Potential was, as said above, is, energy per unit mass or charge. So, we see that, in defining the higher quantity energy or potential (higher therefore closer to action, hence more fundamental or unified) we have to INTEGRATE the lower variable, here, Force, Field or (Any ) 3-vector. This entails therefore arbitrariness into the Physical solution when we solve for these quantities. These physical problems, as they involve differentials or integration, leads to a differential equation. Under further suitable physical conditions called eg laws of nature or physics, become whats called a wave-equation or for particles, equation of motion. We can say equation of motion for particles or equation of motion for waves if they are separate.
Now that we understand what are potential, field, vector and gradient and integral in relation to each other, comes requirements called as symmetry or laws of nature or laws of physics or in simple, boundary conditions to these differential equations known as, wave equation or equation of motion of particles OR waves. ( — which are separate so far )
These equations constrained by the conditions or restrictions which are attributes of physical observation, must therefore unite these variables (potential, field) into one entity which would satisfy the wave or particle equations of motion, the differential equations of motion in PARTICULAR ways only, known as Laws of Nature or Physics. So they become, from their 3-vector or scalar attributes, 4-vectors (or still higher, Tensors).