Open status, sources, and significance
September 9, 2026. Author: Henry Zweiman. Status: complete internally checked proposed infinite counterexample family; historical priority and outside mathematical assessment remain provisional.
The published conjecture
Gao and Thangadurai, Davenport constant and non-Abelian version of Erdos-Ginzburg-Ziv theorem, Definition 3 and Conjecture 4 on PDF page 4, define the holes as nonzero omitted subsequence sums and require them to lie in a coset alpha+H with alpha outside H. Their conjecture is D(G)=nu(G)+2. This is a primary source coauthored by the conjecture's originator. The publisher's volume-2 table of contents identifies the chapter at page 57 of the 2006 Ramanujan Mathematical Society collection.
Geroldinger and Yang, On a classical zero-sum invariant, arXiv:2608.19090v1, August 19, 2026, equation (1.1), introduction, and Section 6, give the same definition and explicitly retain the conjecture as open. Their small Davenport constant d equals D-1. The manuscript uses this convention consistently.
The original Gao 2000 article, DOI 10.1016/S0012-365X(00)00010-8, was identified, but its publisher PDF returned 403. The definition and conjecture were checked in the two primary sources above; no claim is made to have read the inaccessible original article body.
Essential external theorem
Baayen's original Report ZW 1969-006, printed page 1 / PDF page 3, states that every sequence of length 2n+3 in C2^3+C2n has a nonempty zero-sum subsequence for odd n. The familiar basis sequence gives the matching lower bound, so d(G)=2n+2. The original report was retrieved and its statement checked directly. Schmid's 2011 introduction and Geroldinger-Yang Proposition 2.1 independently record the same formula. P30 credits this established Davenport result.
The coset obstruction, exact six-hole computation, sparse-construction parity criterion, and extremal extensions have self-contained hand proofs in the manuscript. The finite checks do not supply the Davenport upper bound or replace a universal proof.
Why the conclusion addresses the conjecture
The constructed sequence has length d(G)-1, not a shorter length. Its missing set is exactly six nonzero elements. An elementary difference argument excludes every coset with representative outside its subgroup; it is not merely an index-two test. At length d(G), all nonzero elements occur as subsequence sums, which proves the exact value nu(G)=d(G). Thus this is a proposed negative answer to the universal conjecture, rather than an unresolved sufficient condition or a failure of only its stronger refinement.
The holes do lie in a proper subgroup. Allowing the zero coset would change the question and would avoid this particular obstruction. Both audited definitions explicitly exclude that interpretation.
The result also determines the index-two refinement and gives maximal zero-sum-free sequences with the larger local invariant. These consequences belong to the same paper. Even n, all-group classification, and unrelated conjectures of Gao remain outside scope.
Priority and research value
Exact invariant/coset/counterexample and group-family searches did not locate a prior counterexample. The latest primary source explicitly states that no example with nu(G)=d(G) was known. Search absence is not an exhaustive priority guarantee, and outside review remains necessary.
If upheld, the theorem gives infinitely many counterexamples to a longstanding conjecture using a uniform six-hole obstruction. It provides a concrete restriction for any revised classification of omitted subsequence sums and their coset coverings. The proof is short once the construction is found; no extra papers are counted for the individual n or the supporting lemmas. No citation-count prediction is made.
All work was reviewed by the originating agent. No human review, separate-agent review, formal proof-assistant verification, or journal acceptance is claimed. Publication adds one proposed manuscript; it does not automatically certify one significant solution toward the 45-paper goal.