The Two Ontologies of Investigation: From the Classical Paradigm to Transductive Architecture

THE TWO ONTOLOGIES OF INVESTIGATION: FROM THE CLASSICAL PARADIGM TO TRANSDUCTIVE ARCHITECTURE An Epistemological Review of Reality, with Application to the Navier–Stokes Problem Author: Cláudio Vicente da SilvaIndependent Researcher — Londrina, Paraná, BrazilPublication date: November 2026 This article presents an epistemological and ontological investigation of two fundamentally distinct paradigms for understanding and investigating reality: the Classical Paradigm, grounded in the Ontology of Being, and the Transductive Paradigm, grounded in the Ontology of Becoming. The study examines the conceptual foundations, methodological assumptions, and limitations of these two approaches, proposing a transition from the investigation of reality as a pre-existing and static whole to an understanding of reality as a generative process of continuous transformation, relational organization, and emergence. Within the Classical Paradigm, investigation is characterized by the identification, isolation, and assembly of previously determined elements. Reality is approached as an already constituted structure, while the investigator occupies the position of an external observer. In contrast, the Transductive Paradigm understands reality through the passage between configurations, in which relational memory, latent potency, and manifested states participate in the continuous organization of systemic structures. The article formalizes this second approach through the Systemic State: Σ = (S, M, P) where S represents the manifested configuration, M represents relational memory, and P represents latent potency. The transition between configurations is expressed through the transduction relation: Σ_f = T(Σ_i) This conceptual architecture is subsequently applied to the Navier–Stokes regularity problem, one of the Clay Mathematics Institute's Millennium Prize Problems. The investigation introduces the Emergence Operator (ℰ), the reconstruction of three-dimensional configurations from fundamental closed structures, and a proposed dynamical compatibility relation: 𝒩₃ ∘ ℰ = ℰ ∘ 𝒩₁ These formulations establish a transductive framework in which regularity is interpreted through the relational coherence and continuity of transformations across different structural levels. The study further explores the interpretation of apparent singularities as possible moments of topological reorganization, the role of relational memory in the evolution of dynamical systems, and the epistemological implications of participation in the investigative process. A central contribution of the article is the distinction between investigation as an archaeology of an already constituted reality and investigation as transductive architecture, in which each configuration participates in the organization of subsequent configurations. This distinction is developed through the concepts of the Ontology of Being, the Ontology of Becoming, epistemology of dissection, epistemology of participation, and the notion of the unacceptable as a dimension that challenges the assumptions of an established investigative paradigm. The article proposes a fundamental epistemological reorientation: the transition from a static and predominantly reductive conception of reality toward a relational, dynamic, and generative conception based on the Theory of Transduction. The central proposition is that understanding complex phenomena requires investigating not only the configurations that manifest, but also the relational processes through which configurations emerge, transform, preserve continuity, and generate new organizational possibilities. This work contributes to the broader development of Relational Architecture and the Theory of Transduction, bringing together philosophical foundations, epistemological considerations, mathematical structures, and theoretical interpretations of physical phenomena. Keywords Ontology of Being; Ontology of Becoming; Classical Paradigm; Transductive Paradigm; Theory of Transduction; Relational Architecture; Epistemology of Participation; Systemic State; Relational Memory; Latent Potency; Emergence Operator; Navier–Stokes Equations; Fluid Dynamics; Mathematical Physics; Topological Reorganization; Dynamical Systems; Scientific Paradigm Shift. Research Areas Philosophy; Epistemology; History and Philosophy of Science; Ontology; Mathematics; Mathematical Physics; Fluid Mechanics; Dynamical Systems; Topology; Complexity Theory; Relational Architecture; Theory of Transduction. Document Scope This publication develops a theoretical and epistemological framework connecting the ontology of reality with the architecture of scientific investigation. Its application to the Navier–Stokes problem is presented through a proposed transductive interpretation involving relational continuity, structural memory, emergence, and topological reorganization. The article is part of the author's broader research program on Relational Architecture and the Theory of Transduction, which investigates the organization and transformation of systems across philosophical, mathematical, physical, biological, computational, and cognitive domains.