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. More broadly, this work illustrates how alternative representations can reveal information-theoretic organization that remains less apparent in conventional numerical descriptions of arithmetic phenomena.

翻译：素数传统上通过数论、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，并运用信息论与随机过程工具进行分析。利用$N = 5 \times 10^9$以内的所有素数（共计234,954,223个素数），我们构建了基于孪生素数出现与素数间隔记录的事件语言，并通过有限阶马尔可夫模型、训练/测试验证、互信息分析及信息视野测量研究其统计特性。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，对应约0.0054比特的信息增益。该增益在样本外验证中保持稳定，反映了真实的统计结构而非过拟合。独立进行的互信息分析验证了马尔可夫模型结果，显示可测量依赖性几乎完全集中于滞后1阶：互信息从滞后1阶的约$5.96 \times 10^{-3}$比特下降至滞后2阶的约$5.07 \times 10^{-7}$比特（约11,700倍衰减，降幅超过四个数量级）。在滞后2阶之后，残余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，也不具备强可预测性；相反，它们展现以弱但可复现的短程统计结构为特征，该结构体现为一阶依赖性与约一个事件的有效信息视野。更广泛而言，本研究揭示了替代表征方式如何能展现传统算术现象数值描述中难以显现的信息论组织规律。