To solve the beamforming bootstrap problem without prior synchronization, we employ an asymmetric scanning protocol. Designate transceiver A as the fast scanner and B as the slow scanner.

Let the total cells in each search grid be:

\[ N_a = i_{\max,a} \times j_{\max,a}, \quad N_b = i_{\max,b} \times j_{\max,b} \]

The time for A to complete one full boustrophedon sweep:

\[ T_{\text{scan},a} = N_a \cdot \tau_a \]

For B to guarantee its beam is on the correct line-of-sight cell when A sweeps past, B must dwell on each cell for at least one full scan of A:

\[ \boxed{\;\tau_{b,\text{opt}} = N_a \cdot \tau_a = \frac{N_a}{R_a}\;} \]

where \(R_a\) is A's scan rate in cells/s. The worst-case discovery time (hit on B's last cell):

\[ \boxed{\;T_{\max} = N_b \cdot \tau_{b,\text{opt}} = N_a \cdot N_b \cdot \tau_a\;} \]

With equal scan rates (symmetric protocol), the beams may enter a deafness cycle where they perpetually miss, with no bounded convergence guarantee.

When a coarse hit is detected (beam gain ≥ threshold for both parties), the radios establish initial communication and exchange coordinates. They then switch to a synchronized jitter protocol to maximize beam overlap, converging via gradient ascent with simulated annealing:

\[ \mathbf{p}_{k+1} = \mathbf{p}_k + \alpha(\mathbf{c} - \mathbf{p}_k) + T_k \cdot \boldsymbol{\eta}, \quad T_{k+1} = \gamma \, T_k \]

where \(\mathbf{c}\) is the optimal pointing direction, \(\alpha\) is the convergence rate, \(T_k\) is the annealing temperature, \(\gamma \approx 0.97\) is the cooling factor, and \(\boldsymbol{\eta} \sim \mathcal{U}(-0.5,\,0.5)\) is uniform jitter noise.