# Theory of Faults (ToF): Numerical Quality Management in Complex Systems
## 1 Introduction  
Complex systems are not random, they are teleonomic; they evolve, react, and interact; they are directed by chance and necessity [1]. These systems are treated within general systems theory methodologies [2], albeit global quality management systems are yet to be developed [3], a task to which this article contributes. Complex projects and infrastructure facilities (complex infrastructure systems, CISs) are protected from aleatoric risk by **barriers**: physical ones, such as dams; behavioral ones, such as peer review; and normative ones, such as standards and ethics [4]. A plethora of probability functions describe the stochastic manifestation of aleatoric events. When barriers are penetrated, incoming aleatoric risks cause system responses, here termed \"**faults**\", which are unique to each system, perturbation, and time instance. Systems are injectively determined by their responses [5]; ergo, systems may be deciphered by faults analysis. Empirical evidence suggests that a log-normal distribution best characterizes the statistics of faults [6]. This stipulation has been corroborated here and in all our other works. **Log-normal** (LN) probability distribution is not a rare occurrence; it best fits most construction projects, designs, and infrastructure operation and maintenance (O&M). It is a product of WBS (Work Breakdown Structure), therecursive formations of infrastructure systems, and design work and rework loops [7]. The log-normal PDF is developed mathematically in this work to a power function representing faults' magnitude probability distributions. When their magnitude distributions are described by power functions, phenomena are termed \"complex\". Therefore, complexity is a trait of the construction industry. **Complexity** is manifested on the one hand in the phenomena of Chaos and on the other in trajectories, emergence, and universality. The butterfly effect, for one, stands for small perturbations that cause unpredictable, at times unlimited, effects recreated here in numerical experiment A.1. Black swan events are events of extreme magnitude that cannot be predicted from experience; a black swan event was encountered here in project 19. Emergent systemic behaviors arising from local interactions, such as risk affiliation and conservativeness, are described here. Also of interest are the phenomena of avalanches, which are energy dissipation events that progress unstopped until they reach the ends of networks. The phenomena of avalanches are presented here in project 1. A phase parameter and phase change are discussed here in detail. Here, we portray avalanches as a cause of phase changes. Phase changes may be directly predicted [8], at least for physical systems; in this work, we direct our efforts to bifurcation points and the trajectory prediction of avalanches. Non-linearity, a hallmark of complexity, disrupts supply chains and undermines meticulous software-aided time management efforts [9]. Non-linearity is also generated by small-world interconnected networks that import risk, by sociological factors, and by signal processing effects. These effects are observed here, too; the case studies relate them to the construction industry. Modeling complex systems is precarious; we portray the limitations on modeling in the Discussion section and as pertinent to the development section and experiment A.1; in the context of CPM, we show the imminent collapse of modeling [9]. All systems [10], physical and behavioral alike, face aleatoric and epistemic uncertainties. The aleatoric risks are related to the unpredictability and the statistical nature of the events perturbating systems; the epistemic risk is caused by a lack of data and modeling shortcomings. In complex systems, the unpredictability of perturbation does not entail an unpredictability of outcome, and stochastic perturbation may be useful for systemic learning [11]. Following a behaviorist path, we address modeling shortcomings by deciphering the world, not modeling it [12].