Physics and mathematics are deeply intertwined. While physicists use mathematics in myriad ways to understand phenomena, learners of physics often view mathematics mainly as a technical tool for solving problems. In this study, we examine a possible way to bridge this gap: reframing introductory problems so that the structural role of mathematics in physical discovery becomes apparent. We conceptualize mathematically afforded learner-level discovery as a physically meaningful insight that is new to the learner and enabled by mathematical structure. Drawing on prior work on the roles of mathematics in physics, we present an abductive comparative analysis of nine canonical examples from introductory mechanics, electromagnetism, and waves, chosen for their affordances in this regard. The analysis led us to distinguish between two complementary dimensions of such discovery: discovery types, which characterize what becomes newly available to learners through mathematics, and discovery mechanisms, which characterize the physical-mathematical moves that make such discoveries visible. The identified discovery types include constraints, thresholds, extrema, and optima; dependences, independences, and regularities; emergent physical-mathematical quantities; structural unities across cases; and emergent physical phenomena. The identified mechanisms include, among others, recognizing functional form, examining bounds and limits, restructuring algebraic expressions, mapping across cases, analyzing dimensions, and revisiting model assumptions. The findings suggest that such discovery opportunities are already embedded in the introductory curriculum, but can become pedagogically available only when the relevant mathematical structures are made explicit. The resulting characterization can support teachers, curriculum developers, and teacher educators in explicating the epistemic role of mathematics in physics without replacing the canonical curriculum.