The Registration Graph: Deriving Physical Structure from a Single Axiom — A Graph-Theoretic Formalization (Axiom through Theorem 12)

We present a formal graph-theoretic framework in which a single axiom — that existence requires registration — generates core structures of physical theory as necessary consequences. From one axiom with four interdependent conditions (existence requires registration, substrate uniqueness, continuity of capacity, discreteness of events), we derive through strict logical dependency: an irreversible generative process with a minimal quantum of distinction, Lorentzian signature from the structural asymmetry between an append-only irreversible axis and reversible degrees of freedom, a least-resistance principle as the economy of registration rather than a variational postulate, energy conservation as invariance of registration flow, the second law of thermodynamics from necessary lossy compression, superposition and entanglement as undiscriminated paths in the graph, the quantum of action with identification of c, ℏ, and k_B as dimensional encodings of a single graph-level identity, the uncertainty principle from finite registration capacity per event, the Born rule from orientation invariance and the self-relational structure of registration, and the unique selection of three spatial dimensions as the condition for topologically stable self-reinforcing structures (knotted constraint loops). Every derivation proceeds forward from the axiom; no result is justified by appeal to known physics. The formalization uses graph-theoretic primitives — vertices, directed edges, partial orders, and path spaces — with an explicit derivation dependency graph making every logical step independently verifiable. The paper identifies precisely where derivations are complete, where honest gaps remain, and where subsequent work should focus. An accompanying framework document (v5) extends the formalization to derive matter as torus knot constraint loops, gravity as constraint-geometry curvature, and a parameter-free formula for the bare fine structure constant α₀ = 3/(130π) ≈ 1/136.14, within 0.7% of the measured value.