AI systems rest on software with low integrity mechanisms, leaving AI systems exposed across every stage from data acquisition to final inference. This paper makes the AI supply chain a first-class object of analysis, decomposing it across four architectural layers: data acquisition, model training, model inference, and a cross-cutting substrate. Within these layers, we identify four structural gaps that traditional supply chain mechanisms do not address: verifiability, versioning, observability, and traceability.Current AI systems fall short on all of them: they carry undeclared behavioral couplings that no resolver enforces; they cannot be reverted back to known working assemblies; they degrade silently rather than surfacing breaking changes; and their lineage can hardly be approximated. To illustrate the scale of the software supply chain of AI, we measure a reference stack of 48 production-grade open-source projects, which declares 4,664 direct dependencies, resolves to 11,508 transitive packages, and totals roughly 392M lines of code.