Four-current in two frames

What you're seeing. In the top row, the same particle worldlines are drawn in two inertial frames. In S (left) the region R is a vertical strip x ∈ [x_a, x_b]; in S′ (right) the same spacetime worldtube appears as a tilted parallelogram, because boost mixes t and x. Positive charges are green, negative red.

The bottom row plots the running spacetime averages ε̄_R(τ) = ¹⁄_|R| ∫_R J⁰ dt dx and J̄_R(τ) = ¹⁄_|R| ∫_R J¹ dt dx, one pair per frame. On the S′ plot the dashed orange ghost is Λ(β) applied to the S curve; the solid lines are the direct measurements inside the boosted parallelogram. They agree because (ε, J) is a 2-vector.

The ∂_μJ^μ readout is a discrete conservation check: the change in charge inside the spatial window [x_a, x_b] at the current slice equals the signed count of boundary crossings. Deviation from zero is a bug.