The read-only-workspace diagonal machine halts

Compute $s(n)$ in $O(s(n))$ work space, simulating the constructibility machine on a virtual unary input of length $n$ if necessary. For varying descriptions, reject when the universal simulator's entire encoded configuration and scratch storage would exceed $s(n)$ cells. A configuration below this cap is described by $O(s(n))$ bits, together with $O(\log (n+2))$ bits for input and program addresses. Thus the number of possible configurations is at most $2^{O(s(n)+\log (n+2))}=2^{O(s(n))}$. The cap may reject a computation before this bound is reached.

Use the uniform encoded-storage simulator and a binary counter of $O(s(n))$ bits to count through this configuration bound, stopping at expiry or an earlier storage overflow. A deterministic computation that remains within the cap and has not halted by then must repeat a configuration. The forced answer therefore makes the simulation total while the simulator, counter, and address bookkeeping together use $O(s(n))$ work space. For a fixed machine the visited-cell version of the supplier gives the analogous bound.