Geometric Transduction and the Reinterpretation of the Riemann Hypothesis
Year: 2026
Type: preprint
Abstract: GEOMETRIC TRANSDUCTION AND THE REINTERPRETATION OF THE RIEMANN HYPOTHESIS Mobile Relational Equilibrium Between Zero and One, Transductive Memory, and the Three-Dimensional Configuration of the Zeros of the Zeta Function This article extends the application of Geometric Transduction Theory to the investigation of the Riemann Hypothesis by proposing a relational and dynamic interpretation of the critical line as a manifestation of mobile equilibrium between two fundamental reference poles: zero and one. The central proposal considers the conventional position (1/2) not as an isolated geometric reference, but as the particular configuration resulting from equality between the relational weights associated with the poles 0 and 1. When these weights are equal, equilibrium occurs at the midpoint of the interval. When their relative contributions change, the equilibrium position shifts toward the pole with greater weight. The article introduces a mathematical architecture based on variable relational weights, a mobile equilibrium function, a dynamic critical line, a transductive real coordinate, and an expanded three-dimensional representation of the zeros of the Riemann zeta function. This framework incorporates geometric memory as a record of changes in relational weights, equilibrium displacement, and the transformation trajectories of the system. The investigation also integrates the functional equation of the zeta function and its symmetry properties with the proposed alternation of relational weights. The Euler product is incorporated as the multiplicative structure connecting the zeta function to prime numbers, allowing the relationship between multiplicative organization, relational configurations, and the distribution of zeros to be examined within a unified architecture. The proposed interpretation expands the conventional two-dimensional representation of nontrivial zeros by introducing an additional relational dimension associated with transduction and memory. Each state is considered in relation to its projected coordinates, its equilibrium position, its relational weights, and its transformation history. The central contribution is the formulation of a dynamic perspective in which the critical line at (1/2) represents the condition of equal relational contributions, while the broader architecture encompasses the movement of equilibrium between the poles 0 and 1. The Riemann Hypothesis is consequently examined through the organization of relational states, the alternation of weights, geometric memory, multiplicative structure, and transductive transformation. By bringing together analytic number theory, prime-number structure, relational geometry, and Geometric Transduction Theory, the article presents an expanded conceptual framework for investigating the geometry and organization of the nontrivial zeros of the Riemann zeta function. Keywords: Riemann Hypothesis; Riemann Zeta Function; Nontrivial Zeros; Geometric Transduction; Mobile Equilibrium; Relational Weights; Zero and One Poles; Relational Memory; Dynamic Critical Line; Euler Product; Prime Numbers; Three-Dimensional Geometry; Transductive Architecture; Relational Mathematics. Author: Cláudio Vicente da SilvaIndependent Researcher — Londrina, Paraná, BrazilPublication date: November 2026
Author: Vicente da Silva, Cláudio
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225655750
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