## Appendix B Detailed Longer Proofs of Theorems, Lemmas and Corollaries  
### Proofs for Section 2  
**Proof of Corollary 4. ** Some of the results were proven in the text above. What remains to complete here is the proof of (134) for UNIP eDACs (of any topology and (non-)uniformity):  
Let \(\left\{\langle I_{\text{s}}^{\text{UNIP}}\rangle\right\}_{\text{s}=0}^{S-1}\) be the \(S\) mean source currents of a UNIP eDAC of any of the considered types. In (122) we have seen that for a generally non-uniform UNIP eDAC,  
\[\sigma_{\text{{J}}^{\text{UNIP}}}^{2}=\sigma_{u}^{2}\langle I_{u}\rangle \cdot\left\langle\mathbf{C}_{\text{UNIP}}\mathbf{I}^{\text{UNIP}}\right\rangle =\sigma_{u}^{2}\langle I_{u}\rangle\cdot\left\langle\mathbf{J}^{\text{UNIP}}\right\rangle \tag{11}\]  
Let us now evaluate the (arithmetic) average of this variance vector,  
\[ \begin{array}{l}\overline{\sigma_{\text{{J}}^{\text{UNIP}}}^{2}}=\frac{1}{ \mathcal{C}}\mathbf{1}_{\text{C}}^{\forall}\cdot\sigma_{\text{{J}}^{\text{ UNIP}}}^{2}=\frac{1}{\mathcal{C}}\left(\mathbf{1}_{\text{C}}^{\forall} \right)^{T}\sigma_{\text{{J}}^{\text{UNIP}}}^{2}=\frac{1}{\mathcal{C}}\left( \mathbf{1}_{\text{C}}^{\forall}\right)^{T}\sigma_{u}^{2}\langle I_{u}\rangle \langle\mathbf{C}_{\text{UNIP}}\mathbf{I}^{\text{UNIP}}\rangle\\ \overline{\mu_{4}}^{2}\frac{1}{\mathcal{C}}\sigma_{u}^{2}\langle I_{u}\rangle \langle\left[(\mathbf{1}_{\text{C}}^{\forall})^{T}\mathbf{C}_{\text{UNIP}} \right]\mathbf{I}^{\text{UNIP}}\rangle=\frac{1}{\mathcal{C}}\sigma_{u}^{2} \langle I_{u}\rangle\langle\left[\frac{1}{2}C\left(\mathbf{1}_{\text{S}}^{ \forall}\right)^{T}\right]\mathbf{I}^{\text{UNIP}}\rangle\\ \overline{\mu_{6}}^{2}\frac{1}{\mathcal{C}}\sigma_{u}^{2}\langle I_{u}\rangle \langle\left(\mathbf{1}_{\text{S}}^{\forall}\right)^{T}\mathbf{I}^{\text{UNIP }}\rangle=\frac{1}{\mathcal{C}}\sigma_{u}^{2}\langle I_{u}\rangle\langle I_{ \text{tot}}^{\text{UNIP}}\rangle\end{array}  \tag{12}\]  
where #1 is an evident way to describe the mean of a vector as an inner product of the vector with \(\frac{1}{\mathcal{C}}\mathbf{1}_{\text{C}}^{\forall}=\frac{1}{\mathcal{C}} \left\{1,1,\ldots,1\right\};\#2\) casts the inner product as matrix left-product by the corresponding row vector; #3 has (11) substituted in for \(\sigma_{\text{{J}}^{\text{UNIP}}}^{2};\#4\) is a matrix manipulation bringing the row inside the mean, past the multiplicative scalar. At this point we may verify the following property of BiWgt UNIP \(C\times S\) code matrices:  
\[\left(\mathbf{1}_{\text{C}}^{\forall}\right)^{T}\mathbf{C}_{\text{UNIP}}= \frac{1}{2}C\left(\mathbf{1}_{\text{S}}^{\forall}\right)^{T} \tag{13}\]  
In fact, the left-multiplication of \(\mathbf{C}_{\text{UNIP}}\) by the row vector \(\left(\mathbf{1}_{\text{C}}^{\forall}\right)^{T}\) amounts to summing up the elements in each of successive columns of \(\mathbf{C}_{\text{UNIP}}\); the sums are then tallied in an \(1\times S\) row vector. But since the elements of \(\mathbf{C}_{\text{UNIP}}\) are \(0\,1\), summing up the elements in any column yields the \(\#\)-of-ones in each column of \(\mathbf{C}_{\text{UNIP}}\). But for each of the BiWgt code matrices of dimension \(C\times S=2^{B}\times B\), it may be verified that half the elements in each column are \(1\) s (the rest are zeros). The property that there are C/2 ones in each column is readily verified for the UNIP PAM4 code matrix (10) and for the UNIP PAM8 code matrix (14). The general case may be formally proven by induction starting with PAM4 case.