The Bandwidth Collapse Theorem – A Universal Structural Limit for Hierarchical Interconnect Architectures

This work introduces the Bandwidth Collapse Theorem, a formal and general result establishing a fundamental upper bound on the effective per-pair bandwidth in hierarchical interconnect architectures. We prove that, independently of the underlying technologies, the observable bandwidth asymptotically converges to that of the weakest level of the hierarchy. The result is extended to arbitrary multi-level systems, covering modern AI training clusters, HPC systems, PCIe fabrics, network-on-chip architectures, and hyperscale datacenters. A set of derived corollaries establishes the implications of this structural limit across performance, energy consumption, collective communication time, and cost scaling. In particular, we prove the Collapse Irreducibility Theorem, demonstrating via max-flow/min-cut arguments that no software optimization, routing strategy, or communication protocol can asymptotically eliminate the phenomenon. The theorem thus defines an architectural invariant — a conservation law governing the design of large-scale computing systems — with direct implications for the future of AI infrastructure and accelerator-centric architectures.