Exact closure for H3, Q3, and A2 via alternative Lagrangian multiforms

Determine whether there exist alternative discrete Lagrangian 2-form constructions for the ABS equations H3, Q3, and A2 whose action over any elementary cube in Z^N vanishes exactly (closure), rather than only modulo 4π^2, thereby achieving the exact closure property without relying on the present integer-field augmentation or with a modified formulation.

Background

The paper proves that the trident Lagrangian multiform yields corner equations equivalent to the ABS quad equations and establishes closure on solutions for H1, H2, A1, Q1, and Q2. For H3, A2, and Q3, the authors show that the action around an elementary cube is only determined modulo 4π^2 and present numerical evidence of non-zero multiples, suggesting that exact closure does not hold for their current formulation.

To accommodate this, they propose a multiform principle allowing integer fields that depend on the chosen surface and relaxing the closure condition to congruence modulo 4π^2. The explicit open question seeks alternative multiforms for H3, Q3, and A2 that would restore exact closure (action equal to zero).