> Define a function z(n) = 1 - f(n)(n).

I don't understand the notation f(n)(n). Is it related to f_{nn} in LaTeX notation? Your later text suggests maybe it was aiming at f(n,n) so I will assume that.

I recognise a form of this argument and I might have tackled it in the supplementary materials I created that are referenced in the article. Let me know.

> However, z(k) = 1 - f(k)(k). Yet f(k) = z, so z(k) = 1 - z(k).

I'm assuming this was intended to be: z(k) = 1 - f(k). Yet f(k) = z, so z(k) = 1 - z(k).

For some k, z(k) = 0.5. f(k) = 0.5. Seems Ok.