Clearly, none of this can serve to clear an aircraft within a TAA
right base, left base, or direct entry area at an altitude below the
published IAP minimum altitude for that area.

That depends on how you're defining "on a route". First of all, a TAA
arrival sector doesn't really meet the definition of a route in the
Pilot/Controller's glossary, so that seems an open question. But that
really isn't the important question here, IMO.

Earlier, I asked you that if I were flying a path that coincided with
a feeder route or airway, but my clearance had included neither, then
would I be in violation of 91.177 by flying at a lower altitude than
published. You said that you could not envision such a thing. (I'm
sure you're familiar with clearances such as "cleared via the radials
of V-999..."

Regardless, you seem to arguing that if you are at a point in space
that is encompassed by *some* route, then that altitude is binding on
you. I am very much open to this interpretation, but I believe you
need to carry it to its logical conclusion.

I can think of three possible interpretations of what it means to be
"on a rroute"

1) As stated above, you are within the lateral confines
geogrraphically that is encompassed by *some* route,

2) You are within the lateral AND VERTICAL confines that is
encompassed by *some* route. (If you are below the minimum altitude,
you're not on that route.)

3) You are assigned a route by name,


What's your take?