Partial-Wave Scattering Phase Shifts and the Levinson Theorem for a Sextic Q-Ball: Symmetry Modes Only and a Ramsauer–Townsend Transparency Point

Partial-wave phase shifts δ_l(k) are computed for the field's own excitations scattering off the ω = 0.85 lump of a sextic-potential Q-ball model (single-lump energy E = 336.7905). Levinson's theorem holds with a specific twist: the counts δ_l(0)/π = (1, 1, 0) for l = 0, 1, 2 match a direct bound-state count exactly, and the counted states are the lump's two symmetry zero modes — the U(1) phase mode and the translation mode — sitting at threshold. A direct bound-state search finds no internal bound modes: the lump's entire bound spectrum is its symmetry modes. The s-wave phase shift sweeps through π across a narrow resonance (centre k_r = 2.3595, Γ_k ≈ 0.012) and reaches δ₀ = π exactly at k = 2.3975, where sin²δ₀ = 0 — a point of perfect first-order s-wave transparency, a Ramsauer–Townsend analogue for an extended classical lump. A broad l = 2 resonance (k_r = 0.2271, Γ_k = 0.042, cross-section peak 1474 at k = 0.2250) and a non-resonant l = 3 channel complete the picture. A focused literature search (Oct 28, 2026) found no prior partial-wave phase-shift calculation for a field's own excitations off a sextic Q-ball; in that hedged form, the calculation appears to be the first of its kind.