= Solution
\b[The comparator alone cannot generally reproduce informed <A-star search> priorities. A valid weaker fallback is <uniform-cost search>, namely <A-star search> with $h=0$.] The issue is that ordering $h^*(n)$ and $h^*(m)$ does not supply the numerical difference needed to compare $g(n)+h^*(n)$ with $g(m)+h^*(m)$.
For example, take accumulated costs $g(n)=5$, $g(m)=1$. Remaining costs $h^*(n)=1$, $h^*(m)=2$ rank $n$ ahead under $\kappa$ but give total estimates $6$ and $3$, so informed <A-star search> selects $m$. Remaining costs $1$ and $10$ give the same pairwise comparator ranking but totals $6$ and $11$, selecting $n$. Thus the comparator's ordering cannot simply replace the cardinal <heuristic function> inside $f=g+h$.
If “do <A-star search>” includes choosing any available <admissible heuristic>, set it identically to zero and prioritize by the known accumulated cost $g$. Apply the <goal test> on removal from the <search frontier>. This is complete and optimal under the stated finiteness and step-cost assumptions, but it does not recover the unavailable perfect-<heuristic function> ordering or its search reduction. The distinction is between implementing the informed priority from $\kappa$ and retaining the algorithm's guarantees through this uninformed special case.
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