Formalization Boundary Theory: Computing the Limits of Convergence-Based Description

This preprint introduces Formalization Boundary Theory (FBT), a computational framework for locating the upper bound of convergence-based description in any dynamically stable system. Given a definable equilibrium condition and update rule, FBT derives max—the maximum step size for which convergent formalization is guaranteed—from the eigenstructure of the system's linearized dynamics. Beyond κmax, formalization necessarily fails; the region outside constitutes an irrecoverable structural remainder. The framework was applied to over 48 systems across physics, economics, mathematics, cognitive science, and network science without breakdown. The convergence condition derived for thermal diffusion coincides exactly with the von Neumann stability criterion, providing independent external validation. This paper is currently under consideration for publication in a peer-reviewed journal. Comments and feedback are welcome. Related updates may be available at: patreon.com/NMStructuralTheoryLab