The Registration Graph: Deriving Physical Structure from a Single Axiom

A formal graph-theoretic framework deriving physical structure from a single axiom: that existence requires registration. Beginning from one axiom (four interdependent conditions, zero free parameters), we derive through explicit logical dependency: the arrow of time, Lorentzian signature, a least-action principle, energy conservation, the second law of thermodynamics, superposition, entanglement, the quantum of action, the uncertainty principle, the Born rule, and the unique selection of three spatial dimensions for topologically stable matter. The formalization uses vertices, directed edges, and partial orders as primitives, with no appeal to known physics at any step. Each theorem's proof depends only on the axiom and previously established results, with all dependencies mapped in an explicit directed acyclic graph. The Born rule is derived from three structural requirements (non-negativity, orientation invariance, self-relational structure) rather than postulated. Three spatial dimensions emerge as the unique dimensionality supporting knotted constraint loops — the framework's account of stable matter. Honest gaps in the derivation chain are explicitly catalogued. The accompanying framework document extends to a parameter-free derivation of the bare fine structure constant α₀ = 3/(130π) ≈ 1/136.14.