Bondwise cancellation of a Jordan-Wigner string ($4S_n^x{S}_{n+1}^x=(c_n^{†}-c_n)(c_{n+1}^{†}+c_{n+1})$)

For $S_n^{+}=P_n{c}_n$ with $P_n={\prod }_{m<n}(1-2c_m^{†}{c}_m)$, neighbouring Jordan–Wigner transformation strings differ by only the local parity factor. The canonical anticommutation relations give $(c_n+c_n^{†})(1-2c_n^{†}{c}_n)=c_n^{†}-c_n$, proving the displayed local spin identity. The closing bond of a periodic chain instead retains total fermion parity; cancellation on bulk bonds does not justify discarding that global sector.