= Solution
Substitute $\omega=\alpha_3/\varepsilon+\cdots$ into the cubic denominator. Its leading terms are
$$
1-\omega^2-i\varepsilon\omega^3
=-\frac{\alpha_3^2+i\alpha_3^3}{\varepsilon^2}+O(1).
$$
The nonzero large root obeys $1+i\alpha_3=0$, so
$$
\boxed{\alpha_3=i,\qquad\omega_3\simeq\frac i\varepsilon}.
$$
The sum of the three roots is exactly $i/\varepsilon$. Combining this with the two slow-root expansions gives the refinement $\omega_3=i/\varepsilon+i\varepsilon+O(\varepsilon^3)$.
For physical radiative sign $\varepsilon>0$, the fast <pole> lies high in the upper half-plane. Its homogeneous mode is $e^{-i\omega_3t}\simeq e^{t/\varepsilon}$, a runaway. It prevents a \b[bounded retarded response with the real-axis Fourier prescription] even though the two slow poles are damped. If $\varepsilon<0$, the fast <pole> decays but the slow pair grow. No nonzero real sign puts every <pole> in the lower half-plane.
There are two distinct boundary choices. A strictly causal initial-value <Green function> exists and includes the growing mode for $\varepsilon>0$; it must be represented by a Laplace/<Bromwich contour> above that <pole> rather than the stated ordinary Fourier integral. Alternatively the real-axis inverse gives a bounded <stationary no-runaway response and preacceleration>: for $\tau<0$, upper closure encloses the fast <pole> and yields a nonzero response before the <force>. Its leading contribution is
$$
G(\tau)\simeq\varepsilon e^{\tau/\varepsilon},\qquad \tau<0,\quad\varepsilon>0.
$$
This follows from the upper-pole <residue> $\operatorname{Res}G\simeq-i\varepsilon$ and the factor $i$ from upper contour closure. Eliminating runaway growth by a future <boundary condition> therefore sacrifices strict retardation on the short scale $\varepsilon$.
The singular root also shows why a perturbative expansion around only $\omega=\pm1$ misses the problem. For a low-frequency effective description, <order reduction of a radiation-reaction oscillator> uses $\dddot x\simeq\dot f-\dot x$ in the small correction and gives $\ddot x+\varepsilon\dot x+x=f+\varepsilon\dot f$. For positive $\varepsilon$ this approximate second-order equation has a causal damped response and discards the fast mode outside its intended regime; it does not make the original third-order equation simultaneously bounded and retarded.
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