# complexity theory – Time constructible function T in equivalence of Turing Machines

I’m reading Computational Complexity by Arora and Barak and I had a doubt regarding a statement made about the equivalence of Turing machines:

For every $$fcolon {0, 1}^∗ → {0, 1}$$ and time-constructible $$Tcolon mathbb N → mathbb N$$, if $$f$$ is computable in time $$T (n)$$
by a Turing machine $$M$$ using alphabet $$Gamma$$ then it is computable in time $$4log|Gamma|T (n)$$ by a Turing machine $$tilde{M}$$ using the
alphabet $${0, 1, square, B}$$.

Specifically, what if any role $$T$$ being time constructible has here?

The proof (sketch) encodes $$Gamma$$ in binary, and uses a larger register to simulate $$M$$‘s transition function, and also a counter. But I don’t think $$T$$ being time constructible comes into play. Could someone confirm or tell me if I’m missing something?