The integration of precise static analysis into interactive development environments (IDEs) necessitates algorithms that can update analysis results incrementally within milliseconds. However, maintaining Bidirected Dyck-CFL reachability—the standard formalism for field-sensitive alias analysis—under dynamic graph mutations remains an open challenge. Standard batch algorithms exhibit prohibiting cubic complexity ( \(\mathcal {O}(N^3)\) ), while existing dynamic approaches often lack formal guarantees when handling non-monotonic edge deletions (the “ghost path” problem). In this paper, we present GC-DBDR, a novel incremental framework rooted in Abstract Interpretation. We reformulate the dynamic analysis problem not as graph patching, but as computing Differential Fixpoints over a lattice. By establishing a rigorous Galois Connection between execution traces and reachability relations, we derive update rules that are correct-by-construction. To resolve the asymmetry between monotonic insertions and non-monotonic deletions, we introduce a Counting-Augmented Abstract Domain supported by a Derivation Hypergraph. This structure operationalizes the Inverse Abstraction Principle, ensuring that reachability facts are retracted if and only if their supporting derivation trees are fully invalidated. We provide formal proofs demonstrating that GC-DBDR is sound and complete relative to batch analysis. Empirically, the algorithm achieves an optimal input-output complexity of \(\mathcal {O}(\varDelta )\) , delivering sub-millisecond tail latencies on real-world benchmarks.