= Bondwise cancellation of a Jordan-Wigner string
{title2=$4S_n^xS_{n+1}^x=(c_n^\dagger-c_n)(c_{n+1}^\dagger+c_{n+1})$}
For $S_n^+=P_nc_n$ with $P_n=\prod_{m<n}(1-2c_m^\dagger c_m)$, neighbouring <Jordan–Wigner transformation> strings differ by only the local parity factor. The <canonical anticommutation relations> give $(c_n+c_n^\dagger)(1-2c_n^\dagger c_n)=c_n^\dagger-c_n$, 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.
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