Fourth-power coarse-block value from a pair of square constituent values

Component values of the fourth power, assembled from a pair of square components.

Work over a field $K$ and fix the Coppersmith--Winograd parameter $q$. The tensor square $CW_q^{\otimes 2}$ carries a fifteen-shape decomposition, each shape $s$ recording a triple of per-mode grades $(\mathrm{shapeX}(s), \mathrm{shapeY}(s), \mathrm{shapeZ}(s))$; the fourth power $CW_q^{\otimes 4}$ carries the coarser nine-grade decomposition indexed by $\sigma = (\sigma_0, \sigma_1, \sigma_2)$.

Let $p = (p_1, p_2)$ be a pair of square shapes that is compatible with $\sigma$, meaning the grades add mode by mode:

Let $X$ and $Y$ be tensors restricting to the square blocks of shapes $p_1$ and $p_2$ respectively, and suppose each has a strict value endpoint at exponent $\tau$: $X$ has tau-value at least $V$ for every $0 \le V < e_X$, and $Y$ has tau-value at least $V$ for every $0 \le V < e_Y$, with $e_X, e_Y > 0$. Then the fourth-power coarse block of grade $\sigma$ has tau-value at least $W$ for every

This is the analysis of component values of the fourth-power laser method: the fine-to-coarse restriction says that the coarse block of grade $\sigma$ dominates the Kronecker product of any compatible pair of fine square constituents, and values are multiplicative under Kronecker products, so every compatible pair contributes a lower bound $e_X e_Y$ on the value of the coarse block. Taking the best pair over all decompositions of $\sigma$ is what turns the known square component values into the fourth-power component values that the global entropy optimisation then consumes.

Formalization note. The two inputs are mme_dwz_fourth_pair_factor_restrictions_to_coarse, which produces the restriction of TensorObj.kron X Y into the coarse block, and mme_HasTauValueAtLeast_kron_of_each_strict_below_product, the binary multiplicativity of the tau-value; the value then transports along the restriction by mme_HasTauValueAtLeast_mono_restrict. Endpoints are kept in strict form throughout, since no step of the extraction ever produces a value at an endpoint.