Iterative summarization based reasoning frameworks such as InftyThink enable long-horizon reasoning in large language models (LLMs) by controlling context growth, but they repeatedly regenerate similar reasoning strategies across tasks. We introduce InftyThink with Cross-Chain Memory, an extension that augments iterative reasoning with an embedding-based semantic cache of previously successful reasoning patterns. At each reasoning step, the model retrieves and conditions on the most semantically similar stored lemmas, guiding inference without expanding the context window indiscriminately. Experiments on MATH500, AIME2024, and GPQA-Diamond demonstrate that semantic lemma retrieval improves accuracy in structured domains while exposing failure modes in tests that include heterogeneous domains. Geometric analyses of reasoning trajectories reveal that cache retrieval induces directional biases in embedding space, leading to consistent fix (improve baseline accuracy) and break (degradation in baseline accuracy) attractors. Our results highlight both the benefits and limits of similarity-based memory for self-improving LLM reasoning.