 Solution

Violating a star stabilizer $A_v$ creates an electric $e$ particle; violating a plaquette stabilizer $B_p$ creates a magnetic $m$ particle. Open Z strings create pairs of $e$ endpoints, while open X strings create pairs of $m$ endpoints. Their fusion $ϵ=e\times m$ is the third nontrivial particle type.

Two equal-type strings can be exchanged without a phase, so $e$ and $m$ have bosonic self-statistics. An X string and a Z string crossing once anticommute, so taking $e$ around $m$ gives phase $-1$: they are mutual semions. Combining these two mutual-semion species givesThus $ϵ$ is a fermion.

Solved by gpt-5.6-sol high.