Winding-number staircase of the effective Lorentz factor and an analogue horizon in a ⁸⁷Rb ring Bose–Einstein condensate: two Bragg-resolvable predictions and a switched-shell null test

Persistent currents in ring-shaped Bose–Einstein condensates are quantised: the azimuthal flow speed can take only the discrete values v = Nħ/(mR) set by the winding number N. We point out that this turns the approach to the sonic (analogue-gravity) horizon into a reproducible staircase: in a ring of radius R = 20 µm with Bogoliubov sound speed c_s = 1.0 mm/s (peak density n₀ ≈ 2.8×10¹³ cm⁻³), one circulation quantum corresponds to Δv ≈ 36.5 µm/s, the effective Lorentz factor of the phonon metric, γ_eff = [1 − (v/c_s)²]^(−1/2), climbs in discrete steps — 1.46 at N = 20, 2.46 at N = 25, 6.1 at N = 27 — and the first supersonic step, N = 28, creates an analogue horizon on the ring. We derive two quantitative, Bragg-resolvable predictions: (i) the step-resolved staircase of γ_eff itself, scanned reproducibly by preparing successive winding numbers, and (ii) a splitting Δf = v/(πR) between co- and counter-propagating phonon modes — 15.7 Hz at N = 27, approaching 15.9 Hz as v → c_s — an analogue of frame dragging that extends to the horizon the Doppler readout of persistent currents already demonstrated at low winding number. Both are accessible to established mode-resolved phonon spectroscopy. We compare the ring-BEC platform with a superfluid ⁴He giant-vortex alternative, and we propose a low-cost null test in which a macroscopic switchable current shell (kV–MV class) surrounds the shielded cell and is modulated 1 min ON / 1 min OFF: square-wave demodulation of Δf over one day bounds any anomalous fractional response at |δΔf|/Δf < 1.5×10⁻³ (1σ), improving as 1/√τ. The experiment is table-top scale and uses only established techniques.