Mersenne Block Dynamics: A Framework for the Collatz Conjecture

This paper introduces Mersenne Block Dynamics, a structural framework for analyzing the accelerated Collatz or Syracuse map on odd integers. The approach decomposes orbits based on the 2-adic valuation of the successor of an odd integer, effectively measuring the length of the trailing run of ones in its binary expansion, termed the Mersenne tail. This decomposition partitions the dynamics into deterministic blocks where the tail length decreases by exactly one bit at each step, creating a rigid wedge pattern in the binary representation. The framework defines a coarse-grained block map that transitions directly between the starts of successive blocks, isolating all arithmetic complexity into a specific exit exponent. The study derives explicit closed-form transition identities and exact time-scale bookkeeping for these block jumps. Furthermore, it establishes that the block length and exit parameters follow independent geometric distributions in terms of natural density. Under a heuristic assumption of orbit mixing, this intrinsic statistical model predicts a net negative expected logarithmic drift, recovering the classical probabilistic prediction for the Collatz conjecture within a precise structural coordinate system.