ALADIN Plasma Stability Law v0.4Invariant QMHD Threshold for Self-Stabilization in Current-Driven Plasmas

This record presents v0.4 of the ALADIN Plasma Stability Law — a first-principles criterion for self-stabilization in current-driven Z-pinch plasmas. The law is anchored to a single measured parameter: the axial current density J₀ = 1.000 × 10¹⁸ A/m². Core Law Statement A current-driven plasma column with uniform axial current density J₀ is stable against sausage (m=0) and kink (m=1) instabilities if and only if \boxed{ \max!\left[ \left(\dfrac{B_z}{B_\theta}\right)^2,; \dfrac{S}{\gamma_m} \right] \ge 1 } with exact definitions: • B_θ = μ₀ J₀ a / 2 • γ_m = v_A / a • v_A = B_θ / √(μ₀ ρ_eff) • ρ_eff = μ₀ J₀² a² / (8 c²) • Π = μ₀ J₀² a² / (ρ_eff c²) = 8 at the magnetic-rest energy balance point QMHD Extensions The law incorporates quantum magnetohydrodynamic effects: Schwinger pair production (induced E_z from Faraday during collapse), quantum degeneracy pressure P_q, and effective density ρ_eff_QMHD = ρ_eff + ρ_pair + P_q / c². Falsification The law is falsified if Π < 4 and no suppression occurs (γ a / v_A > 0.5 or q < 1) in observed dynamics for more than 10 Alfvén crossing times (τ > 10 a / v_A). Regime of Validity Ideal/low-resistivity MHD, β ≲ 1, approximate axial symmetry, resistive diffusion time ≫ instability growth time, subdominant reconnection during collapse phase. Scope This is the frozen core law. The included script demonstrates 1D collapse with emergent B_z dynamo ramp, axial shear S, and QMHD back-reaction. Plots illustrate stability evolution (Π(t), growth rates, pair density, v_A/c). Previous versions v0.2: 10.5281/zenodo.18306470v0.3: 10.5281/zenodo.18313105