Q Quantum Theory III-A: Quantum Gravity -- Foundations -- Whole-Q Sources, Deformation Geometry, Collective Dressing, and the Metric Branch

Q Quantum Theory takes the complete quantum object Q as the primitive physical entity. This paper develops its gravitational foundation. In the local representation $Q=\sqrt{\rho}\,e^{i\theta}Z$, the complete Q state carries energy--momentum, stress, spin and orientation structure; gravity is introduced as the universal physical response to that spacetime/Poincar\'e content. The gravitational variable is one invertible tensor $A^a{}_{\mu}\in GL(4,\mathbb R)$ on a physical Minkowski reference spacetime. In a local Lorentz-polar chart $A=OP$, the dilation--shear factor induces $g=P^T\eta P$, while the full field determines the transported connection and its Levi--Civita-relative distortion. Curved spacetime is therefore retained exactly as the metric-visible effective geometry of a real gravitational interaction field, rather than taken as the primitive microscopic ontology. Variation with respect to $A$ defines the Q gravitational source; an isolated Q carries source-conditioned dressing, many Qs source one shared nonlinear response, and elimination of the positive static response gives a negative relational cross-energy for ordinary positive-energy sources. The weak branch reproduces Einstein/Newtonian gravity and the two helicity-two radiative modes, while the harmonic/Fock spherical solution provides a nonlinear classical benchmark. Finite coherence is postulated at the interaction level: unbounded Q support, finite relative-response width and experimental resolution are distinct. The local regime is an unresolved-response limit, not exact microscopic point response. The companion Part III-B establishes a conditional theorem: under explicit source--response compatibility, boundary-completion and integrability hypotheses, the finite-response coefficients are ultraviolet finite at arbitrary fixed perturbative order; that conditional theorem is not used as a premise here. Part III-C treats the quantum-corrected Fock self-energy.