# Theory of Faults (ToF): Numerical Quality Management in Complex Systems
## 1 Introduction
Info-systems act as suppliers of feedback, practices, and up-to-date code, drilling systems and performing audits. In the Time section, we lay out mathematics that shows how corrective means may proliferate malicious effects. Corrective means are relayed to Case Study 1 and remedial actions to Case Study 2; the concept of learning and feedback is developed in detail. The **mitigative means** suggested in the literature for faults are behavioral; the application philosophies are of two primary schools: human in the loop and automated. The automated response is based on sensor networks, some wireless, some IT [30], and the other school promotes personnels' energetic, rapid response, with open-mindedness [31] and a trouble-shooting culture [32; 33]. We believe the two disciplines are complementary. Both are implemented here; the degree of automation is dependent on entropy and code. Usually, there is time to stop pernicious events before they avalanche. Physical faults propagate in physical networks, while the corrective means are developed in information networks. The Debora factor, which is the relative reaction time of the two systems, is the deciding factor. An example of the work of the two systems is the notorious pedestrian bridge collapse over SW 8 th Street, Miami [34]; it is explained by our A.4 numeric experiment. The graphical abstract (Figure 1) tracks the trajectory of faults in complex construction systems; it stipulates a system approach [10]. Projects and infrastructures, whether in design or maintenance, construction, or facade management, are viewed here as systems. All the modeled systems display similar topologies with common topological traits. Similar topologies are a cornerstone of the general systems theory and the grounds of the universality of our theoretical framework and of the application of similar means to all disciplines [10]. On the other end, general systems theory implies being part of ever more extensive systems that impose their effects and eventually limit the effectiveness of managerial planning and of our models [35]. The similarities of the failure modes, propagation, and avalanche of, for instance, power supply lines and road traffic, leave much room for expertise in relaying mathematics to the specifics of each discipline--in this example, a blackout vs. a roadblock or transformer fires vs. car accidents. The multiplication and propagation of faults on temporal maps is portrayed in numerical experiment A.2 in activity space, in numerical experiment A.3 in faults space, in numerical experiment A.4 in designer space, in Case Study 1 for open faults and in projects space, and in Case Study 2 in design discipline space. In summation of the above, the entities explored are **complex systems**. Complex systems are protected from aleatoric risk by **barriers**. When barriers are penetrated, systems respond by **faults**. The statistics of faults are characterized by a log-normal PDF and **power function** magnitude probability distributions. Complex systems may be presented as **scale-free networks** of local variables. These systems are **information systems**; they generate threshold phenomena, and their threshold is the **IC**. Data signal processing augments the information load. Our goal is to provide databases for data analysis and a generate theoretical understanding and a means of practical applicable **quality management**. This manuscript **contributes** to bridging both informational and theoretical gaps. We provide a mathematical analysis of the effects of risk in complex systems, conduct numerical experiments to investigate and predict systems' response, link the theory to the reality of design and construction management using case studies, and provide support for mitigation, and corrective actions. ToF frameworks offer a means to locate barrier penetrations and suggest corrections, a post factum responsibility allocation, claims, and theoretical research. In this article, we apply a men-in-the-loop paradigm.