= Solution
The stability convention matters. The https://www.damtp.cam.ac.uk/user/na/PartIB/Lect10.pdf[Cambridge numerical analysis notes] define a <strict linear stability domain> through decay $y_n\to0$. For a multistep recurrence, robust decay requires every <amplification root> to lie strictly inside the <unit disk>. With that convention we can prove the stronger result that \b[this method's strict linear stability domain is empty], and therefore bounded. Mere boundedness of oscillatory sequences gives a different answer, described below.
On the <Dahlquist test equation>, $g(y)=\lambda^2y$. With $z=h\lambda$, the <amplification polynomial of a multistep method> is
$$
D\xi^2-16z\xi-A=0,\qquad D=z^2-7z+15,\qquad A=z^2+7z+15.
$$
Exclude $D=0$, where the intended two-step update cannot be solved. If both <polynomial roots> have <modulus> below one, their product requires $|A|<|D|$. But
$$
|A|^2-|D|^2=28\operatorname{Re}z\,(|z|^2+15),
$$
so necessarily $\operatorname{Re}z<0$. The <complex quadratic Schur criterion> also requires
$$
|\overline D(-16z)-(-A)(-16\overline z)|<|D|^2-|A|^2.
$$
Its left-hand side equals $32|\operatorname{Re}z|(|z|^2+15)$, while for $\operatorname{Re}z<0$ its right-hand side is only $28|\operatorname{Re}z|(|z|^2+15)$. The inequality is impossible. Thus
$$
\boxed{\mathcal D_{\rm decay}=\varnothing}.
$$
For comparison, the <bounded-root stability set of the symmetric two-derivative formula> is unbounded. On $z=iy$, rotate the <amplification polynomial> into a real reciprocal quadratic. Its two roots are distinct and on the <unit circle> precisely when
$$
(15-y^2)^2-15y^2>0,
$$
equivalently $y^2<(45-15\sqrt5)/2$ or $y^2>(45+15\sqrt5)/2$. In particular, arbitrarily large imaginary parameters give bounded, undamped numerical oscillations. At the two equality thresholds, repeated <unit-circle roots> produce unbounded solutions. Hence the question's bounded-domain claim holds for the strict decay convention; it does not hold if the phrase is interpreted using the non-growing <root condition for a multistep method>.
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