A Block Criterion for Extremal Jointly Square-Zero Tuples

Let T=(T_1,...,T_d) be a tuple of contractions on a complex Hilbert space with T_i T_j=0 for all i,j. Relative to the common kernel F and its orthogonal complement E, write T_i=[0 0; W_i 0]. We prove that T is extremal among all commuting contractive extensions if and only if every W_i is a coisometry and the map (x_i) -> (W_i x_j-W_j x_i)_(i<j) is injective. Every failure has a nontrivial extension of codimension one. For nonzero finite-dimensional spaces, extremality implies 2 <= q <= p <= (d-1)q/2, where p=dim E and q=dim F. An explicit four-tuple of 5 by 5 matrices with entries in {0,1,-1} attains the upper bound and the smallest possible odd dimension within this class. We also give a quantitative bound on extension columns for nearly coisometric blocks. The unitary special case and block method are credited to Timko. This is a complete scoped theorem, not a classification of all commuting contractive tuples in OWR-3470-005. The preprint is AI-assisted, self-audited and unrefereed; novelty and absolute priority are not certified.