The finite barycentric evaluator terminates with an explicit status

The procedure first checks the length of $X$ and, if it is correct, scans its $n+1$ node candidates for membership in $\mathbb{F}$ and strict order; any failure returns invalid_nodes. If those checks pass, it checks the length of $Y$ and, if it is correct, scans its $n+1$ data values and the query for membership; any failure returns invalid_data. These checks are finite. Otherwise the exact-node branch returns a data value in $\mathbb{F}$. For an off-node query, the procedure next checks whether $1\in \mathbb{F}$: if not, it returns range_failure before initializing any arithmetic with $1$; if so, it proceeds to weight construction.

After the guard in step 1.1, $1\in \mathbb{F}$, so each initial $d_j=1$ and each reciprocal numerator $1$ is in the checked-operation domain. Every successful checked operation returns an element of $\mathbb{F}$, preserving that domain for the next operation. Weight construction visits at most $(n+1)n$ node pairs and performs finitely many such operations; any failure returns range_failure immediately, so this stage terminates.

If weight construction completes, the evaluation loop starts with accumulators $N=D=0\in \mathbb{F}$ and visits at most $n+1$ terms. Every checked operation either returns a value in $\mathbb{F}$ or exits with range_failure. If the loop completes, a zero denominator returns range_failure; otherwise the checked final division either returns range_failure or returns success with a value in $\mathbb{F}$.

These cases exhaust the procedure's branches and every loop is finite, so every input terminates with a declared status.