## 4 Framework Dimensions and Measurement
### Destruction Resistance Index (R)  
The Destruction Resistance Index (R) evaluates a system's ability to endure and maintain functionality under sustained adversarial pressure by focusing on absorbing damage through resource redundancy and defensive countermeasures. The index captures the dynamic tension between a system's internal strengths and external threats. It evolves over time to reflect both sudden shocks and gradual erosion, allowing it to quantify endurance against both acute and chronic stressors.  
Theoretically, R(t) is modeled as a nonlinear survival function integrating resource resilience and adversarial dynamics:  
\[R(t)=R_{0}\cdot e^{\beta t}\big{(}\alpha\cdot\frac{S_{\text{path}}}{S_{\text{ crit}}}-\beta\cdot\frac{C_{\text{attack}}}{\text{c}_{\text{defense}}}\big{)}dt \tag{5}\]  
Here, \(R_{0}\) is the baseline resilience coefficient (tied to initial redundancy/fault tolerance). The integral aggregates a growth factor from resource redundancy (\(\alpha\cdot\frac{S_{\text{path}}}{S_{\text{crit}}}\), where \(S_{\text{redu}}\) is excess capacity over critical threshold \(S_{\text{crit}}\), and \(\alpha\) is calibrated by empirical data on distributed systems) and a decay factor from adversarial pressure (\(\beta\cdot\frac{C_{\text{attack}}}{\text{c}_{\text{defense}}}\), where \(C_{\text{attack}}\) quantifies attack complexity (e.g., via MITRE ATT&CK tactics) and \(C_{\text{defense}}\) reflects defensive coverage (0-1), with \(\beta\) balancing attack sophistication against defense breadth based on historical incident data). The exponential form captures nonlinear degradation when defensive gaps widen. The integral allows \(R(t)\) to account for both instantaneous impacts and cumulative wear (e.g., DDoS resource exhaustion), offering a dynamic alternative to static metrics like MTBF by incorporating adversarial adaptation and system fatigue.  
Practically, R is operationalized through measurable indicators for resource redundancy, defense coverage, and system availability under stress, using mature technologies. Resource redundancy is assessed by indicators like redundant component availability (percentage of critical systems with backups/failover, measured via tools like Zabbix, Prometheus, or Nagios using system logs/CMDBs) and dual-active system switchover success rate (from load balancer logs like F5/HAProxy or drill reports). Defense coverage is quantified by the MITRE ATT&CK coverage ratio, comparing implemented defenses to known tactics using data from SIEMs (e.g., Splunk, Elastic Stack) by mapping controls to attack patterns via security policies and audit logs. Flexibility in tooling is prioritized.  
These indicators inform the computation of \(S_{\text{redu}}\), \(S_{\text{crit}}\), \(C_{\text{attack}}\), and \(C_{\text{defense}}\) for estimating \(R(t)\). Raw metrics are collected, normalized (e.g., to a [0, 1] scale), and combined via weighted ratios aligned with the model. For instance, 90% redundant component availability and a 0.8 MITRE ATT&CK coverage normalize to 0.9 and 0.8, respectively, for integration. This methodology ensures practical, adaptable measurements leveraging existing systems.