Q Quantum Theory III-B: Quantum Gravity -- Quantum and Ultraviolet Structure

Part III-A defines Q gravity through a single full $A=OP$ response field on a physical Minkowski reference spacetime. This companion develops its finite-order quantum structure as one Batalin--Vilkovisky (BV)/contact theory. In the minimal metric-coupled branch the local orientation directions are bulk dynamical redundancies; a propagating polar--log/Hodge sector is included only when supplied by a separately declared complete action. The first-order auxiliary distortion remains exactly Schur equivalent and introduces no additional physical pole. The finite coherent interaction law is lifted to perturbation theory as one graph-independent response on complete gauge-related graph/source groups. In the canonical cycle/worldline realization, active-channel nondegeneracy together with regular strictly positive inherited response density and scale produces a positive response Gram on every surviving loop component and hence inverse-proper-time exponential suppression with a positive lower bound on each compact forest-shape cell. The positive-density inheritance is a microscopic realization hypothesis, not a consequence of ultraviolet damping. On the resolved forest compactification, joint finite-part coefficient projectors commute on compatible normals and the inherited response contacts form an exact contraction cocycle, giving path-independent rank-changing contractions. A single heat kernel of the Euclidean gauge-fixed BV Hodge operator supplies the common physical, ghost, reducible, nonminimal and antifield coincidence regulator; frame/density changes are transported by similarity and source descendants by Duhamel differentiation of the same family. Accordingly, at every fixed finite perturbative order, any gauge/BV-complete coefficient whose graph/source group satisfies the stated response-inheritance, source-role/density, contact, finite-boundary, regularity, common-domain, differentiability and ultraviolet/infrared-separation hypotheses has an absolutely ultraviolet-finite Euclidean/Schwinger representation without ultraviolet-divergent curvature subtraction. This coefficientwise theorem does not imply convergence or Borel summability of the full perturbation series, and Lorentzian reconstruction remains a separate requirement. Under the stated Hermitian-parent hypotheses, closed-time-path response and state-resolved heat-semigroup sewing preserve finite-order largest-time/cutting identities.