= Solution
Since $\sigma_\phi^2=I$, the <rotation gate> is
$$
\boxed{R_\phi(\theta)=\begin{pmatrix}\cos(\theta/2)&-i e^{-i\phi}\sin(\theta/2)\\-i e^{i\phi}\sin(\theta/2)&\cos(\theta/2)\end{pmatrix}.}
$$
For two chronological $\pi$ pulses, later operations multiply on the left:
$$
\boxed{R_{\phi_2}(\pi)R_{\phi_1}(\pi)=-\begin{pmatrix}e^{-i(\phi_2-\phi_1)}&0\\0&e^{i(\phi_2-\phi_1)}\end{pmatrix}=-R_z[2(\phi_2-\phi_1)].}
$$
Thus equatorial pulses generate arbitrary rotations about the $z$-axis. The minus sign is an overall <global phase>.
For the other chronological sequence, $T^4=Z$, and
$$
\boxed{H T^4=HZ=\frac1{\sqrt2}\begin{pmatrix}1&-1\\1&1\end{pmatrix}=R_y(\pi/2).}
$$
This equality has no additional overall <phase> under the standard definitions of $H$ and $T$. Rotations about two nonparallel axes, such as $x$ and $y$, generate all single-qubit operations by an <Euler angles> decomposition. Equivalently, the available equatorial rotations include $R_y(\pi/2)$ and arbitrary $R_z$ rotations constructed above; for example $H=R_y(\pi/2)Z$ and $T=e^{i\pi/8}R_z(\pi/4)$. Hence these rotations provide a universal single-qubit set. Overall <phases> do not affect any <probability> or isolated-register observable; a <phase> controlled by an additional <qubit> would instead be relative and could not be discarded without accounting for it.
The minimum number of distinct fixed rotation gates for approximate universality is \b[two]. One fixed gate and its powers generate a commuting subgroup and cannot approximate all noncommuting single-qubit operations. Two suitable equatorial gates suffice: for example $R_x(\pi/\sqrt2)$ and $R_y(\pi/\sqrt3)$. Each has an angle irrational relative to $2\pi$, so its powers densely approximate rotations about its axis; combining the two supplies the Euler decomposition. Noncommutation alone is insufficient: two Clifford quarter-turns can generate only a finite subgroup. If continuous angles are regarded as available controls, two nonparallel rotation axes likewise suffice, now for exact Euler synthesis.
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