Subplanes of order 3 and maximal arcs in projective planes of order 12

The existence of a projective plane of order 12 is open. A subplane of order 3 of such a plane attains Bruck's bound n ≥ m² + m with equality. The counts in this equality case show that the 27 points on no line of the subplane form a maximal arc of degree 3, and that together with the subplane they form a maximal arc of degree 4. Its secants carry a Steiner system S(2,4,40), and Thas's construction yields partial geometries pg(8,9,6) and pg(9,8,6), whose existence is listed as open. If the 27 points form the affine space AG(3,3) with the tangent classes as directions, the plane extends the points and lines of PG(3,3), and Hall and Roth, and Prince, showed that no such plane exists. We show that in this case already three lines of the subplane lead to a contradiction. More generally, AG(3,3) is not the design of a maximal arc of degree 3 in any plane of order 12. The incidence structure formed by all points and the lines that meet the subplane has no automorphism of prime order other than 3 that preserves the subplane, and one type of order 3 is excluded as well. For maximal arcs of degree 3 or 4, with no subplane assumed, the analogous structure has no automorphism of order 13. With the classification of Kirkman triple systems of order 27 with an automorphism of order 13 by Janko and van Trung, whose count we correct from 661 to 657, we also exclude the case where the 27 points carry such an automorphism compatible with the labelling of their parallel classes. Finally, suppose that every line ℓ of the subplane splits the 27 points into three sets of 9, two points lying in the same set exactly when their joining line meets ℓ in a point of the subplane. Then the 27 points form a Hall triple system, hence AG(3,3), and the plane does not exist. The computer-assisted steps, including a proof of Prince's theorem that PG(3,3) has no covering set of spreads, are certified by LRAT proofs, which are also checked in Lean 4. The exceptions are the results on automorphisms of order 13 of maximal arcs and of the triple system of the 27 points, which rest on exhaustive searches reproduced by independent programs. Mathematics Subject Classification (2020): 51E15, 05B25; 05B07, 51E14, 51E21, 51E23, 68V05, 68V15.