The study of triacylglycerols (TAGs) in their molten state is of fundamental importance for a deeper understanding of the TAG crystallization processes, being highly relevant for both manufacturing and medical applications. Although different models have been proposed to explain the nanostructured nature of the fluid state of TAGs, none of them are fully satisfactory. In this paper, we propose a new model consisting of positionally uncorrelated lamellar TAG assemblies embedded in an isotropic medium that assist as prenucleating structures. This model was validated by applying a novel global fitting method, resulting in an excellent agreement with the small-angle X-ray scattering data. A deeper analysis of the scattering patterns at different temperatures, both in cooling and heating directions, allowed us further to detect the crystalline traces of TAGs even after heating to 40 degrees C and record, on cooling, the onset of crystallization at 30-25 degrees C. The application of the presented novel model not only explains the outstandingly structured fluid of molten TAGs, but also lays the basis for analyzing first the crystallization steps in greater detail, which is outlined in our follow-up paper "Global Small-Angle X-ray Scattering Data Analysis of Triacylglycerols in the alpha-Phase (Part II)".