A Uniqueness Theorem for Distributed Computation under Physical Constraint

Abstract

Foundational models of computation often abstract away physical hardware limitations. However, in extreme environments like In-Network Computing (INC), these limitations become inviolable laws, creating an acute trilemma among communication efficiency, bounded memory, and robust scalability. Prevailing distributed paradigms, while powerful in their intended domains, were not designed for this stringent regime and thus face fundamental challenges. This paper demonstrates that resolving this trilemma requires a shift in perspective - from seeking engineering trade-offs to deriving solutions from logical necessity. We establish a rigorous axiomatic system that formalizes these physical constraints and prove that for the broad class of computations admitting an idempotent merge operator, there exists a unique, optimal paradigm. Any system satisfying these axioms must converge to a single normal form: Self-Describing Parallel Flows (SDPF), a purely data-centric model where stateless executors process flows that carry their own control logic. We further prove this unique paradigm is convergent, Turing-complete, and minimal. In the same way that the CAP theorem established a boundary for what is impossible in distributed state management, our work provides a constructive dual: a uniqueness theorem that reveals what is \textit{inevitable} for distributed computation flows under physical law.