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Abstract

In the present paper we give a principled derivation of Elias' Omega code by combining a constrained variational formulation of prefix coding with a renormalization flow on codeword distributions. Starting from a Lagrangian that minimizes average code length under the Kraft-McMillan constraint, we show that the implied distribution is a fixed point of a coarse-graining map, yielding the canonical iterated logarithm length, up to an additive constant. This establishes completeness and asymptotic optimality, and connects universal integer coding with coarse-grained entropy, uncertainty-type bounds, and multiplicity relations familiar from statistical physics. The renormalization operator induces a discrete flow that converges to the Elias fixed point for any admissible initialization, up to a bounded error, offering a clean bridge between information-theoretic constraints and RG-style scale invariance.