A direct charge-4e superconductor exhibits coherent four-electron order while every charge-2e pairing channel remains uncondensed. We establish three complementary results. First, building on the η-clustering states and bipartite parent of Yoshida and Katsura, we formulate and exactly solve a minimal two-term parent on any connected graph. Its fixed-number ground states have quartet off-diagonal long-range order (ODLRO) without charge-2e ODLRO, while an exact mapping to classical hard-core exclusion dynamics yields the full fixed-sector gap and a branch of quartet-density modes. Nonzero quartet stiffness and vanishing inverse quartet compressibility identify this z=2 parent as a phase-separation boundary. Second, for a finite-range fermionic family with explicit quartet transfer and sufficiently large onsite penalty, we rigorously prove quartet ODLRO without charge-2e ODLRO at half quartet filling, both at the hypercubic XY point for d≥2 and throughout a finite XXZ interval on the square lattice. Third, we derive a strong-coupling realization using only electron hopping and two-body interactions. With local gap U₀, pair hopping K generates quartet motion at order K²/U₀, whereas electron hopping t first contributes at order t⁴/U₀³; charge-2e excitations remain gapped at O(U₀). On bipartite lattices, positive inverse quartet compressibility opens an asymptotically controlled homogeneous z=1 regime with short-ranged pair correlations, while negative curvature drives phase separation.