Lemma 1: exact agreement on 40 random rational point sets (d<=3, n<=4, p in 2,4,6); negative control (sign +1/n) detected as different in 40/40 cases
MC d=2 n=3 p=4 M=1000000: lattice 0.0088085 +- 0.0000157 (exact 0.0087822, z=+1.68); iid 0.0080617 +- 0.0000121 (exact 0.0080453, z=+1.35)
MC d=2 n=4 p=4 M=1000000: lattice 0.0055223 +- 0.0000115 (exact 0.0055418, z=-1.70); iid 0.0046169 +- 0.0000071 (exact 0.0046181, z=-0.17)
MC d=2 n=5 p=4 M=600000: lattice 0.0039854 +- 0.0000122 (exact 0.0039787, z=+0.55); iid 0.0029890 +- 0.0000060 (exact 0.0029911, z=-0.35)
MC d=2 n=3 p=6 M=1000000: lattice 0.0030131 +- 0.0000088 (exact 0.0030227, z=-1.09); iid 0.0023876 +- 0.0000060 (exact 0.0023930, z=-0.91)
