Combinatorics and Graph Theory

EO-45 (Essence-Orbit 45) Model: Complete Formalization of 52-Element Dynamics in an 8×7 Static Grid

Authors: Nikolai Makeev
Comments: 26 Pages. In Russian

This work presents the complete formalization of a 52-element dynamical system within a static 8×7 grid — a deterministic system with historical roots. Using the developed Immersion Analysis Method (IAM), we reveal the full internal structure of the system's 90-step cycle. The system decomposes into three invariant classes: three absolutely fixed elements, two exchange pairs (4 elements) that swap at the cycle's midpoint, and a cyclic group of 45 elements (Essence-Orbit 45). We provide the exact predictive formula Element(j, n) = S[(φ(j) + n) mod 45], where S is an experimentally determined sequence of 45 elements, and φ(j) is the initial phase for cell j.All results are verified by computational experiments over 90,000 iterations. A complete Python implementation is provided, making this the first fully reproducible formalization of this system.
Category: Combinatorics and Graph Theory