The evolution of Large Language Models (LLMs) from static instruction-followers to autonomous agents necessitates operating within complex, stateful environments to achieve precise state-transition objectives. However, this paradigm is bottlenecked by data scarcity, as existing tool-centric reverse-synthesis pipelines fail to capture the rigorous logic of real-world applications. We introduce \textbf{LOGIGEN}, a logic-driven framework that synthesizes verifiable training data based on three core pillars: \textbf{Hard-Compiled Policy Grounding}, \textbf{Logic-Driven Forward Synthesis}, and \textbf{Deterministic State Verification}. Specifically, a Triple-Agent Orchestration is employed: the \textbf{Architect} compiles natural-language policy into database constraints to enforce hard rules; the \textbf{Set Designer} initializes boundary-adjacent states to trigger critical policy conflicts; and the \textbf{Explorer} searches this environment to discover causal solution paths. This framework yields a dataset of 20,000 complex tasks across 8 domains, where validity is strictly guaranteed by checking exact state equivalence. Furthermore, we propose a verification-based training protocol where Supervised Fine-Tuning (SFT) on verifiable trajectories establishes compliance with hard-compiled policy, while Reinforcement Learning (RL) guided by dense state-rewards refines long-horizon goal achievement. On $τ^2$-Bench, LOGIGEN-32B(RL) achieves a \textbf{79.5\% success rate}, substantially outperforming the base model (40.7\%). These results demonstrate that logic-driven synthesis combined with verification-based training effectively constructs the causally valid trajectories needed for next-generation agents.