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At this point you will discover that the earth's axis has
precessed by a nontrivial amount since the year 2000. The ICRS
declination of the zenith is not equal to your latitude, and the
ICRS right ascension of the zenith is not equal to the RA of the
southern horizon (azimuth 180°), which is just nuts. The
meridian really should be a great circle of constant RA.
The discrepancy is not super-huge, less than a minute of arc,
but it would show up in the analemma diagram. Seriously wrong
and ugly. Guess how I know.