Quantum Gravity and the Unification of Quantum Mechanics and General Relativity

The foundational incompatibility of quantum mechanics and general relativity is usually located in a missing common dynamics. This paper isolates a prior, structural incompatibility with a single locatable cause: quantum mechanics requires a fixed background against which system identity is referred, while general relativity makes that same background — the metric — a dynamical variable. One object is thereby forced to be, on one level, both the invariant identity reference and the freely varying quantity; by the Frame/Module exclusion this is contradictory. Supplying the missing identity standard — a structure distinct from the metric — removes the contradiction: the paper identifies the exact contradictory propositions and shows, line by line, that after the same admissibility reference is added they are no longer contradictory. The common carrier is constructed, not inferred from isomorphism, and it carries the typed two-level identity — the persistence-class relation Q2a and the Q2a-constrained profile relation Q2b, built before any realization enters (§6): quantum mechanics arises from the pre-metric admissibility structure by the GNS construction, and general relativity by the metric projection, both from the same F_0 — so the metric belongs exclusively to the general-relativistic projection and no object is both fixed reference and dynamical variable. The competing quantum-gravity programmes are then exhibited as realizations of one admissibility architecture, and proposed resolutions of the information paradox and the singularity become classifiable under one identity criterion. The claim is structural throughout: what is unified is the identity/admissibility layer. A common quantum-gravitational dynamics is not derived here and cannot be, since the architecture constrains the admissible class without selecting a member; §13 exhibits one member of that class explicitly, to show it non-empty and computable, and marks it as a candidate rather than a result. No equation is modified and no value is delivered.