The 4×4 minors of a 5×n matrix are a tropical basis

We compute the space of 5×5 matrices of tropical rank at most 3 and show that it coincides with the space of 5×5 matrices of Kapranov rank at most 3, that is, the space of five labeled coplanar points in the tropical torus. We then prove that the Kapranov rank of every 5×n matrix equals its tropical rank; equivalently, that the 4×4 minors of a 5×n matrix of variables form a tropical basis. This answers a question asked by Develin, Santos, and Sturmfels.