We propose a unified mathematical framework for rate-distortion theory, lattice quantization, and modern error-correcting codes by emphasizing their variational and convex-analytic structure. First, we establish a Gibbs-type variational formulation of the rate-distortion function and show that optimal test channels form an exponential family, with Fullback-Leibler divergence acting as a Bregman divergence. This yields a generalized Pythagorean theorem for projections and a Legendre duality that couples distortion constraints with inverse temperature parameters. Second, the reverse water-filling metaphor is extended to distributed lattice quantization, deriving distortion allocation bounds across eigenmodes of conditional covariance matrices. Third, inference is formalized as decoding by showing that belief propagation in LDPC ensembles and polarization in polar codes can be interpreted as recursive variational inference procedures. These results unify compression, quantization, and decoding as convex projections of continuous information onto discrete manifolds. Extensions to neural compression and quantum information are sketched as corollaries, illustrating the universality of the framework. Illustrative connections to other scientific fields are also presented. Finally, complementary numerical examples and scripts are located in the appendix

翻译：我们通过强调其变分与凸分析结构，提出了一个统一率失真理论、格点量化与现代纠错码的数学框架。首先，我们建立了率失真函数的吉布斯型变分形式，并证明最优测试信道构成指数族，其中Fullback-Leibler散度充当Bregman散度。这导出了投影的广义勾股定理以及将失真约束与逆温度参数耦合的Legendre对偶性。其次，将反向注水隐喻扩展至分布式格点量化，推导出条件协方差矩阵特征模态间的失真分配界。第三，通过证明LDPC码集合中的置信传播与极化码中的极化过程可解释为递归变分推断过程，将推断形式化为解码。这些结果将压缩、量化与解码统一为连续信息到离散流形上的凸投影。作为推论，概述了该框架在神经压缩与量子信息中的扩展，以阐明其普适性。同时展示了与其他科学领域的示例性联系。补充数值示例与脚本详见附录。