Proposition: Binary mesh values

The fine-graining covering radii at capacities $4,5,6$ are

Including symmetry certificates gives

Proof. The binary fine-graining nets at capacities $4$ and $5$ are, respectively,

At capacity $6$, the net is

These lists follow directly from (eq:fgcost); their largest gaps are $1/3,1/4,1/5$.

For capacities $3\le N\le 6$, symmetry includes every grid point $j/N$, so all gaps in the union are at most $1/N$. The cost formula shows that fine-graining introduces no positive point below $1/N$ in these four cases. The initial gap is therefore exactly $1/N$.