No four coplanar in the cube: nineteen certified configurations for n <= 29, and what they do not establish (OEIS A280537)

Abstract

a(n) is the maximum number of points in the n×n×n grid no four of which are coplanar — OEIS A280537. The published data stop at a(8) = 20. This version extends the previous one (n ≤ 7) to n ≤ 29. We exhibit nineteen configurations, each verified by two programs sharing no code and testing different statements: one computes the 3×3 determinant of every quadruple in exact integer arithmetic, the other forms plane normals from triples and tests the scalar product against every remaining point, and also checks that the points are distinct and inside the cube. Before each run the verifier is given a deliberately corrupted witness and must reject it. The nineteen configurations certify eighteen independent lower bounds: a(9) ≥ 23, a(10) ≥ 26, a(11) ≥ 28, a(12) ≥ 31, a(13) ≥ 32, a(14) ≥ 34, a(15) ≥ 35, a(16) ≥ 38, a(18) ≥ 41, a(19) ≥ 43, a(20) ≥ 45, a(21) ≥ 47, a(22) ≥ 49, a(23) ≥ 50, a(24) ≥ 52, a(25) ≥ 53, a(27) ≥ 56, a(29) ≥ 59. What is not established. Every number above is a lower bound. No upper bound beyond the trivial a(n) ≤ 3n is known to us for n > 8, and we expect several of the values to be improved. Two statements above n = 8 are proved, and both concern the cyclically invariant subspace only, not a(n): the symmetric maximum is 23 at n = 9 and 26 at n = 10, each exhausted independently by two implementations sharing no code (1.52 × 10^9 nodes and 8.93 core-hours for n = 10, with the coverage recorded share by share). That subspace need not contain the true maximum — at n = 6 it does not, its exhausted maximum being 15 while a(6) = 16. The note also records a bound of our own that these witnesses refuted, the reason it looked convincing for seven consecutive values before it failed, and a measurement showing why the growth rate of a(n)/n cannot be settled by search under any fixed time budget. Package: the note (PDF and TeX), both verifiers, all nineteen witnesses, and the journals recording what was exhausted and what was only searched.