Abstract. The Everettian interpretation of quantum mechanics removes the collapse postulate but inherits an obligation: to de- fine wavefunction branches precisely. We propose a definition rest ing on five axioms: three from standard quantum mechanics, plus two new ones. First, the monitoring structure of the Hamiltonian —the system-apparatus-environment decomposition and coupling structure — is a necessary physical input. Second, branches are connected components of a coherence graph after environmental decoherence fragments it. The branch partition is identified by the Fiedler eigenvector of the coupling graph Laplacian, a polynomial time spectral computation with no optimisation or variational free dom. From these axioms we derive branch formation at an explicit timescale, partition universality (insensitivity to threshold, initial state, and intra-block details within the regime where the monitor- ing conditions hold), and show that the definition satisfies all four criteria identified by Riedel (2025): forward-in-time tree structure, quasiclassical eigenstates, effective collapse, and bounded entan- glement with area-law scaling. We verify the definition against Stern–Gerlach, double-slit, and Bell experiments, and validate it numerically across random Hamiltonian ensembles, random ini- tial states, systematic violations of the monitoring conditions, and multi-branch (k = 3) scenarios. This paper does not attempt to solve the measurement problem. It provides a precise, computable definition of one of its central objects. The monitoring structure is not derived but taken as input; its necessity is demonstrated by showing that