Local optimality of rank-47 schemes for 4×4 matrix multiplication over F2

The tensor ⟨4,4,4⟩ of 4×4 matrix multiplication has rank at most 47 over F2, and as it is not known whether 47 is optimal, we ask how far the known decompositions of rank 47 are from one of rank 46. A decomposition is locally optimal to depth D if no D or fewer of its rank-one terms can be replaced by fewer rank-one terms. We prove that five pairwise inequivalent decompositions of rank 47 over F2 are locally optimal to depth 18, as is every decomposition equivalent to one of them, so a decomposition of rank 46 over F2 shares at most 28 terms with any decomposition in these five classes. The five classes include the scheme of Fawzi et al. and every public decomposition of rank 47 over F2 known to us; one of them, found in our flip-graph searches, is new. The proof rests on a peeling lemma and, for four sets of terms that resist the other bounds, on a refinement of the substitution method, which we call substitution over subspaces with lifting. Local optimality to depth 18 is also proved in Lean 4.

The Lean proofs check the refutations of the SAT instances used for these four sets with the verified LRAT checker of Lean core. Their trusted base is the Lean kernel together with the compiled code of their native_decide evaluations (decision procedures run as compiled code rather than in the kernel).

The programs, the stored certificates and the Lean proofs are in the archive at https://doi.org/10.5281/zenodo.23652770.

Mathematics Subject Classification (2020): 68Q17, 15A69 (primary); 68V05, 68V15, 68V20 (secondary).