Why do I get this extra factor when working out the dynamics of an adiabatic quantum computation?

I was trying to revise my understanding of adiabatic quantum computation via a simple example. I’m familiar with the overall concept — that you have an overall Hamiltonian
$$
H(s)=(1-s)H_0+s H_f
$$
where $s$ is a function of time, starting from $s=0$ and finishing at $s=1$. You prepare your system in the ground state of $H_0$ (known) and, provided the Hamiltonian changes slowly enough, your final state will be close to the ground state of $H_f$. The "slowly enough" condition is typically phrased in terms of the energy gap between the ground and first excited state, $Delta$. A typical assumption is
$$
frac{ds}{dt}=epsilonDelta^2
$$
for small $epsilon$ (I haven’t been back to pick through more detailed statements).

So, my plan was to test
$$
H(theta)=-costheta X-sintheta Z,
$$
starting from $theta=0$ and the system in $|+rangle$. For all values of $theta$, the gap is a constant (2) so, in essence, I have the conversion $theta=4epsilon t$ and the evolution time will be $pi/(8epsilon)$ and should leave the system in the state $|0rangle$. However, when I tried to do this calculation, I didn’t get this answer. Am I screwing up, or is there some condition that tells me I shouldn’t expect the adiabatic theorem to hold in this case?

---

Details of how I tried to do the calculation:
Let
$$
|y_{pm}rangle=(|0ranglepm i|1rangle)/sqrt{2}.
$$
We have that
$$
H|y_{pm}rangle=pm ie^{mp itheta}|y_{mp}rangle.
$$
We can decompose any state in this basis,
$$
|psirangle=a|y_+rangle+b|y_-rangle,
$$
so we can talk about the time evolution of the coefficients
$$
frac{d}{dt}begin{bmatrix} a b end{bmatrix}=4epsilonfrac{d}{dtheta}begin{bmatrix} a b end{bmatrix}=begin{bmatrix}
0 & e^{-itheta} -e^{itheta} & 0
end{bmatrix}begin{bmatrix} a b end{bmatrix}
$$
I can perform a variable transformation
$$
tilde a=e^{itheta/2}a,quad tilde b=e^{-itheta/2}b
$$
which then satisfy
$$
4epsilonfrac{d}{dtheta}begin{bmatrix} tilde a tilde b end{bmatrix}=ibegin{bmatrix}
2epsilon & -i i & -2epsilon
end{bmatrix}begin{bmatrix} tilde a tilde b end{bmatrix}
$$
This is (finally!) something I can do something useful with! Taking the small $epsilon$ limit, I essentially have
$$
frac{d}{dtheta}begin{bmatrix} tilde a tilde b end{bmatrix}=ifrac{1}{4epsilon}Ybegin{bmatrix} tilde a tilde b end{bmatrix}
$$
I can put in my initial conditions and compare what I expected at the end. The problem is that I have an extra phase of the form $e^{ipi/(8epsilon)}$ floating around, and the taking of the small $epsilon$ limit seems problematic as this varies rapidly between all possible values.