Functional connectivity (FC) is one of the most common inputs to fMRI-based predictive models, due to a combination of its simplicity and robustness. However, there may be a lack of theoretical models for the generation of FC. In this work, we present a straightforward decomposition of FC into a set of basis states of sine waves with an additional jitter component. We show that the decomposition matches the predictive ability of FC after including 5-10 bases. We also find that both the decomposition and its residual have approximately equal predictive value, and when combined into an ensemble, exceed the AUC of FC-based prediction by up to 5%. Additionally, we find the residual can be used for subject fingerprinting, with 97.3% same-subject, different-scan identifiability, compared to 62.5% for FC. Unlike PCA or Factor Analysis methods, our method does not require knowledge of a population to perform its decomposition; a single subject is enough. Our decomposition of FC into two equally-predictive components may lead to a novel appreciation of group differences in patient populations. Additionally, we generate synthetic patient FC based on user-specified characteristics such as age, sex, and disease diagnosis. By creating synthetic datasets or augmentations we may reduce the high financial burden associated with fMRI data acquisition.