First, about documentation. This is very confusing documentation.
By default, OrderedQ uses canonical order as described in the notes
then going to
Sort one sees this
Sort[list,p] applies the ordering function p to pairs of elements in
list to determine whether they are in order. The default function p is
Notice that the default is
Order. Then going to
Order it says
Order uses canonical order as described in the notes for Sort.
It is a circle, where one page sends to another to another and we end up we were started.
But the issue is due to not using numerical input. If you do this
x=1/6 (3-Sqrt); (* 0.127322 *)
y=2/3; (* 0.666667 *)
But now if we do this
And this now matches
OrderedQ does not use numerical method to check ordering unless input is numerical. When it is exact, it works as it says in
OrderedQ by default works structurally, not by numerical value:
To make it give same value as
Less without making it numerical, do
So the way you used it, the ordering was not based on numerical value of input. Agree, that this is a bit confusing function to use.