## Appendix B Detailed Longer Proofs of Theorems, Lemmas and Corollaries
### Proofs for Section 2
**(vii)**: as it is given that \(v_{\text{c}}+v_{\text{c}_{\text{t}_{\text{C}}}}=k\), for \(c=1\), \(2\), \(3\), \(\ldots\), \(C\), then \(v_{\text{c}}-\frac{1}{2}k=-(v_{\text{c}_{\text{t}_{\text{C}}}}-\frac{1}{2}k)\)  
The vector \(\left\{v_{\text{c}}-\frac{1}{2}k\right\}_{c=1}^{C}\) is recognized as a translation by \(-\frac{1}{2}k\mathbf{\Gamma}_{\text{C}}^{\forallin particular, for \(c=1\) and for \(c=C\) we have (with second equalities stemming from \(1_{\mathsf{TC}}=C\)):  
\[\overline{\mathbf{u}}=\frac{1}{2}\big{(}u_{1}+u_{1_{\mathsf{TC}}}\big{)}=\frac{1 }{2}(u_{1}+u_{\mathsf{C}}),\ \ \ \overline{\mathbf{v}}=\frac{1}{2}\big{(}v_{1}+v_{1_{\mathsf{TC}}}\big{)}=\frac{1 }{2}(v_{1}+v_{\mathsf{C}}) \tag{101}\]  
Then, as the FSRs of \(\mathbf{u},\mathbf{v}\) are specified to be equal, then \(u_{1}=v_{1}\), \(u_{\mathsf{C}}=v_{C}\); together with (101) this implies \(\mathbf{u}=\mathbf{v}\). **(ix)**: in light of (viii) it suffices to assume \(\overline{\mathbf{u}}=\overline{\mathbf{v}}\) for the two given raised-skew-centrosymmetric vectors, then show that that implies that \(\mathbf{u}-\mathbf{v}\) be skew-centrosymmetric, i.e.,  
\[[\mathbf{u}-\mathbf{v}]_{c}=-[\mathbf{u}-\mathbf{v}]_{c_{\mathsf{TC}}} \Leftrightarrow u_{c}-v_{c}=-(u_{c_{\mathsf{TC}}}-v_{c_{\mathsf{TC}}}). \tag{102}\]  
From (103) (implied by \(\mathbf{u},\mathbf{v}\) assumed raised-skew-centrosymmetric) along with \(\mathbf{u}=\mathbf{v}\), it follows that \(\frac{1}{2}(u_{c}+u_{c_{\mathsf{TC}}})=\overline{\mathbf{u}}=\overline{ \mathbf{v}}=\frac{1}{2}(v_{c}+v_{c_{\mathsf{TC}}})\). Then  
\[\therefore u_{c}+u_{c_{\mathsf{TC}}}=v_{c}+v_{c_{\mathsf{TC}}}\therefore u_{c}-v_ {c}=v_{c_{\mathsf{TC}}}-u_{c_{\mathsf{TC}}}=-(u_{c_{\mathsf{TC}}}-v_{c_{ \mathsf{TC}}}) \tag{103}\]  
\(\therefore u_{c}-v_{c}=-(u_{c_{\mathsf{TC}}}-v_{c_{\mathsf{TC}}})\ \ \Box\). **(x)**: derivation of \(\
abla v_{\mathsf{\tau}(C-1)}=\
abla v_{\mathsf{C}}\) (180) for a raised-skew-centrosymmetric vector, \(\mathbf{v}\):  
\[ \begin{array}{l}v_{c}\ +v_{c_{\mathsf{TC}}}=2\overline{\mathbf{v}},\ \ v_{c+1}\ +v_{(c+1)_{\mathsf{TC}}}=2 \overline{\mathbf{v}}\Rightarrow v_{c}\ +v_{c_{\mathsf{TC}}}=v_{c+1}+v_{(c+1)_{ \mathsf{TC}}}\\ \Leftrightarrow v_{c}\ +v_{c_{\mathsf{TC}}}=v_{c+1}\ +v_{c_{\mathsf{TC}}-1}\\ \Leftrightarrow v_{c_{\mathsf{TC}}}v_{c_{\mathsf{TC}(C-1)}}=v_{c+1}-v_{c} \Leftrightarrow v_{c_{\mathsf{TC}(C-1)}+1}-v_{c_{\mathsf{TC}(C-1)}}=v_{c+1}- v_{c}\Leftrightarrow\ \
abla v_{\mathsf{\tau}(C-1)}=\
abla v_{c}\end{array}  \tag{104}\]  
where the first line stems from (ii); the first equivalence in the second line uses the fact that the ordering of indices reverses under reverse-operator reflections (proven using definition (167)):  
\[(c+1)_{\mathsf{TC}}=c_{\mathsf{TC}}-1. \tag{105}\]  
In the last line in (104), the mirror-image index definitions were recalibrated from a C-vector to a \(C-1\)-vector (with \(C-1\) is the length of the step vectors, \(\
abla\mathbf{v}\)), based on the following identities, stemming from definition (167) of mirror-image index:  
\[c_{\mathsf{TC}}=c_{\mathsf{\tau}(C-1)}+1,\ \ c_{\mathsf{TC}}-1=c_{\mathsf{ \tau}(C-1)} \tag{106}\]  
The last expression in (104) is our sought result. **(xi)**: inspecting the vector \(\mathbf{v}^{\mathsf{aff}[a,b]}=a\mathbf{v}+b\mathbf{\Gamma}_{C}^{\mathsf{V}}\) when \(\mathbf{v}\) is (skew)-centrosymmetric it is readily seen that \(a\mathbf{v}\) is (skew)-centrosymmetric, respectively. If \(\mathbf{v}\) is centrosymmetric, then so is \(a\mathbf{v}\) and adding the constant vector \(b\mathbf{\Gamma}_{C}^{\mathsf{V}}\) (which is centrosymmetric) preserves centrosymmetry (as centrosymmetry is readily seen to be additive). If \(\mathbf{v}\) is skew-centrosymmetric, then so is \(a\mathbf{v}\) centrosymmetric and adding the constant vector \(b\mathbf{\Gamma}_{C}^{\mathsf{V}}\) yields a raised-centrosymmetric vector. \(\Box\)  
**Proof of Theorem 3. **  
**(i)**: let us first present the derivation for a BiWgt UNIP eDAC. Using (5) and (8),  
\[