Physical significance of the non-integrable sector

Determine what physical significance, if any, can be attributed to the non-integrable sector of the standalone Herglotz theory, whose scale 1-form is not required to be exact.

Background

The paper distinguishes between an integrable sector, in which the scale 1-form is exact and the dynamics can reproduce Palatini gravity, and a non-integrable sector in which that exactness condition is absent. The non-integrable sector enlarges the solution space of the standalone Herglotz theory beyond the solutions associated with Palatini gravity.

The authors explicitly state that the physical interpretation of this additional sector has not been established. This is an unresolved question concerning whether the mathematically permitted non-integrable configurations correspond to physically meaningful gravitational structures or merely arise from the broader contact-geometric formulation.

References

It is not currently clear what, if any, physical significance may be attributed to the non-integrable sector.

— Towards a Connection Formulation of Action-Dependent Palatini Gravity  (2608.24140 - Bell et al., 25 Aug 2026) in Section 5, “Integrable and Non-Integrable Sectors”

Our analysis was somewhat brief, and so further exploring whether these non-integrable configurations could be realised within a physical theory remains an open and interesting question.

— Towards a Connection Formulation of Action-Dependent Palatini Gravity  (2608.24140 - Bell et al., 25 Aug 2026) in Section 9, “Summary and Discussion”