The OP also showed that the var isn't named "^X" and that it is named "\cX" before you claimed the opposite.

Furthermore, perlvar doesn't contradict anything I said. In fact, it says exactly the same thing I said. perlvar specifically says that $^X is a "single-control-character names" and that Perl "understands ^X (caret X) to mean the control-X character". Yes, perlvar says that $^X is just an alias for $. It even specifically uses $^X as the example!

>perl -le"print qq{print qq{\$\cX}}"
print qq{$}

>perl -le"print qq{print qq{\$\cX}}" | perl -l
...\perl.exe