The tensor adjoint formula and \(\gamma _j^*=\gamma _j\), \(\nabla _j^*=-\nabla _j\) give \((\gamma _j\otimes \nabla _j)^* =-\gamma _j\otimes \nabla _j\); adding proves \(D^*=-D\). In the square, every diagonal term is \(\gamma _j^2\otimes \nabla _j^2=I_S\otimes \nabla _j^2\). For \(i<j\), the paired off-diagonal terms are \[\begin {gathered} \gamma _i\gamma _j\otimes \nabla _i\nabla _j +\gamma _j\gamma _i\otimes \nabla _j\nabla _i\\ =(\gamma _i\gamma _j+\gamma _j\gamma _i) \otimes \nabla _i\nabla _j=0, \end {gathered}\] using commutation of the differences and anticommutation of the distinct gamma matrices. This proves the square formula. Expanding \(\nabla _j^2\) gives \((T_j^2-2I_H+T_j^{-2})/4\). Finally \(\langle \psi ,-D^2\psi \rangle _K =\langle D\psi ,D\psi \rangle _K\ge 0\).

Let \(\mathcal R=\prod _j\{0,\ldots ,N_j-1\}\) and, for \(r\in \mathcal R\), define the normalized character function \[\chi _r(x)=M^{-1/2} \exp \left (i\sum _{j=1}^d p_jx_j\right ).\] Choose integer representatives for \(x_j\). Changing one by \(N_j\) multiplies the exponential by \(\exp (2\pi i r_j)=1\); changing \(r_j\) by \(N_j\) likewise multiplies it by \(\exp (2\pi i x_j)=1\). Thus the function depends only on the residue classes.

For \(r,t\in \mathcal R\), put \[\zeta _j=\exp \left (\frac {2\pi i(t_j-r_j)}{N_j}\right ).\] The counting inner product factors: \[\langle \chi _r,\chi _t\rangle _H =M^{-1}\prod _{j=1}^d\sum _{k=0}^{N_j-1}\zeta _j^k.\] If \(t_j=r_j\), then \(\zeta _j=1\) and the inner sum is \(N_j\). Otherwise the chosen range of labels gives \(\zeta _j\ne 1\) and \(\zeta _j^{N_j}=1\), so \[(1-\zeta _j)\sum _{k=0}^{N_j-1}\zeta _j^k =1-\zeta _j^{N_j}=0.\] Division by \(1-\zeta _j\) makes that sum zero. Consequently \(\langle \chi _r,\chi _t\rangle _H=\delta _{rt}\). There are \(M\) such functions, and \(\dim _{\mathbb C}H=M\), as witnessed by the vertex indicators. Orthonormality gives linear independence, so these functions form a complete basis.

For a spinor-valued field, the Fourier transform and inverse therefore are \[\begin {aligned} \widehat \psi (r) &=M^{-1/2}\sum _{x\in \Lambda } e^{-i\sum _jp_jx_j}\psi (x),\\ \psi (x) &=M^{-1/2}\sum _{r\in \mathcal R} e^{i\sum _jp_jx_j}\widehat \psi (r). \end {aligned}\] For scalar fields the coefficient is \(\widehat f(r)=\langle \chi _r,f\rangle _H\); applying that basis expansion in each standard coordinate of \(S\) proves the two vector-valued formulas. Thus \[K=\bigoplus _{r\in \mathcal R} S\otimes \mathbb C\chi _r.\] The forward shift has \(T_j\chi _r=e^{ip_j}\chi _r\), hence \(\nabla _j\chi _r=i\sin p_j\,\chi _r\). Each displayed summand is invariant under \(D\), with block \(D(p)=i\sum _j\gamma _j\sin p_j\). Expanding its square using the same anticommutators gives \(D(p)^2=-s(p)I_S\).