= Solution
Apply the <method of multiple scales> with independent fast time $t$ and <slow time> $T=\epsilon^2t$. A preliminary <slow time> $\epsilon t$ produces no resonant forcing at first order, so its leading amplitudes are constant; the first nontrivial modulation is at $T$. Write $x=x_0+\epsilon x_1+\cdots$, with
$$
x_0=A(T)\cos t+B(T)\sin t.
$$
At order $\epsilon$, the <inhomogeneous linear differential equation> is $(\partial_t^2+1)x_1=-\cos t\,x_0$. Absorbing its homogeneous part into the amplitudes gives
$$
x_1=-\frac A2+\frac A6\cos2t+\frac B6\sin2t.
$$
At order $\epsilon^2$, the forcing is $-2\partial_t\partial_Tx_0-kx_0-\cos t\,x_1$. Its resonant <Fourier series> coefficients are
$$
\left[-2B_T+\left(\frac5{12}-k\right)A\right]\cos t+
\left[2A_T-\left(k+\frac1{12}\right)B\right]\sin t.
$$
The <solvability condition in the method of multiple scales> sets both coefficients to zero, giving the <amplitude equations>. Define
$$
a=\frac{k+1/12}{2},\qquad b=\frac{5/12-k}{2},\qquad
M=\begin{pmatrix}0&a\\b&0\end{pmatrix}.
$$
For bounded <initial conditions> $x(0)=x_*$, $\dot x(0)=v_*$, the requested leading approximation is
$$
\boxed{x(t)=A(\epsilon^2t)\cos t+B(\epsilon^2t)\sin t+O(\epsilon),\qquad
\begin{pmatrix}A(T)\\B(T)\end{pmatrix}=e^{TM}\begin{pmatrix}x_*\\v_*\end{pmatrix},\qquad \beta=2.}
$$
This holds on bounded intervals of $T$, hence $t=O(\epsilon^{-2})$, with the constant in the error allowed to depend on the bounded $T$ interval. If desired, retain the displayed $\epsilon x_1$ and adjust $A(0),B(0)$ by $O(\epsilon)$ to match <initial conditions> at that order. No particular <initial conditions> were supplied.
Since $M^2=abI$, the <matrix exponential> is $I\cosh(\sigma T)+(M/\sigma)\sinh(\sigma T)$ when $\sigma^2=ab>0$. When $ab<0$, put $\omega=\sqrt{-ab}$ and replace this with $I\cos(\omega T)+(M/\omega)\sin(\omega T)$. At $ab=0$ it is $I+TM$. Thus the leading <second instability tongue of a weak Mathieu oscillator> has exponentially growing solutions for $-1/12<k<5/12$, and the leading modulation is bounded for every <initial condition> precisely when
$$
\boxed{k<-\frac1{12}\quad\text{or}\quad k>\frac5{12}.}
$$
At either displayed endpoint, $M$ is a nonzero <nilpotent operator>; some <initial conditions> give a <secular term> proportional to $T$. This leading calculation alone must not be mistaken for an exact finite-$\epsilon$ stability decision at the endpoints.
For an exact small-$\epsilon$ interpretation, put $t=2s$. The standard <Mathieu equation> parameters are $a_{\rm M}=4+4k\epsilon^2$ and $q=-2\epsilon$. Its two relevant <Mathieu characteristic values> are obtained by projecting $-d^2/ds^2+2q\cos2s$ onto even and odd <Fourier series> modes. Near the unperturbed value $4$, the even modes $1,\cos2s,\cos4s,\cos6s$ have diagonal entries $0,4,16,36$ and adjacent couplings $\sqrt2q,q,q$ in an orthonormal basis; the odd modes $\sin2s,\sin4s,\sin6s$ have diagonal entries $4,16,36$ and couplings $q,q$. Substituting $4+c_2q^2+c_4q^4$ into their <characteristic polynomials> gives
$$
a_2(q)=4+\frac5{12}q^2-\frac{763}{13824}q^4+O(q^6),\qquad
b_2(q)=4-\frac1{12}q^2+\frac5{13824}q^4+O(q^6).
$$
Higher modes of the <Fourier series> first contribute at higher order. Therefore the <fourth-order edges of the second Mathieu instability tongue> are
$$
k_-(\epsilon)=-\frac1{12}+\frac5{3456}\epsilon^2+O(\epsilon^4),\qquad
k_+(\epsilon)=\frac5{12}-\frac{763}{3456}\epsilon^2+O(\epsilon^4).
$$
<Floquet theory> gives boundedness of every solution outside the closed gap, $k<k_-(\epsilon)$ or $k>k_+(\epsilon)$, in this fixed-$k=O(1)$ neighborhood. At the actual gap edges the second independent solution is generically unbounded. In particular, if $k$ is fixed exactly at $-1/12$ or $5/12$, it lies just outside the true gap and is stable for each sufficiently small positive $\epsilon$. The strict boxed criterion is the robust leading-order, uniformly separated criterion; the endpoints require this higher-order refinement, and their amplification bounds need not remain uniform as $\epsilon\to0$.
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