Lowering the scalar-dressing exponent

Establish whether Q-exact corrections can soften the poles of the single-pair rational operators sufficiently to lower the scalar-dressing exponent below the sufficient bound K=13, ideally to the dressing level realized by the SU(3) brane-fused operator O^3_{30}.

Background

The construction proves polynomiality using the sufficient denominator-clearing bound K≥13, but the authors explicitly note that this bound may not be minimal. They suggest that Q-exact corrections might cancel poles and permit substantially smaller dressings, potentially matching the lower-charge SU(3) example, but this has not been established.

References

The exponent K\ge13 is a sufficient bound from counting denominators, not a minimal dressing. Lowering the hair, ideally down to the value realised by O3_{30}, requires Q-exact corrections that soften the poles of O_S, rather than only better choices of the weights \Delta_SK; we expect this to be the case, but have not proved it. We will return to this elsewhere.

— Fortuitous BMN Cohomology at Every $N$  (2610.21592 - Koch et al., 15 Oct 2026) in Section 8, Discussion

It remains to identify which other dressings have this property and which can sustain an infinite sequence of fortuitous classes.

— Non-Coulomb $Q$-Cohomology and Quantum Lifting  (2610.22332 - Kibe et al., 15 Oct 2026) in Section 5.4, “Restrictions on graviton dressings”