Unified Algebraic Absorption of Finite-Blocklength Penalties via Generalized Logarithmic Mapping

Abstract: In finite-blocklength information theory, evaluating the fundamental limits of channel coding typically relies on normal approximations and Edgeworth expansions, which introduce additive polynomial corrections for skewness and higher-order moments. This paper proposes an alternative approach: rather than appending external error terms to a Gaussian baseline, we absorb these finite-length penalties using a generalized $q$-algebraic framework. By introducing a dynamic scaling law $1-q_n = αn^{-1}$ for the tuning parameter, we prove that the $q$-generalized information density corresponds to macroscopic higher-order fluctuations. Specifically, by setting this scaling constant to $α= T/(3V^2)$ (where $V$ is the varentropy and $T$ is the third central moment), our framework recovers the third-order coding limit, absorbing the $O(1)$ non-Gaussian penalty without relying on Hermite polynomials. Furthermore, we demonstrate that the $k$-th degree term of our algebraic expansion matches the $O(n^{1-k/2})$ asymptotic order of the $(k+1)$-th moment Edgeworth correction. This approach unifies classical probabilistic approximations within a single algebraic structure, establishing a mathematical connection between finite-blocklength analysis and generalized logarithmic mappings.