Toric code distance is L

Let $E=i^a{X}^x{Z}^z$ be a Pauli in the normalizer. Commuting with every star forces the Z-support $z$ to have even incidence at every primal vertex; commuting with every plaquette forces the X-support $x$ to have even incidence at every dual vertex, by [F1] and [F4]. Thus $z$ is a primal cycle and $x$ is a dual cycle.

If both cycles were boundaries, their supports would be products of plaquette and star checks respectively, so $E$ would be a scalar-phase stabilizer. Hence a nontrivial logical Pauli has at least one nonboundary cycle by [F2] and [F3]. That cycle has a nonzero winding parity. For a horizontal winding, each of the $L$ disjoint vertical cuts is crossed an odd number of times; for a vertical winding, each of the $L$ disjoint horizontal cuts is crossed an odd number of times. In either case its support contains at least one distinct edge from each cut, hence at least $L$ edges. The Pauli support contains the support of each nonzero X or Z chain, even when both bits occur on one qubit, so [F7] gives $\mathrm{wt}(E)\ge L$.

The operator ${\bar{Z}}_1$ from [F5] has exactly $L$ distinct edge factors and is logically nontrivial. Thus the set defining [F6] is nonempty and contains a weight-$L$ element. The code encodes two qubits by [F8], so [F6] applies; combining this upper bound with step 2.1 gives $d(S)=L$.