Complete Certificate Closure for a Constraint-Native Gravity–Quantum Realization under AASC

Overview Complete Certificate Closure is the terminal FEP-1.0 capstone of the gravity–quantum unification ARC. The preceding standalone modules discharge the constructive obligations left open by Structural Unification of Einstein and Schrödinger Dynamics: Module A constructs the operational metric–source attachment. Module B derives gravitational and inertial mass from one conserved Casimir charge. Module C constructs the common upstream continuation and proves the Einstein and Schrödinger projection intertwinings. Module D completes the exact target fibres and proves unique lawful occupancy modulo certificate-preserving equivalence. The capstone imports these accepted endpoints and proves that they form one dependency-faithful, interface-complete realization certificate. Its task is not to replace the standalone proofs or manufacture missing occupants by integration. It establishes that the independently proved components retain their precise theorem content when assembled, share the same physical references, close every required interface, and jointly realize the frozen FEP-1.0 target. Conceptual Significance The capstone completes the transition from common-origin structure to certified joint realization. The ARC begins from the claim that determinate physics already presupposes admissibility, standing, reference, and irreversible fixation. Within one admissible physical interior, Einstein and Schrödinger dynamics emerge as distinct first-response projections. The later modules then construct the structures required to turn that common origin into a complete realization: one admissible interior⟶one operational source⟶one conserved mass charge⟶one upstream continuation\text{one admissible interior} \longrightarrow \text{one operational source} \longrightarrow \text{one conserved mass charge} \longrightarrow \text{one upstream continuation}one admissible interior⟶one operational source⟶one conserved mass charge⟶one upstream continuation ⟶two lawful dynamical projections⟶complete target occupancy⟶one certificate orbit.\longrightarrow \text{two lawful dynamical projections} \longrightarrow \text{complete target occupancy} \longrightarrow \text{one certificate orbit}.⟶two lawful dynamical projections⟶complete target occupancy⟶one certificate orbit. The result is neither an additive juxtaposition of gravity and quantum mechanics nor a declaration that one reduces to the other. It is a dependency-controlled common-origin construction in which both descriptions occupy distinct but coordinated roles inside one complete standing-bearing certificate. Complete Certificate Closure therefore supplies the terminal theorem of the gravity–quantum unification ARC: the accepted Einstein, Schrödinger, source, mass, continuation, record, boundary, and certificate structures integrate without dependency regression into one quotient-faithful FEP-1.0 realization. Place in the Completed ARC The completed gravity–quantum sequence is: Gravity as Structural Necessity — establishes the necessary physical constraint role; Einstein Dynamics — derives the unique minimal first-response metric projection; Schrödinger Dynamics — derives the corresponding minimal Hamiltonian quantum projection; Structural Unification — proves that the gravitational and quantum descriptions are projections of one admissible physical interior; Module A — completes operational metric–source attachment. Module B — proves common mass–charge matching. Module C — constructs common upstream continuation and projection intertwining. Module D — establishes exact certificate-fibre exhaustion and unique lawful role occupancy. Complete Certificate Closure — integrates the full FEP-1.0 gravity–quantum realization certificate. The capstone is therefore the endpoint of the completed ARC rather than another standalone local module. Scope and Nonclaims The manuscript does not claim: an independent rederivation of the universal AASC kernel; that Einstein and Schrödinger dynamics are mathematically identical; reduction of gravity to quantum mechanics or quantum mechanics to gravity; a new mixed Einstein–Schrödinger field equation; unrestricted uniqueness across different targets, source packages, sectors, topologies, boundary conditions, or calibrations; raw uniqueness of all mathematical presentations; quantization of the Einstein metric as a primitive step; construction of a Wheeler–DeWitt equation; a nonzero interacting quantum-gravity field theory; recovery of every conventional quantum-gravity formalism; derivation of all Standard Model fields, masses, or couplings; global existence through every singular or nonregular regime; elimination of gauge, phase, chart, or representative freedom; that reversible evolution alone produces irreversible records; that certificate integration can replace missing standalone proofs; closure beyond the exact theorem interfaces and frozen FEP-1.0 comparison class. The theorem is complete relative to its declared target, imports, regularity domain, calibration package, and certificate-equivalence relation.