> ... (when lat/lon is ambiguous sometimes).

mmm, debatable.   lat/lon always defines an exact point on the surface
of the globe.  What's breaking down is the projection of that globe
being mapped onto a surface of larger area.   That doesn't help you at
all in itself, but it might trigger some thinking about alternative
approaches to whatever it is you are trying to acheive ...
* The pixels you're trying to locate at the corners represent
imaginary spaces at low zooms, i.e. whether they ought to map to some
arbritary lat/lon at all is debateable.
* The "area" of the low-zoom display is meaningless in square miles,
because part of it represents imaginary and/or duplicated areas; not
sure if that's what your after.
* Some pixels in the body represent the same unique lat/long twice or
more, if the map is wrapped.
* Wrapping is an artefact of the display technology and algorithms, it
doesn't happen with paper maps.
* Many hundred pixels in the body represent a single lat/long in the
case of a pole, now that does happen on paper maps .... it's all
getting a bit philosophical!

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