A sharp lower bound for the sextic large sieve

We establish a matching lower bound for the sextic large sieve over the Eisenstein integers in an unbalanced range, with both indices restricted to squarefree ideals. Let ${\mathrm{\Xi }}_6(A,B)$ denote the squared operator norm of the sextic residue-character matrix with row and column norms at most $A$ and $B$, respectively, outside a fixed finite set of primes containing those above $6$. For every fixed $0<\delta <3\mathrm{/}11$ and every $\epsilon >0$, we prove ${\mathrm{\Xi }}_6(A,B){\gg }_{\delta ,\epsilon }(AB{)}^{-\epsilon }{A}^{5\mathrm{/}6}{B}^{1\mathrm{/}3}$ for $B^{16\mathrm{/}11+\delta }\le A\le B^{2-\delta }$. Consequently, the full-cutoff norm satisfies ${\mathrm{\Xi }}_6(X,X){\gg }_{\epsilon }{X}^{17\mathrm{/}16-\epsilon }$. The main difficulty is to preserve the large second moment supplied by Gauss sums when the column moduli are restricted to be squarefree. We prove a second-moment formula uniform in the prime exclusions and character twists arising from Möbius inversion, and use it to show that the squarefree projection retains a positive proportion of the original moment. A separate argument combining cubic and sextic large-sieve bounds then transfers the lower bound to squarefree rows.