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

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 ≠ 3, any rank-≤ 3 lift of the incidence matrix of a projective plane of order q ≥ 3 forces Ω(q⁸) distinct admissible 2 × 2 zero rectangles with nontrivial residue cross-ratio. We further prove that for q ≥ 6 no monomial rank-≤ 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 × 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.