Edge is Everything: A Theory of Everything from One Axiom

Starting from a single, self-proving axiom — "binary relations exist" — we derive the complete structure of fundamental physics with zero adjustable parameters. The logistic function f(x) = 1/(1+e⁻ˣ), identically the Boltzmann distribution for binary variables, yields universal constants f(0) = 1/2 and f'(0) = 1/4. Three principles and seven derived corrections determine all 26 Standard Model parameters (median error 0.19%; 14 of 23 non-exact parameters within 1%). The 25 dimensionless ratios are derived from the axiom alone; the one absolute mass scale reflects the current age of the universe. Beyond parameter determination, the framework derives: the action principle (S = H_joint), Einstein's equation (∂H/∂J = ∂Φ/∂J, where curvature R_ij ≡ ∂H/∂J_ij is the Ollivier–Ricci curvature defined on any metric space), conservation laws, spin, Lorentz invariance, gauge symmetry, CPT conservation, charge quantization, anomaly cancellation, and the unification of gravity with quantum mechanics. The edge propagator f'(u) = 1/(2+2 cosh u) is UV-finite by construction, reduces to the standard 1/q² in the continuum limit, and reproduces the exact QED cross section σ(e⁺e⁻ → μ⁺μ⁻) = 4πα²/(3s) at tree level. At loop level, the bounded propagator matches QED integrals to O(m_e²/M_Planck²) ~ 10⁻⁴⁴; the vacuum polarization, coupling running Δ(1/α) = 9.085, Schwinger term F₂(0) = α/(2π), and electron g−2 are computed, yielding agreement with experiment to 0.042% — consistent with the framework's fundamental precision limit ε₀ d_t ≈ 0.039%. Testable predictions: δ_PMNS = 202.5° (DUNE/Hyper-K), m_ν₃ ≈ 0.049 eV (Project 8), no 4th generation at any energy. Paper I presents the full theory; Paper II provides complete proofs, the propagator derivation, tree-level and loop-level QED calculations, and explicit correspondences with statistical mechanics, the Feynman path integral, gauge theory, the Dirac equation, and the renormalization group. Numerical verification codes are included.