Any rank-≤3 lift of a projective plane incidence matrix in char ≠3 forces Ω(q^8) admissible 2x2 zero rectangles with nontrivial residue cross-ratio, and no monomial lifts exist for q≥6.
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Residue Constraints in the Rank-Three Lifting Problem for Projective-Plane Incidence Matrices
Any rank-≤3 lift of a projective plane incidence matrix in char ≠3 forces Ω(q^8) admissible 2x2 zero rectangles with nontrivial residue cross-ratio, and no monomial lifts exist for q≥6.