Geometric Decomposition of Mersenne Exponents: A Novel Additive Representation System

We introduce a novel representation system where each Mersenne exponent can be expressed as a sum of smaller, unique Mersenne exponents. Through exhaustive analysis of all 52 known Mersenne exponents, we reveal a structured decomposition pattern with three exceptional exponents that defy the unique-sum property. The decomposition exhibits geometric properties, digital root patterns, and hierarchical organization suggesting an underlying lattice structure in the distribution of Mersenne primes. In this work, the classical one‑dimensional ordering of the integers is replaced by a two‑dimensional spiral embedding. Instead of viewing exponents as lying on a straight line, we map each exponent to a point with polar coordinates , where is a monotone function of (such as ) and encodes modular or digital information (for example, remainders modulo selected Fibonacci numbers). This simple change of viewpoint transforms additive and modular patterns into visible geometric features such as rays, layers, and defects. In particular, the Mersenne exponents and certain Collatz orbits form a structured lattice on this spiral, revealing regularities that are difficult to detect on the usual number line.