Paper V: The Tikbon Field: Spin-1 Gravity, Composite Graviton Emergence, and the Recovery of General Relativity

Within the Quantum Geometrodynamics (QGD) framework, we derive the complete gravitational gauge theory from the S³ ≅ SU(2) topology of the spatial universe. The gravitational mediator — the Tikbon field Aμ — is identified with the Hopf fibre component of the Maurer–Cartan form on SU(2), establishing it as a geometric necessity rather than a postulate. Its field strength tensor Fμν = ∂μAν − ∂νAμ unifies the gravitoelectric field g⃗ (acceleration) and the gravitomagnetic field B⃗g (frame-dragging frequency) into a single antisymmetric object, in exact structural analogy with the electromagnetic tensor. The central result is the composite graviton theorem: the spin-2 metric perturbation hμν = (2Φ/c²) kμkν arises algebraically from the spin-1 Tikbon field via the Kerr–Schild construction. The null vector kμ is derived from the Hopf fibration and the scalar potential Φ from the static Tikbon solution. The nullity condition kμkμ = 0 ensures exact linearisation of the Einstein equations — nonlinear self-interactions vanish identically, not approximately. We prove four consistency theorems: The Ward–Takahashi identity is satisfied at one loop, confirming quantum gauge consistency; The Boulware–Deser ghost is absent through three independent mechanisms; The spectral representation of the effective propagator is manifestly positive, guaranteeing unitarity; The compact S³ geometry quantises spatial momenta, rendering the one-loop vacuum polarisation finite without renormalisation — a mechanism that extends topologically to all orders. The Newtonian limit yields G = gT²/(4π), and the far-field quadrupole radiation formula recovers the General Relativity prediction exactly, ensuring consistency with the Hulse–Taylor binary pulsar measurements to within 0.2%. The Tikbon field completes the Tri-Unity of fundamental gauge interactions: electromagnetism (U(1)EM), gravity (U(1)G), and thermodynamics (U(1)T) share identical tensorial architecture. Falsifiable predictions: A sub-dominant scalar breathing mode in gravitational waves, detectable by third-generation detectors; Vector-mode polarisation signatures in LISA extreme-mass-ratio inspiral observations; A phase coherence between h₊ and h× that vanishes in pure General Relativity. Keywords: graviton, spin-1 mediator, Kerr–Schild, Hopf fibration, gravitoelectromagnetism, Boulware–Deser ghost, UV finiteness, Tri-Unity, gravitational waves Related papers: Foundational axioms in Paper I. Kerr metric in Paper II. Tri-Unity in Paper IV. Dark matter application in Paper VI. Dimensional analysis in Paper 0.