= Solution
When $\widetilde b=0$, integration by parts makes the kinetic term $-\widetilde a\partial_\mu V_\nu\partial^\mu V^\nu$. The Lorentzian contraction over the field index gives the time component and the three spatial components opposite kinetic-energy signs. Reversing the sign of $\widetilde a$ merely exchanges which component is a <ghost field>; no nonzero choice makes all four energies positive. At $\widetilde a=0$ there is no healthy kinetic term. Thus the Hamiltonian is never positive definite.
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