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.

翻译：传统上，素数通过数论、概率论和分析框架进行研究。本文提出素数事件语言的概念，将算术现象表示为符号事件序列，并利用信息论和随机过程工具进行分析。利用截至 $N = 5 \times 10^9$ 的所有素数（234,954,223 个素数），我们基于孪生素数出现事件构建事件语言，并记录素数间隔事件。通过有限阶马尔可夫模型、训练/测试验证、互信息分析和信息视界测量，研究其统计性质。对于孪生素数事件语言，一阶马尔可夫建模将测试集交叉熵从 0.325350 比特降低至 0.319949 比特，信息增益约 0.0054 比特。该增益经受了样本外验证，反映的是真实统计结构而非过拟合。互信息分析独立验证了马尔可夫模型的结果，并表明可测依赖几乎完全集中在滞后 1 处。互信息从滞后 1 处的约 $5.96 \times 10^{-3}$ 比特降低至滞后 2 处的约 $5.07 \times 10^{-7}$ 比特（约降低 11,700 倍），降幅超过四个数量级。滞后 2 之后，残差信息在统计噪声基底附近波动。这些结果表明，素数事件语言既非完全无记忆性，亦非强可预测性，而是展现出以 1 阶依赖和约一个事件的有效信息视界为特征的弱但可复现的短程统计结构。