An everywhere-total functional has a computable use bound

For fixed $n$, effectively search for length $m$ and time $t$ such that every binary string $\sigma$ of length $m$ makes ${\mathrm{\Gamma }}^{\sigma }(n)$ halt by $t$ without querying outside $m$.

If this search never succeeded, the finitely branching tree of strings whose finite-oracle simulation has not supplied such a transcript would have nodes at every length. König's lemma yields an infinite oracle on which $\mathrm{\Gamma }(n)$ never halts, contradicting totality.

Therefore the search halts; let $b(n)$ be its first successful length. The finite exhaustive test makes $b$ computable, and its defining property forces every oracle computation on $n$ to use only positions below $b(n)$.