Question as regards a proof of the Time Hierarchy Theorem
Your arguments is as follows:


If $N$ rejects $[N]$ then $([N],[N]) \in H_f$.
Since $([N],[N]) \in H_f$, $N$ accepts $[N]$ in $f(m_n)$ steps.


This is indeed a contradiction, so we conclude that $N$ accepts $[N]$. Then we argue again:


Since $N$ accepts $[N]$ then $([N],[N]) \notin H_f$.
Since $([N],[N]) \notin H_f$, it is not true that $N$ accepts $[N]$ in $f(m_n)$ steps.


Since the running time of $N$ is $f(2m_n+1)^3$ in your case, there is no contradiction. After $f(m_n)$ steps the machine $N$ might still be running.