= Solution
Insert an exact smooth <solution of a differential equation> and expand about $t=t_{n+2}$. The unscaled <local truncation error>, with every term moved to the left, is
$$
d_h(t)=\frac{1-7\alpha}{6}h^3y^{(3)}(t)
+\frac{15\alpha-1}{24}h^4y^{(4)}(t)+O(h^5).
$$
Here the constant, first-<derivative>, and second-<derivative> terms all cancel, using the <chain rule> identity $y''=f'(y)f(y)$. Hence the formal defect convention $d_h=O(h^{p+1})$ gives
$$
\boxed{p=3\quad(\alpha=1/7),\qquad p=2\quad(\alpha\ne1/7)}.
$$
At $\alpha=1/7$, the coefficient of $h^4y^{(4)}$ is $1/21$, so there is no further cancellation. At any other $\alpha$, the cubic coefficient is nonzero.
There is a degeneracy at $\alpha=1$: both the first-<derivative> coefficient and $\rho'(1)$ vanish. The raw defect still has the displayed formal order two, but it cannot be normalized by $(1-\alpha)h$ into an ordinary first-order <consistency of a numerical method> test. Thus a convention requiring this nondegenerate normalization would leave the ODE order undefined at $\alpha=1$. In either convention, it is not a convergent approximation to arbitrary initial-value data; the next part explains why.
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