## Appendix B
\(\square\)  
Proof of Proposition 4.: For \(k_{f}\) expressed by Equation (20) and \(p_{u}=p_{l}-p_{b}\) Equation (8) takes the form  
\[\frac{2p_{u}(1+\varepsilon_{0})}{\Omega\Big{(}k_{f}\Big{)}}\sqrt[3]{\frac{4V _{00}}{\pi}}+\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left( \sqrt[3]{1+\varepsilon_{0}}\right)^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[} \frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]5  
whence the bottom layers overlap ratio is uniquely determined by Equation (27). The overlaps \(\delta\) and \(k_{fb}\) are related by Equation (9). Substituting \(k_{fb}\), Equation (27), into constraints described by Equation (23) leads to inequalities  
\[\frac{2p_{u}(1+\varepsilon_{0})}{\Omega\Big{(}k_{f}\Big{)}}\sqrt[3]{\frac{4V _{00}}{\pi}}+\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left( \sqrt[3]{1+\varepsilon_{0}}\right)^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[} \frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]6  
which can be rewritten in the equivalent form of constraints expressed with respect to integer \(p_{b}\) by Equation (25). Simultaneously, on the basis of Equations (9) and (27), the constraints expressed by Equation (22) are satisfied if and only if  
\[\frac{2p_{u}(1+\varepsilon_{0})}{\Omega\Big{(}k_{f}\Big{)}}\sqrt[3]{\frac{4V _{00}}{\pi}}+\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left( \sqrt[3]{1+\varepsilon_{0}}\right)^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[} \frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]7  
which is equivalent to the inequalities expressed by Equation (26), provided that inequality described by Equation (24) is satisfied. Proposition 4 is derived. \(\square\)  
Proof of Proposition 5.: According to Proposition 1 the solution to Problem 2 exists if and only if there exist integer \(p_{b}\) and overlaps \(\delta\), \(k_{fb}\) satisfying Equation (7), constraints from Equations (22) and (23) and the applicability condition, Equation (21), provided that the denominator of the left hand side of Equation (7) is non-zero, i.e., that inequality from Equation (13) holds, which for \(k_{f}=k_{f,u}\) and \(p_{u}=p_{l}-p_{b}\) can be rewritten as  
\[\frac{2p_{u}(1+\varepsilon_{0})}{\Omega\Big{(}k_{f}\Big{)}}\sqrt[3]{\frac{4V _{00}}{\pi}}+\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left( \sqrt[3]{1+\varepsilon_{0}}\right)^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[} \frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]8  
The above condition can be expressed directly with respect to the overlap ratio as inequality described by Equation (30). By Equation (7), having in mind that \(\Omega\Big{(}k_{f}\Big{)}=1\) and \(p_{u}=p_{l}-p_{b}\),  
the overlap \(\delta\) is uniquely given by equation  
\[\frac{2p_{u}(1+\varepsilon_{0})}{\Omega\Big{(}k_{f}\Big{)}}\sqrt[3]{\frac{4V _{00}}{\pi}}+\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left( \sqrt[3]{1+\varepsilon_{0}}\right)^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[} \frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]9whence direct formula from Equation (34) follows. Thus, the inequalities from Equation (22) take the form  
\[\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left(\sqrt[3]{1+ \varepsilon_{0}}\right)^{2}\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[}\frac{p_{b}}{ 1-k_{fb}}-\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}-\frac{2p_{u}}{\Omega\Big{(} k_{f}\Big{)}}\varepsilon_{0}\Bigg{]}. \tag{10}\]0  
and can be unravelled with respect to \(k_{fb}\) as follows  
\[\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left(\sqrt[3]{1+ \varepsilon_{0}}\right)^{2}\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[}\frac{p_{b}}{ 1-k_{fb}}-\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}-\frac{2p_{u}}{\Omega\Big{(} k_{f}\Big{)}}\varepsilon_{0}\Bigg{]}. \tag{10}\]1  
Simultaneously, for the overlap ratio \(k_{fb}\) inequalities from Equation (23) must be satisfied.