\in {\mathbb{T}}_{j, \tau}$ has long axis in the direction of the center of $\tau$. In particular, our tube $T$ must be nearly vertical, up to an angle of $R_j^{-1/2}$. The tube $T$ intersects $S^1(x,t)$ in two arcs, which we deal with one at a time. Each arc is a graph of the form $ y_2 = h(y_1)$, where $y_1$ lies in an interval $I(T)$ of length $\sim R_j^{-1/2 + \delta}$. Since $T$ intersects $S^1(x,t/2)$, and the tube $T$ is nearly vertical, the function $h$ and all its derivatives are $\lesssim 1$ on $I(T)$.

The following point is crucial for stationary phase. Since $T$ is within an angle $R_j^{-1/2}$ of vertical, and $x \notin 2T$, then the distance from $T$ to the top or bottom points of the circle is $\gtrsim R_j^{-1/2 + \delta}$, and so we get

$$|h'(y_1)| \gtrsim R_j^{-1/2 + \delta} \textrm{ on the interval } I(T).$$

In these coordinates, our integral becomes

$$\int_{I(T)} \eta_T(y_1, h(y_1)) e^{2 \pi i \omega_2 h(y_1)} J(y_1) dy_1,$$

where $J(y_1)$ is the Jacobian factor that relates the arclength on the circle to $dy_1$. Since $h$ and all its derivatives are $\lesssim 1$ on $I(T)$, the same applies to $J$. The function $\eta_T$ is smooth 