Forced Directions for Two Binary Bases

Two linear checks can detect every vector in two prescribed binary bases. Which extra vectors must every such pair of checks miss? These vectors form a subspace, called the forced space. The two papers determine how large this space can be and study how to recognize its members. Forced Directions for Two Binary Bases proves that in dimension d the largest possible forced dimension is d divided by three, rounded down. A deterministic polynomial-time algorithm constructs at most d pairs of checks witnessing the bound. In dimension 3k, equality has a geometric characterization. The columns occur in k successive groups, each becoming a Fano plane with one point missing after the preceding groups are factored out. The missing directions span the forced space. Odd circuits of paired columns give an order-independent way to contract forced directions while preserving every detecting pair. A second rule identifies columns that must receive the same color. The binary Golay code shows the limit of both rules. Removing one of its 23 parity-check columns leaves two bases whose only nonzero forced vector is the missing column, yet neither rule finds it. Detecting One Extra Vector characterizes detection by a partition into a basis and a spanning family with one even circuit. This gives randomized polynomial-time search and an equivalent algebraic test modulo fourth powers. Under a strict density condition, the converse to the zero-Gram obstruction reduces to coloring lifts of the complete graph on five vertices. The converse holds analytically through rank eight, by exhaustive verification in ranks nine and ten, and in rank eleven for lifts with five shared dimensions. The unrestricted converse and deterministic polynomial-time recognition of an arbitrary extra vector remain open. Detection on three prescribed bases is NP-complete.