Open status, dependencies, and significance assessment

The precise target is Geroldinger--Kim--Loper, arXiv:2502.21020v3, Problem 19: characterize the ideals of a Krull monoid with finite v-class group that are conductors of C-monoids with that complete integral closure. The latest arXiv version was checked directly on September 9, 2026; v3 is dated February 28, 2026 and still poses the problem. Initial reconnaissance used v2; the final manuscript cites v3. The corresponding AMS chapter is identified by DOI 10.1090/conm/836/16701, but its publisher page could not be opened in this audit. No direct reading of the final AMS chapter is claimed.

The proposed theorem answers the full monoid formulation, with its empty, whole, reduced, nonreduced and group cases stated explicitly. It gives both a criterion on the ideal alone and a universal explicit realizing monoid. It is distinct from P26's isomorphism question: this theorem concerns which ideals admit any C-monoid realization, and its proof does not depend on P26's reconstruction theorem.

The classical structural input is the dense embedding theorem for C-monoids. It was verified in Geroldinger--Zhong, A characterization of seminormal C-monoids, Section 2, equation (2.4), and the paragraph preceding the proof of Theorem 1.1. That background theorem is explicitly stated for arbitrary C-monoids. No seminormality hypothesis is imported or silently dropped. The source cites Geroldinger--Halter-Koch, Theorems 2.9.11--2.9.12. The book itself was not newly read in full during this audit. Divisor theory, its uniqueness, and Dickson's lemma are classical.

The finite prime-profile/threshold equivalence is an elementary monomial-ideal fact, proved in the paper; it is not advertised as a new automata theorem. The contribution is its necessity for every possible C-monoid realization in the canonical divisor monoid, together with the converse construction and the exact unit obstruction. The necessity proof carefully uses the finite reduced class semigroup rather than incorrectly assuming the full class semigroup is finite.

Reinhart's A note on conductor ideals treats ring conductors relative to a prescribed subring. Its introduction, main criterion and scope were inspected. Those additive conditions differ from the purely multiplicative problem. The construction in this paper is not claimed to produce subrings or orders. The finite-rank conductor results in Geroldinger--Yan--Zhong, DOI 10.1007/s00233-025-10583-6, are related special settings, not a substitute for the infinite-rank necessity argument here.

Targeted searches combined 'C-monoids', 'conductor ideals', 'Krull monoid', 'Problem 19', 'recognizable', 'finite partition' and 'prime profiles'. They located the source problem and related classical work but no later resolution of this precise criterion. This is a record of the search, not a proof of exhaustive historical priority.

Potential significance: a full conductor-existence question is reduced to a concrete invariant of an ideal; the criterion admits infinite families of primes and non-finitely-generated ideals, and gives finite control of the resulting arithmetic class semigroup. It can support later study of arithmetic with prescribed normalization and conductor. Predicted citation counts are not evidence. External proof review, historical priority and significance assessment remain outstanding; the global program does not equate manuscript count with certified significant solutions.