This preprint introduces the Axiomatic Convergence Hypothesis (ACH): an observational claim about convergence behavior in generative systems under fixed external constraint regimes. The paper defines “axiomatic convergence” as a measurable reduction in inter-run and inter-model variability when generation is repeatedly performed under stable invariants and evaluation rules applied consistently across repeated trials. The contribution is a phenomenon-and-protocol disclosure only. It provides: (i) a definition and taxonomy distinguishing output convergence from structural convergence, (ii) a set of falsifiable predictions concerning convergence signatures (e.g., relaxation-like variance decay, threshold effects, hysteresis/path dependence, and universality-class behavior), and (iii) a replication-ready experimental protocol for testing ACH across models, tasks, and domains. This publication intentionally does not disclose any proprietary controller architecture, enforcement mechanism, update rule, persistence/canonization mechanism, memory partitioning design, or operational implementation. The protocol is presented at an observational and measurement level to support independent replication and evaluation using any constraint regime consistent with the category-level template described in the paper.