A 55-Addition Rank-23 Scheme for $3\times 3$ Matrix Multiplication via Exact Two-Sided Minimization

We give a scheme that multiplies two $3\times 3$ matrices with 23 multiplications and 55 additions over any ring, one addition below the previous record of 56 (Sun, 2026) and the first below 56 for exact non-commutative $3\times 3$ multiplication without a change of basis. The scheme forms 13 signed combinations of the entries of A and 14 of B, takes 23 products, and recombines them with 28 further additions; its correctness is machine-checked in Lean 4. The previous records, from 60 down to 56, were obtained by greedy common-subexpression elimination, which only bounds each side of a scheme from above. We minimize both input sides exactly by chain covering, and the output side exactly through the transposition principle: the 9-by-23 output map costs its transpose plus 14, and the transpose is a 9-input map that the input-side minimizer solves exactly. Applied to Perminov’s cn122 scheme (class i19w225c4efh, published at 58 additions) this gives 55 = 13 + 14 + 28, where greedy elimination reaches only 29 on the output side. The same method shows that Sun’s representative is output-optimal at 56, and an exact orbit-wide computation over the Heule-Kauers-Seidl database shows that the record class is the only database class attaining 55.