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I don’t know whether this would hold generally, but if we could show that (a) the universe is S4, (b) there exists a 1-form $A$ defined on $S^4(1/h)$ representing the electromagnetic potential, (c) gravitational field equations are the critical points of an Einstein-Hilbert functional which include the ‘matter Lagrangian’ terms from $F=dA$ for which the gravitational field equations result from the Euler-Lagrange equations of this functional.   Regardless of the explicit forms involved the structure of the problem is such that the metrics that can arise for any ‘physical universe’ are embeddings into $S^4(1/h)$ and therefore will always be (at least partially) determined by $A$.  In this case, gravity is being determined not simply by electromagnetism but also by restricting to an S4 ambient space.  The approach does not require explicitly provided cosmological constant: pick any critical hypersurface and compute the Ricci curvature and the cosmological constant $h^2$ will be included as the Ricci curvature of the ambient sphere which happens to be extremely close to the measured value.