Open status, literature, and significance

Date: September 9, 2026. Author: Henry Zweiman. Status: internally checked proposed theorem for two infinite families; historical priority and outside mathematical assessment remain provisional.

The precise published target

Bashir, Geroldinger, and Zhong, On a zero-sum problem arising from factorization theory, Conjecture 1.1, asks for consecutive factorization lengths over the signed support of every longest atom in a finite abelian group other than the cyclic and elementary two-group cases. The chapter appeared in 2021, DOI 10.1007/978-3-030-67996-5_2.

The equivalent invariant formulation is still explicitly Conjecture 2.3 in Doniyor Yazdonov, On the structure of length sets with maximal elasticity, version 2, revised February 25, 2026. The source's density results and its Property P do not prove the assertion for every individual longest-atom support. Its standing-conjecture statement and the relevant theorem hypotheses were checked directly.

P29 proves the signed-support assertion for noncyclic groups C2^a + C4^b + C8^c whenever a+c is not 1 modulo 3, or c=1. It also proves that only distances 1 and 3 can occur for arbitrary groups of this form. The remaining exponent-eight cases are not declared solved.

Closest established results and dependencies

• The 2021 theorem already covers rank-two and homocyclic cases, a coprime-parameter condition, and two-level p-groups under a divisibility condition on the exponent indices. In particular the order-four/order-eight indices are 2 and 3, so that condition does not cover C4^b + C8 for arbitrary b. Both the preprint theorem and the author's 2021 theorem poster retain that condition.
• Geroldinger and Zhong, Long sets of lengths with maximal elasticity, Canadian Journal of Mathematics 70 (2018), Lemma 3.10, is used as an external structural lemma. Its full proof and exact hypotheses were read. P29 uses its non-half-order relation criterion only with least multiplier 2 and element order 8, and also uses its basis-support conclusion.
• The signed-atom length divisibility, Davenport formula for p-groups, and truncated group algebra are established tools. P29 gives short self-contained arguments for its precise uses of these tools.
• Geroldinger and Yang, On a classical zero-sum invariant, August 19, 2026, was checked through its relevant Section 3 results and proofs. Its group-algebra identities and order-of-terms consequences are credited; no novelty is claimed for the group-algebra method. The result sought here concerns a different signed-support distance invariant.

Targeted searches for the exponent-eight maximal-elasticity case and the C4/C8 support case did not locate a prior theorem settling these full families. This is a primary-source-informed search, not an exhaustive proof of priority. The explicit general conjecture does not itself establish that each subclass was previously open.

Contribution and potential use

The main structural contribution is the classification of order-eight support terms when every atom length is two modulo three, together with the elimination of lower-order terms by extremal augmentation degree. It treats arbitrary numbers of order-two and order-four factors when the order-eight rank is one. The broader filtration bound identifies distance three as the sole residual obstruction for all exponent-eight groups.

The consequence for transfer Krull monoids is a uniform interval description of maximal-elasticity length sets. Thus the result connects a zero-sum support theorem to factorization invariants of monoids and domains. Extending the support classification to several order-eight coordinates is a concrete route for further work. No citation-count forecast is made.

Review limits and counting

The proof and computations were checked by the originating agent. No separate-agent review, human expert review, formal proof-assistant verification, or journal acceptance is claimed. This is one consolidated paper, not one paper per parameter family or lemma. Publication contributes one manuscript to the public inventory; it does not by itself certify a significant open-problem solution toward the requested target of 45.