A toric logical string has constant energy barrier

Suppose $J_e\le J_m$. Apply $Z_{h_{0,0}},Z_{h_{1,0}},\dots ,Z_{h_{L-1,0}}$ in that order. After $k$ factors with $1\le k<L$, the partial string is a primal trail from $q_{0,0}$ to $q_{k,0}$; these endpoints are distinct, so [F3] gives exactly two e defects and no m defects. By [F5], changing one star eigenvalue from $+1$ to $-1$ changes its energy contribution from $-J_e$ to $+J_e$, an increase of $2J_e$; the excess is therefore $4J_e$. After the last factor the path is closed and has zero syndrome by [F3]. This is an elementary path in [F1] whose peak excess is at most $4J_e$.

Suppose $J_m<J_e$. Apply $X_{v_{0,0}},X_{v_{1,0}},\dots ,X_{v_{L-1,0}}$ in that order. The crossed dual edges form a closed horizontal dual loop. After $k$ factors with $1\le k<L$, the dual trail has distinct endpoint faces $f_{L-1,0}$ and $f_{k-1,0}$; [F3] gives exactly two m defects and no e defects. By [F5], each m defect raises the energy by $2J_m$, so the excess is $4J_m$. The final closed trail has zero syndrome by [F3] and ends at ${\bar{X}}_2$. This is an elementary path in [F1] with peak excess at most $4J_m$.

If $J_e\le J_m$, step 1.1 gives $4J_e=4\min (J_e,J_m)$. If $J_m<J_e$, step 1.2 gives $4J_m=4\min (J_e,J_m)$. The bound is independent of $L$.