A Proof of the Logical Impossibility of Formulating the Riemann Hypothesis as a Mathematical Proof Problem

Abstract This paper addresses the Riemann Hypothesis (RH), which has long been sought in the form of a “proof,” by establishing—exclusively and at the level of first principles—that the traditional formulation of RH as a proof problem is itself not logically viable. 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. Furthermore, 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. This selection is not definitional, heuristic, or arbitrary; it is enforced directly by the universality requirement itself. Because these results are completed in a logical form that is not deniable, the following consequences are fixed: 1. If this conclusion remains non-deniable as stated, it constitutes a major turning point for mathematics. 2. If this conclusion can be denied within an equally rigorous logical system, that system itself would constitute a major turning point for mathematics. In this sense, the present work explicitly delineates an unavoidable historical direction of discussion. The reader is therefore encouraged to judge this fact directly by engaging with the paper itself. In doing so, this inescapable line of reasoning opens new possibilities and discoveries across mathematics and mathematical physics.