On the Information Physics of Primes: Foundations of the Mersenne Chain Reactor

Abstract We introduce a physical law for Mersenne numbers through the Mersenne Chain Reactor (MCR): prime means saturation, and otherwise composite. This is the central result of the paper. In the reactor, the answer is not obtained by ordinary divisibility tests or step-by-step arithmetic checking. Instead, the system evolves through discrete temporal states, and the final primality verdict appears as a physical state of the evolving medium. The reactor consists of interacting states moving inside a cyclic topology under deterministic local rules. At the critical time, the system reveals its truth. If full topological saturation is reached, the Mersenne number is prime. If voids remain, it is composite. Thus, the distinction between prime and composite is expressed physically: prime corresponds to complete saturation, while composite corresponds to unsaturated structure. This gives Mersenne primes a new interpretation within reaction systems and natural computing. More broadly, we argue that this result is a first step toward a general physics of prime numbers. The reason is simple: physics does not flatter Mersenne primes. If a genuine physical law emerges here, then this may be the beginning of a wider scientific framework in which primality is studied through physical evolution, interaction, and information flow.