## 3 Methodology and Proposed Innovation
### PA Properties of QUAKE  
Additive synthesis dictates that all metrics participating in the algebraic combination should hold the same order relation, so that the additively composited output can be considered as legitimate. This composition inherits either the monotonic or isotonic properties individually or in combination, if and only if the combined metrics hold such properties themselves. However, as the authors of [46, 47] state in their study, the monotonicity and isotonicity properties of the produced composite additive metric have to be proven nonetheless. Hence, the convergence potential and optimality of the solution is ensured, and the produced synthesized metric's logic is solidified. **Theorem 3** (Monotonicity of QUAKE). : _A composite metric is monotonic, provided all its synthesized elements themselves hold such an attribute._  
**Proof of Theorem 3**. : Let \(\langle PJ,L,RNR^{norm},RCR^{norm}\rangle\equiv\langle\omega_{1},\omega_{2}, \omega_{3},\omega_{4}\rangle\), \(l_{1},l_{2}\in E\) and \(\omega_{1},\omega_{2},\omega_{3},\omega_{4}\in\mathbb{R}^{+}\). Then,  
\[ \begin{array}{c}\stackrel{{ Equation}}{{\rightarrow}}\omega_{1}(l_{1})+\omega_{1}(l_{2})> \omega_{1}(l_{1})\\ \stackrel{{ Equation}}{{\rightarrow}}\omega_{2}(l_{1})+ \omega_{2}(l_{2})>\omega_{2}(l_{1})\\ \stackrel{{ Equation}}{{\rightarrow}}\omega_{3}(l_{1})+ \omega_{3}(l_{2})>\omega_{3}(l_{1})\\ \stackrel{{ Equation}}{{\rightarrow}}\omega_{4}(l_{1})+ \omega_{4}(l_{2})>\omega_{4}(l_{1})\end{array} \right\}(+) \tag{20}\]  
and  
\[A,B\
i\left\{ \begin{array}{c}A=\omega_{1}(l_{1})+\omega_{2}(l_{1})+\omega_{ 3}(l_{1})+\omega_{4}(l_{1})\\ B=\omega_{1}(l_{2})+\omega_{2}(l_{2})+\omega_{3}(l_{2})+\omega_{4}(l_{2})\\ \end{array} \right. \tag{21}\]  
\[ \begin{array}{c}\stackrel{{ Equations}}{{\rightarrow}}\stackrel{{ (\ref{eq:PJ})}}{{\rightarrow}}A<A+B\equiv\omega(l_{1})<\omega(l_{1})+ \omega(l_{2})\stackrel{{ Equation}}{{\Rightarrow}}\\ A<A+B\equiv A\prec A+B\equiv\omega(l_{1})\prec\omega(l_{1}\oplus l_{2}) \left\{\begin{array}{c}\forall\omega\in\mathbb{R}^{+}\\ \forall l\in E\end{array}\right.\end{array}  \tag{22}\]  
Thus, the synthesis of the additive-over-the-path and minimize proposed metrics \(\langle PJ,L,RNR^{norm},RCR^{norm}\rangle\) is proven to be strictly monotonic, inheriting the same algebraic property of its members. Equivalently, the weighted aggregation in the form of the QUAKE metric is also strictly monotonic, since \(w_{i}>0,i=\{1,2,3,4\}\). **Theorem 4** (Isotonicity of QUAKE). : _A composite metric inherits the isotonicity property, provided all its constituent members are isotonic themselves._  
**Proof of Theorem 4**. : Let \(\langle PJ,L,RNR^{norm},RCR^{norm}\rangle\equiv\langle\omega_{1},\omega_{2}, \omega_{3},\omega_{4}\rangle\), \(l_{1},l_{2},l_{3}\in E\) and \(\omega_{1},\omega_{2},\omega_{3},\omega_{4}\in\mathbb{R}^{+}\). Then,  
Table 4 shows which applications of the studied categories were documented in the literature to operate within low, moderate, and high bandwidth values. It is of great interest that XR-based applications utilize significantly greater data rates than other modern technologies, like telesurgery and teleportation of both ACVs and UAVs. Finally, Table 5 summarizes the computational demand boundaries of the examined application groups. It can be distinguished that the modern application categories follow the same pattern for CPU usage ranges as for BW. After gathering the necessary ranging thresholds and classification groups illustrated in Tables 2-5, the materialization of the data into functional weights for the \(PJ\), \(L\), \(BW\), and \(CPU\) measurements takes place. Table 6 illustrates a variety of combinations of the weights \(w_{1},w_{2},w_{3},w_{4}\) of the QUAKE metric.