## Appendix B Detailed Longer Proofs of Theorems, Lemmas and Corollaries
### Proofs for Section 3  
**Proof of Corollary 6. ** Result (294) is obtained by interpreting the LSH of (291) as \(\text{INL}_{c}\). Next, (295) is derived by plugging in the accumulations of DNL samples for \(\text{INL}_{c}\) and for \(\text{INL}_{c+1}\) (per first line in (294)) and simplifying the difference of sums. Finally, (296) is derived by induction. Plugging \(c=1\) into (296) yields the first step of the induction:  
\[\text{INL}_{C-1}=\text{INL}_{C}-\text{DNL}_{C-1}\Leftrightarrow\text{DNL}_{C-1 }=\text{INL}_{C}-\text{INL}_{C-1} \tag{114}\]  
The second equivalent equality is a special case of (295) with \(c=C\) 1. Now assuming (296) is true for index \(c\), it is readily shown that it must also be true for index \(c+1\), i.e., we must prove \(\text{INL}_{C-(c+1)}=\text{INL}_{C}-\sum_{i=C-(c+1)}^{C-1}\text{DNL}_{i}\):\[ \begin{split}&\text{INL}_{\text{C}}-\sum_{i=\text{C}-(c+1)}^{\text{C}- 1}\text{DNL}_{i}=\text{INL}_{\text{C}}-\sum_{i=\text{C}-c}^{\text{C}-1}\text{ DNL}_{i}-\text{DNL}_{\text{C}-(c+1)}=\text{INL}_{\text{C}-c}-\text{DNL}_{\text{C}-(c+1)}\\ &=\text{INL}_{\text{C}-c}-\text{DNL}_{(\text{C}-c)-1}=\text{INL}_ {(\text{C}-c)-1}=\text{INL}_{\text{C}-(c+1)}.\end{split} \] (A45)  
\(\square\)  
**Proof of Corollary 7. ** (i) Affine transformations leave the INL, DNL invariant:  
\[\begin{split}&\text{INL}  
\[\therefore.\,{\rm INL}_{1}=\left.\frac{1}{\
abla J_{c}}(J_{1}-J_{1}^{\rm nom})\right|_{ J_{1}=J_{1}^{\rm nom}}=0,\ \ \therefore\,{\rm INL}_{C}=\left.\frac{1}{\
abla J_{c}}(J_{C}-J_{C}^{\rm nom})\right|_{ J_{1}=J_{1}^{\rm nom}}=0\] (111) \[\sum_{i=1}^{C-1}{\rm DNL}_{i}=\left.\sum_{i=1}^{C-1}{\rm DNL}_{i}\right|_{ c:=C}=\left.{\rm INL}_{c}\right|_{c:=C}={\rm INL}_{C}=0\] (112) where \({\rm INL}_{c}=\sum_{i=1}^{c-1}{\rm DNL}_{i}\) and (i) were used. **(iii)**: from (287) it follows that \({\rm DNL}_{\bf J}\equiv\frac{1}{\
abla\bf J}\
abla{\bf J}-\Upsilon_{C-1}^{\rm \forall}\). Since \(\
abla{\bf J}\) was shown centrosymmetric in (190), \(\
abla{\bf J}=\left(\
abla{\bf J}\right)^{\dagger}\), it then follows using the linearity of the inverse operator:  
\[{\rm DNL}_{\bf J}^{\dagger}\equiv\left(\frac{1}{\
abla\bf J}\
abla{\bf J}- \Upsilon_{C-1}^{\rm\forall}\right)^{\dagger}=\frac{1}{\
abla\bf J}(\
abla{\bf J })^{\dagger}-\left(\mathbf{1}_{C-1}^{\rm\forall}\right)^{\dagger}=\frac{1}{ \
abla\bf J}\
abla{\bf J}-\mathbf{1}_{C-1}^{\rm\forall}={\rm DNL}_{\bf J}. \tag{113}\]  
An alternative derivation: to show that the DNL vector is centrosymmetric use the observation (302) that the DNL is nothing but a certain affine transformation of the form \({\bf J}^{\rm aff[g,g]}\equiv a{\bf J}+b\Upsilon_{C}^{\rm\forall}\) (181) or (301), acting on the eDAC output steps vector, \(\
abla{\bf J}\). Having shown in Theorem 3 (v) (190) that the steps vector \(\
abla{\bf J}\) of the eDAC output constellation is centrosymmetric, and having shown in Corollary 5 (xi) that centrosymmetry is preserved under an affine transformation of the form (181), it then follows that the DNL must be centrosymmetric. **(iv)**: to show that the INL vector is skew-centrosymmetric, recall def (299) of INL:  
\[{\rm INL}\equiv\frac{{\bf J}-{\bf J}^{\rm nom}}{\
abla{\bf J}}=\frac{1}{ \
abla{\bf J}}\left({\bf J}-{\bf J}^{\rm unif[J_{1},J_{C}]}\right) \tag{114}\]  
It suffices to show that \({\bf J}-{\bf J}^{\rm unif[J_{1},J_{C}]}\) is skew-centrosymmetric (as any proportionality transformation, here scaling by \(\frac{1}{\
abla{\bf J}}\), preserves skew-centrosymmetry). We observe that \({\bf J},{\bf J}^{\rm unif[J_{1},J_{C}]}\) are both raised-skew-centrosymmetric (\({\bf J}\) is by Theorem 3 (iv), and \({\bf J}^{\rm unif[J_{1},J_{C}]}\) is by Lemma 5 (ii)); we next invoke Corollary 5 (ix) which states that when two vectors \({\bf u},{\bf v}\) are both raised-skew-centro-symmetric and they satisfy \(u_{1}=v_{1}\), \(u_{C}=v_{C}\), then their difference \({\bf u}-{\bf v}\) is skew-centrosymmetric.