The Integer Lattice of Basis-Exchange Squares of a Matroid and the Square Basis Theorem (Application: A Plücker Criterion for Equivalence under Positive Column Scaling)

We prove that for every matroid M of rank at least 1, the integer kernel ker_Z A_M of the basis incidence matrix A_M has a Z-basis consisting of basis-exchange square relations e_{B+i} − e_{B+j} − e_{B'+i} + e_{B'+j} (the Square Basis Theorem). This is stronger than the statement that the squares generate ker_Z A_M. The proof consists of closure under direct sums, induction by deleting an element together with duality, and a single graph-connectivity statement, which is proved by a minimal counterexample argument. For uniform matroids U_{k,n} we give an explicit basis by a unitriangular construction with respect to the colex order. The proof does not depend on the Lasoń–Michałek saturation theorem and gives an elementary proof of the lattice equality as a corollary. The result is a lattice-level statement and is distinct from White's conjecture. As an application, we give a criterion for equivalence of two full-rank real matrices under positive column scaling, stated only in terms of sign conditions on Plücker ratios and the square conditions. The proof does not rely on computer calculation. Exhaustive checks of the key graph-connectivity theorem for all matroids on at most 7 elements and sample checks of the main theorem are supplementary evidence only. MSC 2020: 05B35, 13F65.