I'm not versed at all into the Hodge star operator but it feels like Maxwell's equations expressed in both systems[1] might locate the answer. With Hodge star:

    dF = 0
    d * F = J

Expressed in GA:

    ∇F = J

The eyeball-difference of which (dF = 0) would be your "how GA then blends all of this together", if I understand correctly. I'd guesstimate that dF = 0 to be akin to Gauss' law; and that maybe GA somewhat incorporates that the curl of a gradient is the zero field.

[1] https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...