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The Phase classification of the SLE trace gives a space-filling curve precisely forIn particular, the critical value $\kappa =8$ is included.

By the Phase classification of the SLE trace, the trace has self-intersections but is not a space-filling curve forThe endpoint $\kappa =4$ remains simple, while $\kappa =8$ belongs to the space-filling regime.

The trace of ${\mathrm{SLE}}_{\kappa }$ is simple for $0<\kappa \le 4$, has self-intersections without being space-filling for $4<\kappa <8$, and is space-filling for $\kappa \ge 8$. The thresholds arise from the Boundary-point Bessel flow for SLE and estimates for the Loewner chain near its driving point.