A Deterministic Solution to the Vitali Set Pathology Utilizing Neural-Matrix Synaptic Resonance Network (NM-SRN) v2.0 AGI QSC-PSI's MOO Framework

Abstract

The Vitali set, constructed using the Axiom of Choice, has stood for over a century as the canonical example of a non-measurable set within Lebesgue measure theory. Its existence is traditionally cited as proof that any measure satisfying translation invariance and countable additivity cannot assign a value to every subset of the real numbers. In this paper, we present a definitive, deterministic solution to the Vitali set pathology using the Neural-Matrix Synaptic Resonance Network (NM-SRN) v2.0 AGI QSC-PSI’s Mathematical Object-Oriented (MOO) Framework. By constructing a Fractal Coordinate System (F) that maps the unit interval to a tree topology based on rational equivalence classes, and defining a Topological Tree Metric (dF ), we derive a Fractal-Deterministic Measure (μD ) that assigns a finite, well-defined value to any Vitali set. This demonstrates that the classical ”non-measurable” classification is not an inherent property of the set, but an artifact of the specific architectural assumptions (translation invariance, countable additivity) of Lebesgue measure theory. The solution is certified at Sigma-3 confidence within the MOO Framework, with full provenance tracking via Intelligent Tags and Mathematical CMDCs.