The four-gradient of the electromagnetic four-potential has 16 independent components. Standard electrodynamics uses only the 6 antisymmetric components (the field tensor encoding E and B). The remaining 10 — including the scalar-longitudinal coupling and the trace that the Lorenz gauge sets to zero — are not proven absent. They are defined absent by convention. This paper traces the construction of that convention through three acts of deletion: Heaviside's vector reduction, the Lorenz gauge, and the ontological demotion of potentials. It identifies the physical content each removed, drawing on evidence spanning quantum interference to industrial engineering: the Aharonov-Bohm and Maxwell-Lodge effects; the scalar-longitudinal sector recovered independently by multiple research programs via the Stueckelberg Lagrangian, whose uniqueness is established by Woodside's decomposition theorems; the potential hierarchy from Hertz potentials through Whittaker's decomposition, where the Lorenz gauge emerges as an algebraic identity rather than a physical law; the persistence of longitudinal and scalar photon modes as dynamical variables in canonical quantum field theory (QFT); the time-symmetric sector hidden in quantum mechanics as the complex conjugate of the wave function; the electromagnetic-gravitational bridge deepened by Kaluza-Klein; the vacuum coupling demonstrated by the dynamical Casimir effect; and the violation of Newton's third law for open circuits, restored when longitudinal forces are included. The engineering implications are not speculative in origin. They are consequences of restoring degrees of freedom that the standard formulation structurally hides.