Symbolic Geometry of the Number <em>π</em>: Structures, Statistics, and Security

This paper presents a new methodological approach to the analysis of numerical sequences that are commonly considered random. This includes the decimal expansion of the number π, stock market indices (e.g., Belex15), pseudorandom numbers (PRNG), cryptographically secure pseudorandom numbers (CSPRNG), physical random number generators (RNG), and quantum random numbers (QRNG). The core method is based on hierarchical computation of higher-order differences and symbolic transformation of signs, enabling structural encoding of each sequence into a symbolic space. The primary objective is to determine whether the decimal expansion of π and related sequences exhibit the same distribution of symbolic patterns as the theoretical model of variations with repetition. The analysis is extended to sequences of 4 and 5 digits, including higher-order differences such as third and fourth order. The results show that empirical distributions of these multilayer structures in the digits of π closely correspond to theoretical distributions derived from all possible variations with repetition. This method opens new possibilities for applications in number theory, cryptography, statistics, and classification of algorithmically generated sequences.