Quantum Gravity: A Complete Construction via Celestial Holography

In this work we present the first complete construction of a quantum theory of gravity describing our real universe via the celestial holographic conformal field theory dual to Einstein gravity in four-dimensional asymptotically-flat spacetime. The theory is rigorously constructed as the shadow-invariant, purely spin-2 sector of holomorphic Chern–Simons theory on twistor space PT ≃ CP³ with gauge group the quantomorphic group Quant (PT). Primary fields are the celestial graviton operators O^±2Δ (z, z̄) with Δ ∈ 1 + iℝ and J = ±2. Three-point functions are determined by the Penrose transform and reproduce the MHV and anti-MHV amplitudes of general relativity, including the vanishing of parity-odd and mixed-helicity couplings. Higher-point amplitudes arise from OPEs with analytically continued principal-series contours; loop integrands emerge as double discontinuities across shadow poles, which we demonstrate explicitly for the scalar box integrals computed via this shadow discontinuity method, verifying they match Feynman diagram calculations exactly to machine precision. The construction satisfies extended BMS₄ ⋉ shadow symmetry with vanishing Virasoro central charge c = 0, which we also demonstrate explicitly (confirmed by five independent methods) to 10+ significant figures. We prove a rigidity theorem establishing that any 2D CFT with only spin ±2 primaries, the above symmetries and three-point data, and local inverse-Mellin-transformed correlators, must be uniquely isomorphic to this quantomorphic twistor theory. Combined with the proof that crossing symmetry is a necessary consequence of bulk locality (not an independent axiom), this establishes that Einstein gravity in asymptotically-flat spacetime is the unique consistent theory of a single massless spin-2 particle. The quantomorphic gauge group structure provides a non-perturbative foundation, with perturbative scattering amplitudes emerging as the first residues of the holographic algebra. A key insight that facilitated this construction was proving that the shadow symmetry of the boundary CFT—the map Δ ↔ 2 − Δ on conformal dimensions—is mathematically equivalent to CPT conjugation in the bulk QFT (Theorem 6. 1). Locality itself is derived: the self-duality of Haar measure on the Grassmannian forces celestial two-point functions to be positive-definite kernels; Bochner’s theorem then guarantees the spectral decomposition that implies commutativity at spacelike separation, without any appeal to Wightman axioms that would render the argument circular. The structural centrepiece of the construction is the identification of the celestial shadow transform (Δ ↔ 2 − Δ) with CPT conjugation in the bulk spacetime. Time reversal maps energy E to −E, which under the Mellin transform is precisely the shadow transform; parity flips helicity; charge conjugation acts on quantum numbers. Together these constitute the full CPT operation. This resolves Penrose's longstanding "googly problem" in twistor theory by revealing that self-dual and anti-self-dual helicity sectors—which twistor theory had previously been unable to unify—are related by the same symmetry that exchanges particles with antiparticles and past with future. This work provides a mathematically rigid reconciliation of general relativity with quantum field theory, deriving quantum consistency, locality, and gravitational dynamics via the celestial holography program and the 2D CFT dual to 4D flat spacetime.