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Almost-closure for Q4

Establish an almost-closure relation for the action over an elementary cube associated with the ABS equation Q4 by generalising the deformation-based closure arguments, including appropriate extensions of Lemma \ref{lemma-Sjump} and a precise analysis of the relation between the multi-affine variables v(t) and the transformed variables V(t) in the limit t → 0, and determine the constant (expected to be related to elliptic half-periods) that replaces 4π^2 in this case.

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Background

The authors do not treat Q4 in this part but suggest their techniques could extend to it. For H3, A2, and Q3 they establish an almost-closure (action determined modulo 4π2), while for Q4 they expect a related but different constant linked to elliptic function half-periods.

They explicitly defer the proof of almost-closure for Q4, indicating it will require non-trivial extensions of a key lemma about branch jumps and a careful paper of the scaling limits relating multi-affine and transformed variables.

References

We are optimistic that these arguments can be generalised to prove almost-closure of Q4, where we expect the $4 \pi i$ will be replaced by a quantity related to the half-periods of the elliptic function underlying this equation. This is left for future work, as it will require non-trivial extensions of Lemma \ref{lemma-Sjump} and a careful study of the relations between multi-affine variables $v(t)$ and transformed variables $V(t)$ in a suitable limit $t \to 0$.

— Discrete Lagrangian multiforms for ABS equations I: quad equations (2501.13012 - Richardson et al., 22 Jan 2025) in Section 5 (Conclusion)