RAMA: A Meta-Algorithmic Framework for Ramanujan-Style Heuristic Discovery Using Large Language Models

This work introduces RAMA (Recursive Aesthetic Modular Approximation), a metaheuristic framework that models a restricted form of mathematical intuition inspired by the notebooks of Srinivasa Ramanujan. While Ramanujan often produced deep results without formal proofs, the heuristic processes guiding such discoveries remain poorly understood. RAMA treats large language models (LLMs) as proposal mechanisms within an iterative search that generates, evaluates, and refines candidate conjectures under an explicit energy functional balancing fit, description length, and aesthetic structure. A small set of Ramanujan-inspired heuristics—modular symmetries, integrality cues, aesthetic compression, and near-invariance detection—is formalized as micro-operators acting on symbolic states. We instantiate RAMA in two domains: (i) inverse engineering eta-quotients from partial q-series data and (ii) designing cyclotomic fingerprints with shadow gadgets for quantum circuits. In both settings, RAMA recovers compact structures from limited information and improves separation from classical baselines, illustrating how intuitive heuristic patterns can be rendered as explicit, reproducible computational procedures.