= Solution
The observed $90^\circ+30^\circ$ substeps sum to $120^\circ$, consistent with three catalytic cycles per revolution of $F_1$-<ATPase>. Surface attachment fixes the $\beta$-subunit frame while the gold-labelled $\gamma$ shaft reports relative rotation. A finite dwell before the smaller step indicates hidden chemical transitions rather than a completely smooth motor.
Suppose two irreversible independent first-order transitions, with rates $k_a$ and $k_b$, must occur sequentially before the $30^\circ$ step. Their waiting times $X_a$ and $X_b$ have <exponential distributions>, and $\tau=X_a+X_b$. <Convolution> gives
$$
P(\tau)=\int_0^\tau k_ae^{-k_at}k_be^{-k_b(\tau-t)}dt
=\boxed{\frac{k_ak_b}{k_a-k_b}(e^{-k_b\tau}-e^{-k_a\tau})},\qquad\tau\ge0.
$$
This <probability density> integrates to one and has $\langle\tau\rangle=k_a^{-1}+k_b^{-1}$. For equal rates its continuous limit is $P(\tau)=k^2\tau e^{-k\tau}$, not a singular distribution. In particular $P(0)=0$ distinguishes two resolved sequential waits from a single exponential wait. Hydrolysis and product release are plausible underlying chemical events, but the waiting-time fit alone neither assigns their order nor uniquely identifies which biochemical event causes each substep. Concentration dependence or independent labelling would be needed for that assignment.
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