Resolve the P₇^(0)-breaking dressed-parent contribution

Determine whether component-resolved ultraviolet parent operators dressed by P₇^(0)-odd vacuum insertions, especially the special 1463_H singlet S, generate a nonzero coupling to the selected dark direction in the separated 1539_H construction.

Background

Moving the dark Higgs to the separated 1539_H parent makes all undressed pure-dark operators containing the selected Xi_D and only dark G(2) gauge-sector insertions vanish by the canonical E₇ charge-flip symmetry. This removes the undressed dimension-nine hook and, more generally, the undressed one-Xi_D tower.

The result does not cover operators containing P₇0-odd ultraviolet spurions. The special 1463_H singlet S is itself odd under the relevant charge flip, and its benchmark VEV is of order the cutoff, so an S-dressed operator would not receive a parametrically small suppression if its parent contraction and selected component are nonzero. Establishing or excluding that matching is therefore necessary for complete ultraviolet protection.

References

Conversely, the charge-flip argument merely permits this dressing; it does not prove that the required $E_7$ parent and selected component are nonzero. That dressed matching coefficient remains open.

— Dark matter glueball candidate from a $G(2)$--$E_6$--$E_7$ exceptional grand unified theory: $G$-parity, spin, mass and stability  (2609.04296 - Masi, 3 Sep 2026) in Section 2, subsection “The first physical pure-dark parent channel is the d=9 hook”; Section 2, subsection “Outcome of the one-copy consistency tests”; Conclusions, paragraph “The separated 1539_H completion”