Inequality Ranking and Inference System ($\texttt{\textbf{IRIS}}$): Giving Mathematical Conjectures Numerical Value

We introduce $\texttt{\textbf{IRIS}}$, a geometric and heuristic-based scoring system for evaluating mathematical conjectures and theorems expressed as linear inequalities over numerical invariants. The $\texttt{\textbf{IRIS}}$ score reflects multiple dimensions of significance—including sharpness, diversity, difficulty, and novelty—and enables the principled ranking of conjectures by their structural importance. As a tool for fully automated discovery, $\texttt{\textbf{IRIS}}$ supports the generation and prioritization of high-value conjectures. We demonstrate its utility through case studies in convex geometry and graph theory, showing that $\texttt{\textbf{IRIS}}$ can assist in both rediscovery of known results and proposal of novel, nontrivial conjectures.