Quantum Ramanujan Machine

Automated conjecture generation is usually implemented as a loop that repeatedly proposes a candidate expression, evaluates it numerically, screens for near matches to a target quantity, refines the proposal distribution, and escalates survivors to high precision checks and proof attempts. The computational bottleneck is frequently the evaluator, especially when the target quantity is an expectation, integral, or simulation derived statistic whose classical estimation cost scales poorly with the required tolerance. This paper defines a “Quantum Ramanujan Machine” as a conjecture engine whose evaluator and screening stages are exposed as explicit operators inside an end-to-end pipeline, enabling quantitative predictions about when a quantum subroutine changes the pipeline bottleneck rather than merely accelerating a subroutine in isolation. The manuscript contributes three coupled elements. First, an operator level decomposition of conjecture generation that isolates the generator, evaluator, screening, verifier, and refinement update as composable maps acting on candidate distributions. Second, a local stability condition for refinement dynamics expressed as a spectral radius bound on the linearization of the update operator, which makes refinement noise and scoring curvature part of the algorithmic design constraints. Third, throughput and channel bounds that relate tolerance, precision, candidate volume, and stage capacities to end to end conjecture yield, and that predict the regime in which quantum mean estimation or amplitude amplified search shifts the limiting stage. The results are formulated to support empirical adjudication through measurable quantities, including yield scaling at fixed verification criteria and bottleneck shift predictions under controlled changes to evaluation accuracy and access costs.