That's just my point. If 11.59 p.m. is immediately before midnight then the only justification for changing the signifier is if you re-start counting the hours as 0.00.
Otherwise you appear to have 4 sections of the day 1.00 a.m to 11.59 a.m./ 12.00 to 12.59 p.m./ 1.00pm to 11.59 p.m. / 12.00 to 12.59 a.m.
In each case, the signifiers change before the numbering system is restarted, and 12.01 a.m. is one hour later than 11.01 p.m. rather than being one hour after 11.01 a.m.

Nigel