THE COLLATZ ATTRACTOR: FROM ARITHMETIC CHAOS TO GEOMETRIC TRANSDUCTION VIA THE UNO SPIRAL AND THE ARITHMETIC FUNNEL

Overview This paper establishes the concept of the Topological Inverse within the framework of the Primitive Architecture and the Cosmos Scenario. In classical arithmetic mathematics, the inverse of a quantity tends toward cancellation or zero (x + (−x) = 0). However, in a relational ontology where absolute emptiness and the void do not exist, the inverse of a geometric structure cannot be nothingness. This work demonstrates that the topological inverse of a geogram — particularly the UNO Spiral, which represents movement accompanied by memory — is its Chirality, or the Inversion of the Spiral. Core Thesis By inverting the discrete orientation state s ∈ {−1, +1} extracted by the Transduction Operator T_H, the system does not collapse into the void. Instead, it alters its topological phase, representing the necessary passage from "Final Transduction" (the dissolution of form) back to "Fecundated Latency" (∘′). Chirality is thus the ontological mechanism through which the Cosmos breathes, ensuring that the end of a finite cycle is the condition of possibility (the seed) for the following cycle. The inverse is not the end of the relation; it is the change of its topological phase. Mathematical Formalization The paper bridges the ontological concept of chirality with rigorous functional analysis. It defines the functional space of spiral geograms G and constructs the Chirality Operator Q. The central theorem proves three fundamental properties: Involution (Q² = I): The topological inverse is a true reflection, not a destruction. Antisymmetry of Transduction (T_H(Qr_s) = −T_H(r_s)): The dynamic orientation is perfectly reversed. Conservation of Structural Memory (A_Q(t) = A(t)): The amplitude profile (the memory of the process) remains strictly invariant under topological inversion. Furthermore, it proves the Topological Invariance of the Skeleton, demonstrating that the unoriented geometric locus remains intact while only the arrow of transduction is flipped. Context and Significance within the Research Program This work resolves the philosophical fear of the end and the fallacy of infinity. By proving that the inverse of a form is its chiral conjugate rather than zero, the architecture establishes that the universe is not a machine of forgetting. Every dissolution carries the structural memory of the form that preceded it. The "nothingness" that follows the end of a cycle is, in fact, a field of Highly Structured Latency, ready for the next Fecundation. This completes the dynamic architecture of the Cosmos Scenario, providing the exact geometric mechanism for the Cycle of Finites and Relational Eternity. Target Audience Researchers and scholars in the philosophy of science, theoretical physics, topology, dynamical systems, geometry, and ontology, particularly those investigating the transition between form and latency, symmetry breaking, and the geometric foundations of physical and biological cycles. About the Author Cláudio Vicente da Silva holds a degree in Philosophy, with postgraduate studies in Higher Education Methodology and Philosophy of Science. He has 24 years of service in the public sector of the State of Paraná, Brazil. As an independent researcher, he conducts investigations at the intersection of philosophy of science, mathematical foundations, geometry, structural transduction, and relational architecture. His works combine philosophical investigation, mathematical formulation, and the construction of theoretical models, published through scientific research platforms and digital repositories, including Zenodo.