Branch Structure as Coherence Graph Fragmentation: A Geometric Framework for Everettian Quantum Mechanics

Abstract
We propose a geometric framework for Everettian quantum mechanics in which branch structure is identified with coherence graph fragmentation. The universal density matrix is represented as a coherence graph: nodes are basis states, coupling edges are Hamiltonian matrix elements Hᵢⱼ, and coherence weights are off-diagonal density matrix elements ρᵢⱼ. These are distinct objects playing distinct roles. A flow current Jᵢ→ᵏ = (2/ℏ) Im(Hᵢᵏρᵏᵢ), derived directly from the von Neumann equation without additional postulates, governs the redistribution of diagonal amplitude weight across the graph. Branch formation is identified with coherence graph fragmentation: the process by which environmental decoherence suppresses off-diagonal coherence weights, shutting down inter-sector flow and isolating amplitude sectors as independent subgraphs. This identification is the central claim of the framework. The framework makes no new empirical predictions and introduces no modifications to quantum formalism. Its contribution is a geometric language for reasoning about branch structure grounded in standard decoherence theory. Key open problems — basis selection, behavior under approximate decoherence, temporal stability of flow channels, the interpretation of diagonal weights as probabilities, and the Born rule — are identified and motivate further development.