Exact Functional Contractions for 21-Term Decompositions of 3×3 Matrix Multiplication over F₂

We study exact necessary conditions for 21-term decompositions of the 3x3 matrix multiplication tensor over F2. The main object is the contraction of the matrix multiplication tensor against nonzero linear functionals on one factor. For every nonzero linear functional lambda in F2^9, any 21-term decomposition necessarily satisfies a lower bound on the number of summands detected by lambda, determined by the rank of the corresponding contracted target matrix. We verify all 511 nonzero functionals independently and obtain agreement between two exact GF(2) rank implementations. In the tensor convention used here, the contraction ranks satisfy rank_F2(T_lambda) = 3 * rank_F2(L_lambda) for all 511 nonzero lambda, where L_lambda denotes the associated 3x3 binary matrix. This yields a finite family of exact, machine-checkable necessary inequalities on the factor multiset of any putative 21-term decomposition. The formulation subsumes the single-slice cover condition and pairwise XOR-slice conditions as special cases. We also describe independent implementation checks and the computational scope of the result. These conditions are necessary but not sufficient. The paper does not provide a rank-21 decomposition and does not prove the nonexistence of rank-21 decompositions. No global tensor-rank claim is made. A German-language version of the manuscript is included in the record.