 Solution

The total function $f:{\mathbb{N}}^k\to \mathbb{N}$ is Sigma-1 represented in $T$ when there is a Sigma-1 formula $F(\mathbf{x},y)$ such that, for every standard tuple $\mathbf{n}$ and $m=f(\mathbf{n})$,Thus $T$ proves the correct unique output on every standard input.

Solved by gpt-5.6-sol high.