Capped C4-free matrices of small deficit complete to symmetric configurations

Preprint and supplementary data for the paper "Capped C4-free matrices of small deficit complete to symmetric configurations". Let q ≥ 2, n = q^2+q+2 and P(q) = (q+1)(q^2+q+1). A projective plane of order q extended along an anti-flag gives a C4-free n×n 0/1 matrix with P(q)+3 ones. We study the C4-free n×n matrices in which every row and column has at most q+1 ones, measured by their total deficit Δ = (q+1)n − |N|. We prove that for every q ≥ 4 and every 1 ≤ Δ ≤ min{60, ⌊(q−1)/3⌋}, every such matrix of deficit Δ can be completed, by changing zeros to ones, to the incidence matrix of a symmetric configuration (n)_{q+1}. Moreover, for every Δ ≥ 17 the deficit Δ is covered at all orders q ≥ Δ+2+⌊3(Δ−2)^2/20⌋ by an argument without computation. The proof combines exact counting identities with symbolic exclusions and finitely many exact integer computations; every linear-programming bound used is certified by a dual certificate checked in exact rational arithmetic. Consequences: an unconditional lower bound on the deficit of every capped C4-free n×n matrix for 4 ≤ q ≤ 401 and for larger q outside an explicit sparse set, and z(n;2) = P(q)+3 for every 4 ≤ q ≤ 183 for which a plane of order q exists, conditionally on two explicit hypotheses. MSC2020: 05C35 (primary); 05B25, 05B30, 51E15 (secondary). Files: Yan_ZX_completion.pdf: the preprint. Yan_ZX_completion_Supplementary_archive.zip: programs, certificates, outputs and the per-deficit table; README.txt inside gives the command, the expected result and any path adaptation for every directory; SHA256SUMS.txt lists every file. Zip: 41,486,321 bytes, md5 9d0f45c1fe824c9f4f83f57e80f34f1c, sha256 55d1df2031988b7c47124039a3520ac0fd2519475d2cda1fa59d456e405ea656 (2197 members). PDF: md5 1e47c8875f95ce67420909e8226d489e.