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Two independent cyclic intervals intersect with probability at most $\min \{1,(2k-1)/d\}\le 2k/d$. For $k\le d/2$, precisely the $2k-1$ starting-point offsets $-(k-1),\dots ,k-1$ can overlap; for larger $k$, the upper bound is automatic. The cyclic interval overlap bound and the preceding chi-squared divergence estimate therefore giveAs in 1(e), the square-root hypothesis is meaningful when $\alpha d\ge 1$. The chi-squared testing lower bound bounds the total variation distance by $\sqrt{2{\alpha }^{\epsilon }}/2$. The worst individual Type II error dominates the error under the uniform mixture model. ThusTaking $\alpha$ small makes this lower bound informative. This argument uses the actual cyclic-interval alternative, without substituting a different family of subsets.