A beam of charged particles at intermediate energies (10-100 MeV) is usually treated as a collection of independent particles. Each follows its own trajectory through the accelerator, deflected by external fields, scattered by residual gas. The particles know about each other only through a smooth, averaged-out space charge potential.
Yee, Collins, Iofin, and Fu (arXiv:2603.10457) show that above a critical density n_c, the beam stops being a beam and becomes a plasma. Collective oscillations appear — Langmuir waves, undamped modes, Friedel oscillations at twice the Fermi wavevector. The beam undergoes a phase transition that belongs to the 3D Ising universality class by renormalization group analysis.
Two results stand out. First, the plasma frequency is distribution-independent: Ω_p² = ne²/(mε₀), fixed by the f-sum rule regardless of whether the beam is Gaussian, uniform, or degenerate. This is a universal property — the total oscillator strength is pinned by particle number, and no amount of reshaping the velocity distribution can change it. Second, a variational autoencoder (Prometheus) trained on structure factor data from particle-in-cell simulations detects the onset of collective behavior without being told the theory. Six validation checks against the analytic predictions all pass.
The structural insight: the boundary between “beam” and “plasma” is a genuine phase transition, not a smooth crossover. Below the critical density, there are no collective modes — Landau damping absorbs them. Above it, undamped waves appear with √(n - n_c) onset. The beam doesn't gradually become plasma-like. It snaps into a qualitatively different state. The transition is as sharp as magnetization in an Ising model, and for the same mathematical reason.