mme_CW_subrank_capacity_lower

Laser-method lower bound on the subrank capacity of the CW tensor at q = 6.

For the Coppersmith–Winograd tensor T_6, the asymptotic laser-method analysis yields

Combined with the border-rank bound $\widetilde R(T_6) \leq 8$ (mme_CW_border_rank_le via mme_degenerates_asymptoticRank_le) and the abstract bridge mme_omega_le_of_subrank_capacity, this yields

Why this constant. $V = 5/2$ at $q = 6$ is chosen for the cleanest top-level reduction: it leaves a comfortable margin under $2376/1000$ and matches the regime where the laser-method bound for T_q becomes effective. The actual sharp value achievable by the canonical CW §7–§8 analysis (via the symmetric tensor square of $T_6$ with refined Salem–Spencer indexing) is somewhat larger; future refinements (Stothers, Vassilevska Williams, Le Gall, Alman–VW) will produce sharper bounds via their own subrank-capacity lower bounds on their own tensors, each as a new theorem node.

Proof status. Open. Recurses into the full Layer-2 abstract laser-method machinery:

1. Block decomposition of $T_q^{\otimes 2N}$: the rank-one support of $T_q$ is 3-graded by the index type $\tau : \mathrm{Fin}(q{+}2) \to \{0,1,2\}$, so $T_q^{\otimes 2N}$ splits as a sum of "block tensors" indexed by triples $(I, J, K)$ of multi-types.

2. Restriction to a Salem–Spencer set: pick $S \subseteq [N]$ with no nontrivial 3-AP; restricting block indices to $S$ kills all collisions between block tensors (each pair of distinct surviving blocks shares no factor coordinate), turning the sum into a direct sum.

3. Each surviving block is a matrix-multiplication tensor $\langle a, b, c\rangle$ of explicit multinomial dimensions.

4. Counting: Stirling / multinomial bounds on the number of surviving blocks plus their dimensions give the explicit value lower bound.

5. Bridge from Mathlib's Behrend bound $|S| \geq N \exp(-4\sqrt{\log N})$ to the $\varepsilon$-form $|S| \geq N^{1-\varepsilon}$.

These Layer-2 leaves are themselves paper-agnostic: replacing $T_q$ by any other tensor with a 3-graded support of the right combinatorial profile yields the same machinery.