Dark matter glueball candidate from a $G(2)$--$E_6$--$E_7$ exceptional grand unified theory: $G$-parity, spin, mass and stability

Abstract: We determine the gauge-invariant identity, mass scale and ultraviolet stability of the dark matter candidate arising from (G(2)\to SU(3)C) in the exceptional (G(2)!-!E_6!-!E_7) construction. The broken (G(2)) sector contains odd (1^{+-}) and (0^{--}) channels, whereas the scalar (0^{++}\sim X\bar X) state is even and unprotected. For (m_X\simeq5.7\times10^{13}\,\mathrm{GeV}), weak-binding reference masses are (M{2X}\simeq1.14\times10^{14}\,\mathrm{GeV}) and (M_{3X}\simeq1.71\times10^{14}\,\mathrm{GeV}), while the exact pole masses remain nonperturbative. Gauge-invariant Fröhlich--Morchio--Strocchi (FMS) operators, Hall--Post bounds, (Y)-junction arguments and the pure-(SU(3)) glue spectrum favor (1^{+-}) in their controlled regimes without excluding a deeply bound (0^{--}) state. We then test whether this dark grading survives the full chiral exceptional embedding. An on-shell analysis finds no independent purely dark odd operator through dimension seven: the first nonvanishing basis appears at dimension nine. So the minimal one-copy exceptional embedding does not provide an exact ultraviolet dark (G)-parity. Moving the dark Higgs from the common (\mathbf{1463}H) parent to a separated (\mathbf{1539}_H) removes the scalar-parent obstruction and the relevant dimension-nine exceptional parents vanish on the selected pure-dark component in the undressed limit. A genuinely gauged or geometric (\mathbb Z{2,D}) would therefore leave a bosonic parity after dark Higgsing. Its extension to the mirror-free theory nevertheless fails because the required (G(2)) conjugation also conjugates color, (\mathbf3_C\leftrightarrow\bar{\mathbf3}_C). The remaining obstruction to exact dark matter stability is therefore ultraviolet and chiral, rather than low-energy or purely scalar.