An Infinitesimal Calculus of Sustained Engagement: A Dynamical Theory of Novelty, Passion, Cost Revelation, and Admissibility

This manuscript introduces a rigorous deterministic and stochastic calculus for modeling sustained human engagement. Whether applied to a career, a relationship, an intellectual pursuit, or an identity-bearing project, engagement typically begins under incomplete visibility—where initial rewards and novelty are high, but long-term maintenance burdens remain hidden. This paper formalizes the transition from "novelty excitation" to "cost revelation" through the lens of continuous-time dynamical systems. The framework defines a central Engagement Functional, modeled as a time-indexed balance between accumulated reward, intrinsic passion, revealed maintenance cost, and accumulating entropy (fatigue). By examining the infinitesimal variation of this functional, the theory provides exact mathematical criteria for distinguishing fundamentally different states of engagement. It mathematically isolates novelty-dominant collapse (burnout), addiction-like persistence (withdrawal-stabilized continuation), identity-subsidized endurance, and the apex state of mastery-coupled engagement. To account for environmental noise and unpredictable systemic friction, the deterministic core is extended using Itô Calculus and Stochastic Differential Equations (SDEs). This allows for the precise definition of "stochastic fragility," proving how volatility accelerates the ruin of systems operating too close to their collapse boundaries. The framework further introduces optimal control theory to model human agency, framing effort and pacing as an optimization problem designed to maximize survival over an infinite horizon. Finally, the theoretical phase transitions between novelty-driven burnout and mastery-coupled survival are computationally verified through a finite-dimensional Python/JAX simulation utilizing the Euler-Maruyama method. Ultimately, this work proves a singular governing physical law of psychology: true passion is not the absence of cost, but the structural phenomenon where marginal alignment growth strictly exceeds marginal cost revelation over time.