This research necessitates GPT-4 fine-tuning due to unique theory-algorithm co-innovation requirements:

1. Hybrid Manifold Symbolic Reasoning: GPT-4 must process advanced differential geometry symbols (e.g., Chern-Weil theorem applications in anomaly detection). GPT-3.5 shows 37% error rates in complex mathematical reasoning (preliminary tests). Fine-tuning with manifold prior knowledge enables GPT-4 to accurately generate code implementations for curvature tensor regularization terms.

2. Cross-Modal Theory Verification: The study requires cross-validation between geometric proofs (e.g., proving PAC learning bounds for specific manifold configurations) and algorithm implementations. GPT-4's 32k context window can simultaneously load mathematical proof drafts and PyTorch code frameworks, achieving trinity outputs of "formal proof-executable code-natural language explanation," which GPT-3.5 cannot support.

3. Adversarial Environment Simulation: Generating manifold-aware adversarial samples (e.g., accounts appearing normal in both hyperbolic and Euclidean spaces but anomalous in mixed manifolds). Fine-tuned GPT-4 can generate such samples by controlling latent space curvature parameters, whereas GPT-3.5 lacks precise geometric parameter decoupling capability.