## Appendix B Detailed Longer Proofs of Theorems, Lemmas and Corollaries
### Proofs for Section 3
\begin{array}{ll}&\Big{(}J_{c}^{\mathrm{unif}[I_{1}J_{c}]}\Big{)}^{ \mathrm{aff}[a,b]}=a\Big{(}J_{c}^{\mathrm{unif}[I_{1}J_{c}]}\Big{)}+b=a\big{(} p_{c}I_{1}+q_{c}J_{c}\big{)}+p_{c}b+q_{c}b\\ &=(p_{c}J_{1}+p_{c}b)+(q_{c}aJ_{c}+q_{c}b)=J_{c}^{\mathrm{unif}[aI_{1}+bJ_{c}] +b]}=J_{c}^{\mathrm{unif}[J_{1}^{\mathrm{eff}[aI_{1}]},J_{c}^{\mathrm{eff}[aI _{1}]}]}\end{array} \] (A61)
**(ii)**:  
\(J_{c}^{\mathrm{unif}[I_{1}J_{c}]}\equiv p_{c}I_{1}+q_{c}J_{c}\) with \(p_{c}:=\frac{C-c}{C-1},q_{c}:=\frac{C-1}{C-1}\). The convex coefficients \(p_{c},q_{c}\) mirror-images at the mirror-image indices are readily evaluated: \(p_{c_{1C}}=\frac{C-c}{C-1}\Big{|}_{c=C+1-c}=\frac{c-1}{C-1}=q_{c},\quad q_{c_{ 1C}}=\frac{c-1}{C-1}\Big{|}_{c=C+1-c}=\frac{C-c}{C-1}=p_{c}\). To recap:  
\[ \begin{array}{ll}&p_{c_{1C}}=q_{c},\quad q_{c_{1C}}=p_{c}\\ &J_{c_{1C}}^{\mathrm{unif}[I_{1}J_{c}]}\equiv p_{c_{1}}I_{1}+q_{c_{1C}}J_{c}= q_{c}J_{1}+p_{c}J_{c}\\ &\cdot\cdot\cdot J_{c}^{\mathrm{unif}[I_{1}J_{c}]}+J_{c_{1C}}^{\mathrm{unif}[I_ {1}J_{c}]}=(p_{c}J_{1}+q_{c}J_{c})+(q_{c}J_{1}+p_{c}J_{c})\\ &=(p_{c}+q_{c})I_{1}+(q_{c}+p_{c})J_{c}=I_{1}+J_{c}\\ &\cdot\cdot J_{c}^{\mathrm{unif}[I_{1}J_{c}]}+J_{c_{1C}}^{\mathrm{unif}[I_{1}J _{c}]}=k|_{k=j_{1}+J_{c}}\end{array} \] (A62)  
Using Corollary 5 (vii) (178) it follows that the vector \(\mathbf{J}^{\mathrm{unif}[I_{1}J_{c}]}\) is raised-skew-centrosymmetric, and its center is \(\overline{\mathbf{J}^{\mathrm{unif}[I_{1}J_{c}]}}=\frac{1}{2}(I_{1}+J_{c})\). \(\square\)  
**Proof of Theorem 4. ** The following three lines (top, mid, bottom lines) are now shown equivalent:  
\[ \begin{array}{l}\delta_{\langle\text{nom}\rangle}\mathbf{J}\equiv\mathbf{J}- \langle\mathbf{J}^{\text{nom}}\rangle=\delta_{\langle\text{nom}\rangle}\langle \mathbf{J}\rangle+\delta\mathbf{J}\\ \delta_{\langle\text{nom}\rangle}\mathbf{J}\equiv\widetilde{\mathbf{J}}- \left\langle\widetilde{\mathbf{J}}^{-\text{nom}}\right\rangle=\delta_{\langle \text{nom}\rangle}\langle\mathbf{J}\rangle+\delta\widetilde{\mathbf{J}}\\ \mathbf{IEV}_{\mathbf{J}}=\widetilde{\mathbf{J}}-\left\langle\widetilde{ \mathbf{J}}^{-\text{nom}}\right\rangle=\ \ \mathbf{IEV}_{\mathbf{J}\rangle}\ \ +\ \delta\widetilde{\mathbf{J}}\end{array}  \tag{62}\]  
where we recall the definition of perturbation \(\delta\mathbf{J}\equiv\mathbf{J}-\langle\mathbf{J}\rangle\). The bottom line here reproduces (342) to be proven, assuming the top line (339) of Lemma 7. The mid line is obtained by unary-normalizing (applying rounded hats) onto the top line, and using the linearity and commutativity of the round-hat with the mean and nominal mappings. By commutativity of \(\sim\) and \({}^{\text{nom}}\) we mean the property \(\widetilde{\text{${\mathsf{mom}}$}}=\widetilde{\mathbf{J}}^{-\text{nom}}\), where the nominal mapping, \({}^{\text{nom}}\), fits a uniform constellation between two end-points equal to the mean LSL and MSL of a given constellation. The order of the two operations (fitting of a uniform constellation between the end-points, vs. unary-normalization, \(\sim\)) is immaterial since \(\mathbf{J}^{\text{nom}}\) is a discrete convex combination of the endpoints of \(\mathbf{J}^{\text{nom}}\), which is a linear operation. Formula (342) is derived from the second line by using the Alt Def of IEV (327). Therefore, \(\mathbf{IEV_{\mathbf{J}}}=\mathbf{IEV_{\langle\mathbf{J}\rangle}}+\ \delta \widetilde{\mathbf{J}}\) in (342) is now established.