Why the Bethe Ansatz Works: A Structural Explanation via Interaction Propagation

The Bethe Ansatz provides exact solutions for certain interacting quantum many-body systems, yet its success is confined to narrow regimes and breaks down abruptly outside them. Despite extensive developments in integrable systems, a structural explanation of this phenomenon has remained elusive. In this paper we give a representation-independent account of both the existence and the failure of Bethe-type exact solvability. We identify a single governing mechanism: the behaviour of interaction propagation. For systems in which propagation terminates after finite depth without encountering structural boundaries, global interaction data factor through finitely many local components, forcing Bethe-type solvability. Conversely, once a structural boundary is encountered, irreducible interaction data arise that obstruct such finite factorization and preclude Bethe-type solutions. This yields a sharp structural dichotomy. Within this regime, exact solvability is not an analytic accident but a rigidity phenomenon, while its failure is governed by intrinsic boundary formation. In this way, the Bethe Ansatz is understood as a consequence of constrained interaction propagation rather than as a model-specific construction.