A universal simulator can realize T steps with machine-dependent O(T log T) overhead

By [L1], the simulator can first decode $e$ into the finite control data of $M$ and copy those data to a read-only program track. It then stores the current simulated configuration in a geometrically blocked tape layout whose block sizes double outward from the simulated heads. This representation keeps recently accessed cells near the center while allowing farther regions to be reached after only logarithmically many block crossings.

One simulated step of $M$ changes only the current state, one scanned tape symbol, and one head position. In the blocked layout from step 1.1, finding the current symbol, updating it, and recentering after a head move costs amortized $O_e(\log T)$ over the first $T$ simulated steps: most steps touch only the central blocks, and a level-$j$ reblocking costs $O_e(2^j)$ but occurs only once per $\mathrm{\Omega }(2^j)$ simulated head moves. Summing over the $O(\log T)$ occupied levels gives the asserted amortized bound. The subscript records the fixed cost of interpreting the finite alphabet, state set, and transition table encoded by $e$; the cited efficient-universality theorem permits exactly this machine dependence. No constructibility hypothesis on the integer $T$ is used.

Summing the amortized $O_e(\log T)$ cost over at most $T$ simulated steps gives total simulation time $O_e(T\log T)$. The initial decode-and-load work contributes $O_e(\mathrm{\mid }e\mathrm{\mid }+\mathrm{\mid }w\mathrm{\mid }+1)$. If the simulation halts with output word $u$, at most $\mathrm{\mid }w\mathrm{\mid }+T+1$ simulated tape cells can belong to its visited output segment. A final scan therefore converts $u$ to the self-delimiting word $\widehat{u}$ in $O_e(\mathrm{\mid }w\mathrm{\mid }+T+1)$ time, which is absorbed by the displayed bound. By [L2], the simulator is universal because it emits exactly $\widehat{u}$ and, when $M$ diverges, continues simulating forever rather than entering a halting state.

Therefore, for each fixed valid code $e$, the machine $U$ has the claimed universal behavior with total running time bounded by $c_e(\mathrm{\mid }e\mathrm{\mid }+\mathrm{\mid }w\mathrm{\mid }+T\log T+1)$ for a suitable constant $c_e$ independent of $w$ and $T$.