Abstract This paper proves the following claim: that formulating the Riemann Hypothesis (RH) as a proof-theoretic mathematical problem is logically impossible. The impossibility established here is neither methodological, historical, nor sociological. It is structural and logical: the very form of the problem, as traditionally posed, cannot in principle support the kind of universal determination that a mathematical proof requires. In particular, when RH is formulated within any framework that demands universal validity, the available modes of verification become intrinsically asymmetric: a single counterexample can refute the statement, yet no accumulation of positive instances—numerical, analytical, or structural—can establish its truth. This asymmetry does not reflect a limitation of current techniques, axioms, or computational power. It follows necessarily from the logical structure of universal propositions themselves. The results derived are as follows: 1. RH is a universal proposition. 2. Finite verification can be used only for counterexample detection. 3. A Weil–Bochner–type reformulation takes a quadratic form Q(f) as the object of judgment. 4. This formulation requires non-negativity for all admissible f. 5. This requirement cannot be logically weakened. As a consequence, RH must be released from the binary classification of “proved” versus “not proved.” What is terminated here is not a conjecture, but the legitimacy of treating RH as a conventional object of proof. This work therefore fixes a definitive boundary: beyond this boundary, RH admits no further development as a proof problem, regardless of refinement in method, axiomatic extension, or analytical sophistication. At the same time, this paper does not merely end in negation or reservation. In the course of fixing the impossibility of the conventional problem-form, it derives a clear and exclusive structural answer to the question RH always carried at its core: what primes are. Among structures extractable from the natural numbers, we show that the only structures capable of maintaining logical closure are non-degenerate binary interference structures, which coincide exactly with the prime numbers. If this result stands as non-deniable, then it constitutes a decisive turning point in mathematics. If, on the other hand, this result is shown to be deniable within some alternative logical framework, then the existence of such a framework would itself constitute an equally decisive turning point. In either case, the consequences established here cannot be dismissed. What has been fixed is not an opinion, but a boundary: a point beyond which the traditional proof-theoretic treatment of the Riemann Hypothesis cannot proceed unchanged. Accordingly, this work does not terminate discussion. It determines, irreversibly, the direction in which further discussion must occur. Engagement with this result is therefore unavoidable. Whether by acceptance or by refutation, the mathematical community must now confront the structural facts established here.