Termination of the Ann-closure algorithm

Prove that the Ann-closure algorithm (Algorithm 6) always terminates without relying on an oracle—equivalently, show that the queue Q in Algorithm 6 eventually empties for every input—or provide a counterexample.

Background

The Ann-closure step is required to compute pseudo-ASL Gröbner bases in general. The current algorithm includes an oracle to check completion, and the authors conjecture intrinsic termination but lack a proof or counterexample.

References

As written, it is conceivable that the queue $Q$ in Algorithm \ref{alg:ann-closure} never empties; we conjecture that this cannot happen, but have been unable to either prove it or find a counterexample.

— Gröbner Bases Native to Term-ordered Commutative Algebras, with Application to the Hodge Algebra of Minors  (2510.11212 - Grochow et al., 13 Oct 2025) in Section 6 Algorithms, Remark (Regarding the oracle assumption)

We have not established general termination or the original amortized update-time bound for this worklist. We do not implement the deamortized or full worst-case construction.

— Scalable Approximate Algorithm for Dynamic Densest Subhypergraphs with Solution-Guided Maintenance  (2609.39778 - Chen et al., 30 Sep 2026) in Appendix, Section “CQ Reconstruction and Evaluation Protocol,” subsection “Deletion completion and implementation scope”