How does this proof show that sequences of $O(1)$ polynomially bounded Kolmogorov complexity are NOT the polynomial computable ones?
Wow, that was sooooo long ago. Something like the early nineties...

The point is, for that T, at the time of writing, the construction proper was assumed to be well-established in the background knowledge of any potential reader. Thus, we did not give details. Essentially, what you do is a Cantor-like proof as in the non-numerability of the reals: for each i, run the i-th DTIME(f) machine on a string of s(i) zeros, and add that string to T if and only if the simulated machine rejects it. (Colloquially that string was often called "the cookie".) Then g is to be any time-constructible function that allows for this process to be completed - for this we will need it considerably larger than f. There is a slight circularity in that the process depends on s, which depends on g, which is defined as sufficient time to run the process. But it is easy to upper bound this time and select a sensible g.

Hope this helps! (And hope this works. It is my first time writing anything into stackexchange.)