SMCEvolve recasts LLM-driven program search as Sequential Monte Carlo sampling from a reward-tilted target distribution, introducing adaptive parent resampling, mixture-of-mutation acceptance, and automatic convergence control. Across math, symbolic regression, and ML research benchmarks the method exceeds prior evolving systems while terminating under self-determined error thresholds and using fewer LLM calls. Jiang et al. (2026) supply explicit complexity bounds on the LLM budget required to reach target approximation error. This corrects the absence of convergence analysis in earlier frameworks such as FunSearch (Romera-Paredes et al., 2024), which reported performance gains without guarantees on sample efficiency. The SMC formulation also aligns with established probabilistic programming techniques in Pyro (Bingham et al., 2019), where resampling and mutation steps have long delivered finite-sample guarantees. SMCEvolve thereby supplies the missing principled components for scaling automated scientific discovery beyond benchmark-specific heuristics.