Prime Event Languages: An Information-Theoretic Investigation of Twin-Prime Event Structure

Abstract

Prime numbers are traditionally studied through numerical, probabilistic, and analytic frameworks. In this work, we introduce the concept of a prime event language, in which arithmetic phenomena are represented as symbolic event sequences and analyzed using tools from information theory and stochastic processes. Using all primes up to $N = 5 \times 10^9$ (234,954,223 primes), we construct event languages based on twin-prime occurrences and record prime-gap events. We investigate their statistical properties through finite-order Markov models, train/test validation, mutual-information analysis, and information-horizon measurements. For the Twin Prime Event Language, first-order Markov modeling reduces test-set cross entropy from 0.325350 bits to 0.319949 bits, corresponding to an information gain of approximately 0.0054 bits. This gain survives out-of-sample validation and therefore reflects genuine statistical structure rather than overfitting. Mutual-information analysis independently confirms the Markov results and shows that measurable dependence is concentrated almost entirely at lag 1. The mutual information decreases from approximately $5.96 \times 10^{-3}$ bits at lag 1 to approximately $5.07 \times 10^{-7}$ bits at lag 2 (approximately 11,700-fold reduction), representing a reduction of more than four orders of magnitude. Beyond lag 2, residual information fluctuates near the statistical noise floor. These results indicate that prime event languages are neither perfectly memoryless nor strongly predictable. Instead, they exhibit weak but reproducible short-range statistical structure characterized by first-order dependence and an effective information horizon of approximately one event.