Balanced orbit defects in half-turn-symmetric no-three-in-line configurations, with new 2n-point configurations for n = 36, 37, 39

Abstract

A 2n-point no-three-in-line configuration on the n×n grid with half-turn symmetry decomposes into orbits of a larger subgroup of the dihedral group that it happens to contain, together with half-turn pairs that are not such orbits — its orbit defect. Balance lemma. Read as arcs on the row classes {i, n−1−i}, the defect pairs form a balanced directed multigraph and hence a union of directed cycles. With one further observation — two half-turn pairs on the same long diagonal are collinear — this classifies every defect of at most three pairs: one pair must be a diagonal loop (Flammenkamp's pseudo-class rct4); two pairs are two loops on different diagonals or a directed 2-cycle, which is an orbit of the diagonal reflections; three pairs form a directed 3-cycle or a loop plus a 2-cycle. Enumeration. The corresponding families are enumerated exhaustively for small n by an exact branch-and-bound program, validated against OEIS A000769 and the class counts of Flammenkamp's database: the 3-cycle family for odd n ≤ 25, 33 and in part 37, 39, 41; the two-loop family for even n ≤ 28 and n = 36; the mixed families for n ≤ 27, all empty. New configurations. For n = 36, the first located 2n-point configurations whose symmetry group is exactly the half-turn (three inequivalent ones); for n = 37 (two) and n = 39 (four), the first located ones of that group outside the rct4 pseudo-class; and a further one for n = 33. "First located" refers to the public files of Flammenkamp's database as of 2026-08-11, confirmed by its maintainer, who has added the configurations to the database. Independent check before publication. All ten configurations were re-verified by a program sharing no code with the search: every one of the C(2n,3) triples tested by integer cross product, and all eight images under the square group compared against the set. Each has 2n points, zero collinear triples, and a stabiliser of exactly {identity, half-turn}. That check covers the configurations themselves and not the balance lemma, the classification, the completeness of the sweeps, or the claim of priority, which rests on an external fact; this is stated in the package rather than left to be assumed. Configurations, programs and sweep journals are public. Disclosure: the computations, the programs and the text were produced by AI agents (Anthropic Claude) under the direction of the author of record, who posed the question, chose what to compute, decided what to claim and is responsible for the content. Version 1.1. The balance lemma and the classification of small defects are now checked on complete material: an independent enumerator (sharing no code with the paper's solver) lists all 2n-point configurations for n = 5..11 — the seven totals match A000755 exactly — and on every one of the 309 half-turn-symmetric configurations among them, for both bases, the defect multigraph is balanced, |D| ≡ n (mod 2) for the C4 base, and every defect with at most three pairs falls into exactly the classified families. Zero failures. The checking script, its enumerator and its output are added to the package. Version 1.2, after an external review. The review verified the lemma, the count formulas, all ten configurations and the maintainer's confirmation independently, and caught a real inconsistency: the mixed-family count at n=27 was quoted as 1956 while the generator yields 2028. Measured, not guessed: exactly 72 sub-classes at n=27 are dead (their defect cells already contain three collinear points), and the earlier text mixed two conventions — the total at n=23 (1210, of which 50 dead) but the live count at n=27. All counts are now totals with the dead stated; emptiness is unaffected, as a dead sub-class is empty a fortiori. The emptiness of the mixed families is now quantified: 8188 live sub-classes over 9 ≤ n ≤ 27 carry no configuration, where the 3-cycle rate on the same range (one per 613) would predict about 13 — e^−13.4 ≈ 1.6·10^−6 under a naive Poisson reading, so the emptiness looks structural and an explanation is worth seeking. Also: the abstract is synchronised with the sweep table (37, 39, 41 complete in both bases, 45 partial; 30–34 in the two-loop family not swept and said so); the V2-base 3-cycle family at even n is named as open; and the phrase "first located" is scoped to its n.