Coherent clique decompositions of punctured affine planes and Baer sublines

Let l_0 be a line of AG(2,Q) and let S be a set of k slopes other than that of l_0, where 2 <= k <= Q-1; two points off l_0 are compatible if the line joining them has no slope in S. A coherent decomposition is a family of Q-cliques of this compatibility graph, any two sharing at most one point, such that every point lies in exactly Q-k of them, the maximum allowed by local counting. We prove an incidence identity for the leave (the uncovered compatible pairs) which forces the affine stabilizer of S to contain a sharply transitive set of maps on S. Together with the theory of directions and character sums it settles several infinite families. No coherent decomposition exists when S is the set of nonzero squares of F_Q with Q >= 5 odd, or an index-two additive subgroup of F2^h with h >= 3 (for h = 3 by exhaustive enumeration). For Q = q^3 and k >= Q-q^2 a coherent decomposition exists if and only if k = Q-1, or Q is a square, k = Q-sqrt(Q) and the directions outside S, including that of l_0, form a Baer subline. For Q = q^2 and k >= Q-q, refining classical results on nets of critical deficiency, it exists if and only if k = Q-1, or k = Q-q and these directions form a Baer subline, and in the Baer case it is unique. Hand proofs and exact computation together settle every slope set with Q <= 9. This record contains the manuscript (PDF), the supplementary note (PDF) and the supplementary archive: code, CNF encodings, LRAT proofs for k >= 3 and for k = 2, Q <= 8, logs and the records of the validation checks, with a SHA-256 manifest and instructions for reproducing every computation. The four LRAT proofs for k = 2, Q = 9 (about 5.5 GB compressed) are the 53 files UG_SUPPLEMENT_Q9_K2_LRAT.tar.part00 to .part52, joined as described in JOIN.txt and checked with SHA256SUMS.txt. All 28 LRAT proofs in the supplementary archive are also accepted by the formally verified checker cake_lpr. The files Yan_Coherent_clique_decompositions_cake_lpr_check.txt, .tsv and .sh record this check: the tool versions, the SHA-256 digests of every CNF and proof, both verdicts for every proof, and the script that ran it. MSC2020: 05B25 (primary); 05B30, 05C70, 11T23, 51E15 (secondary)