Part VII · The compactification scale of SU(7) grand gauge-Higgs unification is bounded

This is a summary written for this page, not the paper's abstract. Part VII has no archived record yet, so nothing here is citable and the wording is ours rather than the archive's.

In gauge‑Higgs unification the compactification scale is tied to the weak scale by α_min = 2 m_W R₅, so the Wilson‑line vacuum is the electroweak hierarchy. Part VI computed the one‑loop curvature of that vacuum for the SU(7) model of arXiv:2503.04090 and reached a sign. Part VII computes the vacuum itself.

Expanding the potential about the symmetric point produces one ladder, indexed by p = 5 − 2k, whose k-th rung pairs the 2k-th moment of the bulk content against ζ(p). Solving the stationarity condition gives α_min as an algebraic function of two moments, and the same expansion gives the Higgs mass — so both columns of their Table 1 follow from the moments with no minimisation anywhere.

The moments are quantised: 8D is an odd integer, and 8D ≡ 2A₄ + 3 (mod 6). Maximising 1/R₅ is therefore an integer program over a rational cone, whose relaxation has a two‑variable dual with enumerable vertices — giving an upper bound on the compactification scale that holds for arbitrary bulk content, attained at the quantum D = 1/8. Since the Kaluza‑Klein excitations of the gauge bosons sit at 1/R₅, the class is bounded above and therefore falsifiable in one machine.

Scope, and the paper states it throughout: one loop, flat space, a free Kaluza‑Klein tower, a single Wilson phase, and every absolute number inherits Part VI's open question about the published α column.