I) By design, nothing regarding Math here

IV, V, VI, VII) More related to a lexical fact more than Math.

XIV) 10x - X = 8.9999....1 for every finite sequence of any length. Unless you prove 8.999... = 9, you didn't prove anything.

My 2 cents. Let's assume 0.9999... periodic is different from 1. Let's define x = 1 - 0.9999... as the difference of the 2 numbers, which is not 0 by assumption.

Now 0.9999... + x = 1 by definition. But that is absurd, because any value of x not 0 will cause 0.9999... + x to be different from 1. So, our initial hypothesis that 0.9999... is different from 1 is false.

Thus 0.9999... must equal 1.