"AI psychosis" or "delusional spiraling" is an emerging phenomenon where AI chatbot users find themselves dangerously confident in outlandish beliefs after extended chatbot conversations. This phenomenon is typically attributed to AI chatbots' well-documented bias towards validating users' claims, a property often called "sycophancy." In this paper, we probe the causal link between AI sycophancy and AI-induced psychosis through modeling and simulation. We propose a simple Bayesian model of a user conversing with a chatbot, and formalize notions of sycophancy and delusional spiraling in that model. We then show that in this model, even an idealized Bayes-rational user is vulnerable to delusional spiraling, and that sycophancy plays a causal role. Furthermore, this effect persists in the face of two candidate mitigations: preventing chatbots from hallucinating false claims, and informing users of the possibility of model sycophancy. We conclude by discussing the implications of these results for model developers and policymakers concerned with mitigating the problem of delusional spiraling.