If the GL(4) incompatibility argument in P1 establishes necessity, it establishes only that some local Lorentz structure is needed, not specifically the vierbein formulation as distinct from equivalent spin-connection approaches.
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incompatibility argument(as used in logic and philosophy of physics)
A logical case for showing that two things cannot both be true at the same time.
local Lorentz structure(used in physics and philosophy of space-time)
A mathematical framework that describes how space and time work in small, localized regions, based on Einstein's theory of relativity.
spin-connection approaches(used in general relativity and theoretical physics)
An alternative mathematical method for describing how spacetime is structured, focusing on how rotations change from place to place.
vierbein formulation(used in general relativity and differential geometry)
A specific mathematical way of describing the structure of spacetime by using four basis vectors at each point—it's one approach among several equivalent methods.