## Appendix B Detailed Longer Proofs of Theorems, Lemmas and Corollaries
### Proofs for Section 2
\begin{array}{l}\mathrm{d}\mathrm{y}\mathrm{d}^{[B]}\cdot\Big{\{}\mathbf{b} ^{\mathrm{BiWgt}}_{\mathrm{UNIP}}[c]+\mathbf{b}^{\mathrm{BiWgt}}_{\mathrm{ UNIP}}[c_{\mathsf{TC}}]\Big{\}}=\mathrm{d}\mathrm{y}\mathrm{d}^{[B]}\cdot \mathbf{b}^{\mathrm{BiWgt}}_{\mathrm{UNIP}}[c]+\mathrm{d}\mathrm{y}\mathrm{d}^{[ B]}\cdot\mathbf{b}^{\mathrm{BiWgt}}_{\mathrm{UNIP}}[c_{\mathsf{TC}}]\\ =(c-1)+(c_{\mathsf{TC}}-1)=c+c_{\mathsf{TC}}-2=C+1-2=C-1=\mathrm{d}\mathrm{y} \mathrm{d}^{[B]}\cdot\mathbf{\Gamma}_{B}^{\mathsf{V}}\end{array}  \tag{107}\]  
where the last equality is readily verified by evaluating the inner product of the dyadic sequence (7) with the all-ones vector,\[\mathrm{d}yad^{[B]}\cdot\mathbf{1}_{B}^{\forall}=\left\{2^{B-1},2^{B-2}, \ldots,4,2,1\right\}\cdot\underbrace{\{1,1,...1\}}_{B\text{ ones}}=2^{B-1}+2^{B-2}+\ldots+4+2+1\] \[=2^{B}-1=C-1\] (A31)  
An alternative derivation of (A31) uses \(\mathbf{b}_{\text{UNIP}}^{\text{BiWgt}}[C]=\mathbf{1}_{B}^{\forall}\) as well as (8), as follows:  
\[\mathrm{d}yad^{[B]}\cdot\mathbf{1}_{B}^{\forall}=\mathrm{d}yad^{[B]}\cdot \mathbf{b}_{\text{UNIP}}^{\text{BiWgt}}[C]=\left.\mathrm{d}yad^{[B]}\cdot \mathbf{b}_{\text{UNIP}}^{\text{BiWgt}}[c]\right|_{c:=C}=\left.c-1\right|_{c:=C }=C-1.\] (A32)  
To recapitulate (A31),  
\[\mathrm{d}yad^{[B]}\cdot\left\{\mathbf{b}_{\text{UNIP}}^{\text{BiWgt}}[c]+ \mathbf{b}_{\text{UNIP}}^{\text{BiWgt}}[c_{\text{TC}}]\right\}=\mathrm{d}yad^ {[B]}\cdot\mathbf{1}_{B}^{\forall}\] (A33)  
But the equation \(\mathrm{d}yad^{[B]}\cdot\mathbf{b}_{1}=\mathrm{d}yad^{[B]}\cdot\mathbf{b}_{2}\), where \(\mathbf{b}_{1},\mathbf{b}_{2}\) are Boolean vectors (i.e., have elements \(0\,11\)), is readily shown to be equivalent to \(\mathbf{b}_{1}=\mathbf{b}_{2}\) (this amounts to the radix-number representation being unique), i.e., \(\mathrm{d}yad^{[B]}\). may be formally \"cancelled\" from both sides (A33), yielding \(\mathbf{b}_{\text{UNIP}}^{\text{BiWgt}}[c]+\mathbf{b}_{\text{UNIP}}^{\text{ BiWgt}}[c_{\text{TC}}]=\mathbf{1}_{B}^{\forall}\), (182) elementwise amounting to  
\[ \begin{array}{l}b_{s}^{\text{BiWgt}-\text{UNIP}}[c]+b_{s}^{\text{BiWgt}- \text{UNIP}}[c_{\text{TC}}]=1,\ \ s=0,1,2,3,\ldots,B-1;\\ \therefore\ b_{s}^{\text{BiWgt}}[c_{\text{TC}}]=1-b_{s}^{\text{BiWgt}}[c]\end{array} \] (A34)  
Thus, if \(b_{s}^{\text{BiWgt}}[c]=0\Big{|}1\), respectively, then \(b_{s}^{\text{BiWgt}}[c_{\text{TC}}]=1-0\Big{|}1=1\Big{|}0\), i.e., we have established the sought one's-complement relation. Now given two bitstrings in one's-complement relation, their XOR is the all-ones bitstring. This proves (182) for BiWgt UNIP. Proceeding to prove (i) for a ThWgt UNIP eDAC, it suffices to inspect the ThWgt UNIP code matrix and verify that the bitstrings (codewords) in \(c-\)th and \(c_{\text{TC}}-\)th rows are in one's-complement relation, \(\mathbf{b}_{\text{UNIP}}^{\text{ThWgt}}[c]+\mathbf{b}_{\text{UNIP}}^{\text{ ThWgt}}[c_{\text{TC}}]=\mathbf{1}_{B}^{\forall}\), from which remainder of the properties in (182) follow. Thus, for either ThWgt \(\mid\)BiWgt eDACs we have  
\[\mathbf{b}_{\text{UNIP}}[c]+\mathbf{b}_{\text{UNIP}}[c_{\text{TC}}]=\mathbf{1 }_{B}^{\forall}\] (A35)  
**(ii)**: to prove the \"zero's complement\" property (184) for BIP eDACs, let the generic codeword of a BIP eDAC (be it of the ThWgt \(\mid\)BiWgt topology) be denoted \(\mathbf{b}_{\text{BIP}}[c]\) and that of a UNIP eDAC be denoted \(\mathbf{b}_{\text{BIP}}[c_{\text{TC}}]\). Resorting to the UNIP-BIP affine transformation \(\mathbf{b}_{\text{BIP}}[c]=2\mathbf{b}_{\text{UNIP}}[c]-\mathbf{1}_{C}^{\forall}\) (42), we evaluate the sum \(\mathbf{b}_{\text{BIP}}[c]+\mathbf{b}_{\text{BIP}}[c_{\text{TC}}]\):  
\[