Efforts to catalogue the structure of metabolic networks have generated highly detailed, genome-scale atlases of biochemical reactions in the cell. Unfortunately, these atlases fall short of capturing the kinetic details of metabolic reactions, instead offering only topological information from which to make predictions. As a result, studies frequently consider the extent to which the topological structure of a metabolic network determines its dynamic behaviour, irrespective of kinetic details. Here, we study a class of metabolic networks known as non-autocatalytic metabolic cycles, and analytically prove an open conjecture regarding the stability of their steady states. Importantly, our results are invariant to the choice of kinetic parameters, rate laws, equilibrium fluxes and metabolite concentrations. Unexpectedly, our proof exposes an elementary but apparently open problem of locating the roots of a sum of two polynomials S = P + Q, when the roots of the summand polynomials P and Q are known. We derive two new results named the Stubborn Roots Theorems, which provide sufficient conditions under which the roots of S remain qualitatively identical to the roots of P. Our study illustrates how complementary feedback, from classical fields such as dynamical systems to biology and vice versa, can expose fundamental and potentially overlooked questions.