Residue Constraints in the Rank-Three Lifting Problem for Projective-Plane Incidence Matrices

Abstract

We study the rank-three lifting problem for incidence matrices of finite projective planes through residue-level determinant constraints invisible to tropical valuations alone. In residue characteristic $\neq 3$, any rank-$\le 3$ lift of the incidence matrix of a projective plane of order $q \ge 3$ forces $\Omega(q^8)$ distinct admissible $2 \times 2$ zero rectangles with nontrivial residue cross-ratio. We further prove that for $q \ge 6$ no monomial rank-$\le 3$ lift exists; in particular, any putative low-rank lift must already involve nontrivial first-order corrections on valuation-0 entries. These results arise from a local analysis of $4 \times 4$ identity-pattern minors, where we derive the leading derangement equation together with its first-order companion and show that every vanished identity-pattern minor contains a cross-ratio-defective admissible rectangle. The unresolved part of the problem is therefore genuinely global: one must decide whether a rank-3 residue model, together with a compatible first-order deformation, can satisfy the full overlapping system of local residue constraints.