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We use a relativistic ionization front to provide various initial transversewakefield amplitudes for the self-modulation of a long proton bunch in plasma.We show experimentally that, with sufficient initial amplitude($\ge(4.1\pm0.4)$ MV/m), the phase of the modulation along the bunch isreproducible from event to event, with 3 to 7% (of 2$\pi$) rms variations allalong the bunch. The phase is not reproducible for lower initial amplitudes. Weobserve the transition between these two regimes. Phase reproducibility isessential for deterministic external injection of particles to be accelerated.

Transition between Instability and Seeded Self-Modulation of a Relativistic Particle Bunch in Plasma