Exact Zarankiewicz numbers one above three plane orders

The Zarankiewicz number z(n;2) is the largest number of ones in an n x n 0/1 matrix with no 2 x 2 submatrix of ones. Let n = q^2+q+2 and P(q) = (q+1)(q^2+q+1). A projective plane of order q together with one extra point and one extra line placed along an anti-flag is a C4-free n x n 0/1 matrix with P(q)+3 ones. We show that, for every q >= 4, a C4-free n x n matrix with P(q)+4 ones leads to one of two structures, provided that every C4-free (q^2+q+1) x q^2 matrix has at most q^3+q^2+1 ones: a symmetric configuration (q^2+q+2)q+1, or an incidence structure with all degrees at most q+1 and total deficit between 1 and q-3 to which no incidence can be added without creating a C4. By a theorem of Payne the configuration does not exist for 4 <= q <= 401. For q = 7, 8, 9 we verify the rectangular bound with exact certificates and exclude the second structure by a partner-counting argument: for q = 7 by hand together with a computer-assisted nullity enumeration for one family of patterns, and for q = 8, 9 by an exact enumeration of 88 and 209 pairs of deficit patterns. Hence z(58;2) = 459, z(74;2) = 660 and z(92;2) = 913. This record contains the manuscript (PDF), the supplementary note (PDF) and the supplementary archive: code, exact certificates, the explicit extremal matrices, logs and the records of the validation checks, with a SHA-256 manifest and instructions for reproducing every computation. MSC2020: 05C35 (primary); 05B25, 05B05, 51E15 (secondary)