An efficient universal simulation with a clock

Fix a multitape universal simulator $U$, using the finite transition-table encoding convention of A fixed effective binary encoding of deterministic one-tape Turing machines, extended to finitely many tapes by recording the tape count and the finite table on tuples of scanned symbols. Here machines have the multitape semantics of The classes DTIME, NTIME, DSPACE, and NSPACE. On a self-delimiting pair $z=⟨M,x⟩$ of total length $n$ and clock value $b\in \mathbb{N}$, $U_b$ first decodes the pair and then simulates at most $b$ steps of $M(x)$, rejecting malformed encodings or a clock expiry. For each fixed $M$, this uses $O(n+b\log (b+2))\text{time}\text{and}O(n+S+\log (b+2))\text{space}\mathrm{.}$ These bounds apply with the binary numeral for $b$ already supplied and when the simulated computation visits $S$ cells; their constants may depend on $M$. The $O(n)$ terms pay for decoding and retaining the encoded input, including when $b=0$ or $b=1$. For a clock constructed from the input length, fix a time-constructible function $h:\mathbb{N}\to \mathbb{N}$ with $h(n)\ge n$ eventually and a constructor $C_h$ as in Time-constructible and space-constructible functions. Run $C_h(1^n)$ to obtain $b=h(n)$ before invoking the simulator. This adds $O(n+h(n))$ time, so total time is $O(n+h(n)\log (h(n)+2))$. If this constructor uses $W_h(n)$ cells, total all-tapes space is $O(n+W_h(n)+S+\log (h(n)+2))$. No constructor-space bound smaller than $W_h(n)$ is implicit. Small externally supplied clocks, including zero and one, remain covered by the first interface.