Vector Logic and the Metaverse: Continuous Structures Beyond Discrete Conventions

This paper proposes a formal transition from discrete axiomatic systems to a continuous mathematical framework termed vector logic. The research addresses the inherent constraints of bivalent and traditional multi-valued logics when modeling complex, evolving systems. By representing system states and hypotheses as vectors within separable real Hilbert spaces, the framework enables a geometric interpretation of logical operations. The core of the proposal is the orthogonal decomposition of reality into three distinct subspaces: variants, variations, and invariants. A complete system of axioms is provided to define negation, conjunction, and implication through geometric transformations. The study demonstrates that classical, fuzzy, and quantum logics are recovered as specific projections or degenerate cases of this unified model. Furthermore, the "metaverse" is formalised not as a virtual medium, but as a topological phase-space construct representing the norm-closure of all reachable state vectors. This approach situates the work within the tradition of many-valued logics while introducing a least-squares optimization method for hypothesis testing. The result is a robust formal tool for analyzing state transitions in high-dimensional environments, including modern artificial intelligence architectures.