Elias' Encoding from Lagrangians and Renormalization

In the present paper we give a derivation of Elias' Omega code from physics principles by combining a constrained variational formulation of prefix coding with a renormalization flow on codeword distributions. Starting from a Lagrangian that minimizes average codelength under the Kraft-McMillan constraint, we show that the implied distribution is a fixed point of a coarse-graining map, yielding the canonical iterated log-sum length, asymptotically up to an additive constant. This establishes completeness and asymptotic optimality, and connects universal integer coding with coarse-grained entropy, uncertainty-type bounds, and entropy relations familiar from statistical physics.