完全統一理論 ― 曲率密度ρによる重力・電磁・量子の構造統一記述 / Complete Unification Theory: Structural Unification of Gravity, Electromagnetism, and Quantum Mechanics via Curvature Density ρ

We present a five-part series establishing a structural unification of gravity, electromagnetism, and quantum mechanics through the curvature density of U(1) connections on the null Kerr screen S² ≅ CP¹.

In Part I, we prove that U(1) connections arising in three domains — the gravitational rotation 1-form, Berry connection, and electromagnetic connection — share the universal representation F = ϱ ω_FS on S², with differences reduced to the scalar density ϱ and the topological sector c₁. We prove c₁[Ω^(rot)] = 0 for general non-expanding horizons (NEH) without assuming axisymmetry, and establish the universality of local connection forms via the Lie algebra isomorphism u(1) ≅ so(1,1) ≅ ℝ.

In Part II, we formulate the Holonomy Variational Principle (HVP) as a variational ansatz that selects physics through stationarity conditions on holonomy functionals. The common Euler–Lagrange equation Δ_{S²}ϱ = ∂V/∂ϱ governs all three sectors, with differences encoded solely in the potential V and topological charge c₁.

In Part III, we construct a Ψ₂ boundary theory on NEH/IH and show that Einstein boundary constraints (cross-focusing equation, rotation 1-form structure equations) are consistently characterized as HVP stationarity conditions in the gravitational sector. The closure problem is systematically addressed within an effective field theory (EFT) framework: the effective closure is justified as the leading screen EFT term determined by S² geometry, spin-weight structure, and rotational symmetry, with quantitative benchmarking via QNM consistency conditions.

In Part IV, we reinterpret the Chern number wall (c₁ = 0 for gravity vs. c₁ = 1 for Berry) as a superselection rule within the unified framework, and propose five theory-specific predictions including the ϱ-no-hair test for black hole spectroscopy.

In Part V, we demonstrate the natural inclusion of electromagnetic fields via Dirac quantization within the stratified curvature density space, confirm compatibility with Dirac spinor structure, and propose a constrained 4D unified action as an extension.

The unification claimed here is structural — not a grand unified theory of interactions, but a demonstration that boundary connection structures across three domains are described by a common geometric grammar. Einstein's equations are characterized, not derived.