The conventional conceptual framework for fundamental physics is built on a tacit construct: the premise of particles being zero-dimensional (0-D) points. There has never been a viable alternative to this, and the Bell-type inequalities preclude large classes of alternative designs with hidden variables. Although they do not absolutely preclude the possibility of particles having non-local hidden-variable (NLHV) designs, there is the additional difficulty of finding a solution within the very small freedom permitted by the constraints. Nonetheless we show that it is possible to find such a design. We propose the internal structures and discrete field structures of this ‘cordus’ particule, and the causal relationships for the behaviour of the system. This design is shown to have high conceptual fitness to explain a variety of fundamental phenomena in a logically consistent way. It provides insights into the fundamentals of matter, force, energy and time. It offers novel explanations to long-standing enigmas and suggests that a reconceptualisation of fundamental physics is feasible. We thus show that the 0-D point premise can be challenged, and is likely to have profound consequences for physics when it falls.