Skin/Proxmity effects near transformer core
It is similar to the proximity effect, but it is not the proximity since there is only one conductor (as you have stated in your long comment thread with Andy aka , but maybe that is too much insisting on semantics since it is basically same phenomena ).
It comes down to Maxwell's equations, but this is the intuition:
By placing a core with high permeability at one side of the conductor your are changing the magnetic field H around the conductor. You can never really separate the inside of the conductor from the outside of the conductor and you have to solve Maxwell's equations for the whole system, by choosing the right boundary conditions i.e. your assumption about the skin effect that it is only an internal effect doesn't quite work because that calculation assumes a certain boundary condition for the surface of the wire.
So the two differential equations have to be solved:
$$
\nabla \times \pmb{H} = \pmb{J} \tag{1}
$$
and
$$
\nabla \times \pmb{E} = - j \omega B \tag{2}
$$
Now assuming the core has a permeability that is much higher than the permeability of free space the H field in (1) is very close to zero on the left side of the conductor (where the core is) and very high on the right side of the conductor (where you have free space). Now deriving the whole expression for this setup is very similar to the skin effect, but the boundary conditions are a bit different since left and right surface of the conductor aren't symmetric anymore.
As an aside: Andy aka has it backwards in their answer &quot;there is no magnetic field on the right of the rectangle and hence current finds it much easier to travel on the right edge of the rectangle.&quot; This is exactly the wrong way around. The current flow is precisely stronger where the magnetic field H is larger. Which you can actually see in the picture they posted with the proximity effect also and you can see in equation (1). You can convince yourself with the right hand rule that when the current goes in the same direction the H field cancels between the two conductors and no current flows there. When the current flows in opposite direction the H fields adds up between the two conductors and the current crowds towards the stronger H field towards the middle of the conductors.