We study alpha factor mining, the automated discovery of predictive signals from noisy, non-stationary market data-under a practical requirement that mined factors be directly executable and auditable, and that the discovery process remain computationally tractable at scale. Existing symbolic approaches are limited by bounded expressiveness, while neural forecasters often trade interpretability for performance and remain vulnerable to regime shifts and overfitting. We introduce FactorEngine (FE), a program-level factor discovery framework that casts factors as Turing-complete code and improves both effectiveness and efficiency via three separations: (i) logic revision vs. parameter optimization, (ii) LLM-guided directional search vs. Bayesian hyperparameter search, and (iii) LLM usage vs. local computation. FE further incorporates a knowledge-infused bootstrapping module that transforms unstructured financial reports into executable factor programs through a closed-loop multi-agent extraction-verification-code-generation pipeline, and an experience knowledge base that supports trajectory-aware refinement (including learning from failures). Across extensive backtests on real-world OHLCV data, FE produces factors with substantially stronger predictive stability and portfolio impact-for example, higher IC/ICIR (and Rank IC/ICIR) and improved AR/Sharpe, than baseline methods, achieving state-of-the-art predictive and portfolio performance.