No Uniform Hanna Neumann Bound for Convex-Core Volume

There is no universal constant K such that V(A intersection B) <= K V(A)V(B) for torsion-free convex-cocompact Kleinian groups, where V is three-dimensional convex-core volume. An explicit rank-two classical Schottky deformation has positive core volume tending to zero and constant systole arcosh(31). Independent parity kernels are distinct proper index-two subgroups of rank three, with rank-five intersection of index four; their volume ratio is exactly 1/V(G). Fixed ranks, fixed indices and a fixed positive systole therefore do not restore a uniform bound. We also give the exact finite-cover double-coset identity and prove a neighborhood-volume inequality by counting lifts. This is a complete counterexample to the universal-constant interpretation of Swarup's Question 3.2 in Bestvina's list, recorded as AMR-010-0302. It does not settle arbitrary ambient-dependent bounds for all infinite-index subgroups of one fixed G, so the broad source record is not claimed wholly resolved. Classical Schottky theory, Bonahon's volume regularity and covering theory are credited. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority certification is claimed.