We develop a statistical mechanics framework for prefix coding based on variational principles, renormalization, and quantization. A Lagrangian formulation of entropy-optimal encoding under the Kraft-McMillan constraint yields a Gibbs-type implied distribution and completeness of the optimal code. A renormalization operator acting on codeword distribution laws produces a coarse-graining flow whose fixed points have iterated-log structure; discrete quantizations of these fixed points include Elias' $ω$ code as a special case. Extending the theory to mixed discrete-continuous source laws, we show how continuous codelength functions can be quantized into countable prefix codes and derive resolution-adjusted entropy bounds together with Heisenberg-type and Boltzmann-type relations. This provides a unified and physically motivated view of universal coding, with Elias' $ω$ code as a guiding example.

翻译：我们基于变分原理、重整化与量子化，为前缀编码建立了一个统计力学框架。在克拉夫特-麦克米伦约束下，熵最优编码的拉格朗日量表述导出了一个吉布斯型的隐含分布以及最优码的完备性。作用于码字分布律上的重整化算子产生了一个粗粒化流，其不动点具有迭代对数结构；这些不动点的离散量子化以 Elias 的 $ω$ 码作为一个特例。将该理论推广到离散-连续混合信源律，我们展示了连续码长函数如何被量子化为可数前缀码，并导出了分辨率调整的熵界以及海森堡型与玻尔兹曼型关系。这为通用编码提供了一个统一且具有物理动机的视角，并以 Elias 的 $ω$ 码作为指导性示例。