Large language models (LLMs) achieve high accuracy on many reasoning benchmarks but remain brittle under structural perturbations of rule-based systems. We introduce a diagnostic framework with four stress tests -- redundant vs. essential rule deletion, contradictory-rule injection, logic-preserving rewrites, and multi-law stacking -- and use it to expose Logic Inertia: the tendency of generative LLMs (Qwen2/3, TinyLlama, GPT-4o, Gemma-3-4B-IT) and the encoder-only BERT baseline to persist along learned deductive trajectories under inconsistent premises. The collapse is sharp: untreated baselines fall from accuracy 1.00 on the base task to 0.00 on contradiction injection (instance-level exact match), and GPT-4o resolves only 56.0% of contradiction cases. We propose Conflict-Aware Fusion, a four-stage training pipeline that enforces verification-before-deduction as a learned structural prior: (i) SFT establishes the verification preamble; (ii) DPO sharpens the halt-on-contradiction decision boundary; (iii) Logical Invariance REgularisation (LIRE) penalises divergence between logically equivalent rule formulations via symmetric KL; (iv) Reinforcement Learning from Verification Feedback (RLVF) uses a symbolic forward-chaining engine as a deterministic oracle reward, jointly optimising invariance and sensitivity. The pipeline saturates all four primary stress tests for both 1.5B and 8B backbones. We further validate a Phase 2 extension that replaces the propositional oracle with a Lean 4 kernel, attaining 99.0% kernel agreement on the 105 classically-derivable (T) questions within a stratified 187-question Lean-translated sample (overall 71.7% across both polarities), providing a sound upgrade path to formally verified RL training. Code and benchmark: https://github.com/14H034160212/lemo