Riemann Hypothesis and the Algebra of Functional Symmetry.

This work proposes an investigation that navigates the boundaries between pure mathematics, theoretical physics, and a novel symbolic structuring of knowledge. Emerging from the conception and development of the operator $J^*$ — whose mathematical nature appears to touch deeper layers of the structural regularity of the universe — a formal framework is outlined, capable of engaging with some of the most persistent questions of modern science: the distribution of prime numbers, the limits of the zeta function, the structure of time, and the behavior of singularities. Approaching the Riemann Hypothesis from a symbolic-functional perspective, the work does not claim to offer a singular, definitive answer. Instead, it presents an alternative interpretative path — rigorous, yet open — which respects the formal foundations of mathematics while extending them into new conceptual territories. In this process, formulas, symmetries, and interpretations arise whose validity rests not only on numerical accuracy, but on structural coherence and the potential for integration across multiple layers of mathematical and physical description. It is not merely about calculating the number of primes more precisely — though that is also achieved — but about transforming the language through which we understand the notions of regularity, chaos, symmetry, and rupture. The operator $J^*$, in this sense, appears to build a bridge between regularization and revelation, between the finite and the asymptotic, between physical time and symbolic time. The text, written in both Portuguese and English, moves through varying degrees of formalism, containing rigorous proofs, computational simulations, and a theoretical structure that invites further expansion. Authored by a young 18-year-old researcher, this article may be read both as a technical proposal and as a symbolic reinterpretation manifesto of deep mathematics. For contact or collaborations, please write to: 📧 [email protected]