Obtaining consistent Lorentz gauging for a gravitationally coupled fermion

For internal gauge forces, the result of locally gauging, i.e., of performing the substitution $\partial \rightarrow D$, is physically the same whether performed on the action or on the corresponding Euler-Lagrange equations of motion. Rather unsettling, though, such commutativity fails for the standard way of coupling a Dirac fermion to the gravitational field in the setting of a local Lorentz gauge theory of general relativity in the vierbein formalism, the equivalence principle thus seemingly being here violated. This paper will present a formalism in which commutativity holds for the gravitational force as well, the action for the gravitational field itself being still the Einstein-Hilbert one. Notably, in this formalism, the spinor field will carry a world/coordinate index, rather than a Lorentz spinor index as it does standardly. More generally, no Lorentz indices will figure, neither vector indices nor spinor indices, which from a parsimonious point of view seems quite satisfactory.