For over 130 years, the Lemoine Conjecture has stood as a resilient mystery in additive number theory, verified by computers but missing a definitive formal proof. This manuscript settles the conjecture by introducing the Lemoine Predictive Process, a constructive method that transforms the search for prime partitions from a trial-and-error problem into a guided logical chain. We identify a governing principle within modular arithmetic that ensures every failed test case inherently contains the mathematical key to the next step. By formalizing this process in the Lean 4 theorem prover, we demonstrate that a dead end is logically impossible, as it would require a fundamental contradiction of the algorithm's recursive rules. This work provides the first mechanized, constructive proof of the conjecture, bridging the gap between historical number theory and modern formal methods.
Jonathan ƒ(n) Reed (Fri,) studied this question.