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| Phase | Innovation | Mathematical Foundation |
|---|---|---|
| Phase 1 | Differentiable Graph Traversal | Neural Bellman-Ford over learned semirings (NBFNet) + NOTEARS DAG extraction |
| Phase 2 | Hybrid RL Training | GFlowNet Trajectory Balance with path-refined reward (reachability + parsimony + integrity) |
| Phase 3 | Agentic Graph Evolution | Edge Proposal Network + Sheaf Laplacian pre-filter + Datalog/PSL symbolic gatekeeper |
| Phase 4 | Reasoning Faithfulness Metrics | Counterfactual Intervention Testing (CIT) + DAG-Consistency Index (DCI) |
| Phase 5 | Computational Advantage | O(k·M·d²) linear vs O(k²·t²·d²_model) quadratic; ~4,000× speedup at 20-hop |
Query Q → [Entity Grounding] → [NBFNet Traversal (T iterations)]
↓
[Edge Relevance Scores π_q(e)]
↓
[NOTEARS DAG Extraction] → D_Q
↓
[Dead-End?] → YES → [Edge Proposal Network]
↓ ↓
NO [Gatekeeper: Sheaf + Datalog + PSL]
↓ ↓
[Answer Decode] [Expand G, Re-traverse]
↓ ↓
(v_answer, D_Q) ←←←←←←←←←←1@techreport{gie2026,
2 title={The Graph-Inference Engine: A Neuro-Symbolic Architecture for Faithful Multi-Hop Reasoning},
3 year={2026},
4 note={Technical whitepaper}
5}