of mme_CW_coupled_tensor_extraction_below_raw together with the admissibility of the floor profile (mme_CW_coupled_floor_pruning), fed into mme_HasTauValueAtLeast_of_cofinal_finite_extractions along the cofinal subsequence $N\mapsto 2N$ with zero error.

Why the sub-base form. The platform's HasTauValueAtLeast T tau V requires, for each fixed $\varepsilon>0$, infinitely many $N$ with an extraction achieving $V^N(1-\varepsilon)$ — a loss that does not depend on $N$. The laser construction only delivers $\mathrm{raw}^N e^{-cN^{3/4}}$, and $e^{-cN^{3/4}}$ eventually falls below any fixed $1-\varepsilon$. So the sharp base $\mathrm{raw}$ is not reachable this way, while every $V<\mathrm{raw}$ is, because $(V/\mathrm{raw})^N$ decays exponentially and therefore swallows the sub-exponential loss. This is why every value theorem along the successful $\omega<2.376$ route is stated in _below form.

General-$q$ form of mme_CW_q6_coupled_raw_cyclic_value_below.