This paper presents a novel deterministic approach to analysing the behaviour of Mersenne numbers Mₚ = 2ᵖ - 1 in the Collatz process. The study is based on a detailed investigation of the binary structure of numbers, introducing the concept of binary block algebra. It is proven that during the pure growth phase, the binary representation of Mersenne numbers preserves a strict structural invariant: a continuous block of ones (the tail) is truncated by exactly one bit per odd step. The peak value is derived explicitly as Pₚ = 2 * 3^ (p-1) - 1, and a hierarchical reduction rule is established: each peak Pₚ ultimately leads to a smaller peak P₂₄₈₋ ( (+₂) /₄), guaranteeing convergence to 1 for all Mersenne numbers. The paper also introduces the matryoshka principle (regeneration of smaller Mersenne sub-peaks in high-order bits) and the binary vise method (mirror induction bounding the most significant bit). Computational modelling for M₅ = 31 confirms all theoretical predictions. The work does not claim a full proof of the Collatz conjecture but offers a rigorous description for the entire class of Mersenne numbers and their generalisations, and provides new tools for further research.
Emma Helmdach (Fri,) studied this question.