Bob Miller wrote:

Maybe one could say the ground side of the balanced line and its
dipole quarter wave are acting as an elevated counterpoise to the
"random wire" on the other side?

Antenna systems are often a lot easier to analyze and understand if you
put aside concepts like "counterpoise" and "ground".

An antenna is a two terminal device, even if it's "end fed". The
transmitter is also a two terminal device. Connect the two together and
you have an electrical circuit. Like any circuit, the current leaving
one terminal has to equal the current going into the other terminal.

So what happens with an end fed antenna? Well, whatever the current
going into the antenna (and current must go into it, since the power
into it is I^2 * R, where R is the sum of radiation and loss
resistance), an equal and opposite current must go somewhere else. If
you succeed in completely choking off the current going somewhere else,
you've also succeeded in choking off the current going to the antenna.
So you don't want to do that.

Conductors don't care what label you put on them -- calling one a
"ground" or "counterpoise" doesn't give it magical properties. When a
current flows on a conductor, it creates a field. This field will
radiate unless canceled by other fields. So the "somewhere else" that
the current flows is just as much an antenna as the supposed antenna is.

If the current has nowhere else to go, it'll go down the outside of the
coax, which will effectively become the other half of a dipole. If you
choke off the current on the outside of the coax, it'll go somewhere
else if it can. But if there is no other place, then the current to the
antenna will drop -- the feedpoint impedance will increase.

If the current goes into two or more radial wires which are
symmetrically placed, the fields from the wires will largely (but not
completely) cancel, so the net radiation from the radials will be small.
This can reasonably called a "counterpoise" -- a place for the current
to flow without creating much radiation. Or you can connect the antenna
to a buried radial field ("ground"), which behaves much the same way,
but with even better field cancellation. But people often put these
names on other configurations, expecting the currents or fields to
behave differently than on other conductors.

But the things to remember are that all antennas have two terminals, and
the current into one equals the current out of the other. And current
flowing along any conductor creates a field, whether you consider it to
be an "antenna" or not. Keeping this in mind helps a lot in
understanding end fed and other antennas.

Roy Lewallen, W7EL