The classical realization problem asks which rings can appear as endomorphism rings of abelian groups. While Baer initiated this question and Corner's theorem resolved it broadly for countable reduced torsion-free groups, the finite rank case remains deeply incomplete. Specifically, we do not have a clean characterization of exactly which rings arise as endomorphism rings when we restrict to torsion-free abelian groups of finite rank, a setting that is far more constrained and combinatorially delicate than the general case. The question sits at the boundary of ring theory, group theory, and model-theoretic definability, making it hard to attack from any single direction. The finite rank restriction forces interactions between the additive and multiplicative structure of the ring that do not arise in infinite rank constructions, and many standard tools like the Black Box or other prediction principles are unavailable or weakened. Current results handle special classes such as strongly indecomposable groups or rings satisfying particular flatness or freeness conditions, but a unified necessary and sufficient condition is missing. Resolving this problem would give logicians and algebraists a precise dictionary between ring-theoretic properties and group-theoretic realizability in the finite rank world, with consequences for the model theory of modules, the classification of Butler groups, and definability questions in second-order arithmetic. It would also clarify which algebraic invariants are genuinely detectable by endomorphism algebras versus which are invisible to them, a distinction relevant to both pure algebra and computable structure theory.