The text is not 100% clear to me (partly because it is a bogus proof), but you seem to redefine the arithmetic operators as

  i ‘+’ j := i + √j
  i ‘-‘ j := i - √j
  i ‘×’ j := i + j
  i ‘÷’ j := i - j

That completely changes the structure of the ‘integers’. Essential properties such as

  i + j = j + i
  (i + j) + k = i + (j + k)
  i + i = 2 × i

no longer hold, so you can’t use the usual rules of arithmetic. For example, the PDF does

  ‘3’ × X ‘+’ ‘4’ = (‘3’ × X) ‘+’ ‘4’ = (‘3’ + X) + ‘4’ = (‘3’ + X) + ‘√4’

and then uses the rule

(‘3’ + X) + ‘√4’ = (X + ‘3’) + ‘√4’

but that rule doesn’t hold in the alternative arithmetic.

Also, ‘even’ and ‘odd’ are completely different from the traditional even and odd.

Fo example, for any even ‘n’, we have

  ‘n’ ÷ ‘2’ = n - 2

so division by two of an ‘even’ ‘number’ always keeps the number ‘even’, and for starting point ‘4’, we get an infinite sequence ‘2’, ‘0’, ‘-2’, etc. (also a direct counterexample to the claim that you always end up at ‘1’)