A factorization is an algorithm

Suppose R blocks add up to the tensor, cell for cell: T_abc = Σ_r U_ra · V_rb · W_rc. Put that into the read-off sum and only the order of summation changes:

C_c = Σ_r W_rc · (Σ_a U_ra A_a) · (Σ_b V_rb B_b)      # R multiplications m_r

Every exact CP decomposition of the multiplication tensor into R rank-one terms is an algorithm that multiplies with R multiplications.

This is section 11’s CP with an equals sign where section 11 had ≈: all 64 cells must match. The smallest possible count is the rank of the tensor. For 2 × 2 the rank is 7 — Winograd proved in 1971 that six multiplications cannot do it.