⚛️ An Upper Bound on the Compactification Scale of SU(7) Grand Gauge-Higgs Unification

…and the dijet angular distribution that tests it.

📄 Paper (EN + ES) and every verification script on Zenodo → https://doi.org/10.5281/zenodo.22087251

That is the concept DOI: it always resolves to the current version. The paper, its sources and every archived run are also in paper/, scripts/ and outputs/.

📚 Part VII of a series

• Part I — Anomaly- and Tadpole-Compatible Fermion Completion of 6D SU(4) GHU → github.com/karlesmarin/ghu-su4-completion · Zenodo 10.5281/zenodo.21432625
• Part II — Three Gates to a Quark Generation → github.com/karlesmarin/su4-sm-cell-criterion · Zenodo 10.5281/zenodo.21432627
• Part III — A Centre-Charge Selection Rule for the Wilson-Line Potential → github.com/karlesmarin/centre-parity-selection · Zenodo 10.5281/zenodo.21438226
• Part IV — Schur Functions at (1,−1,t,t⁻¹) → github.com/karlesmarin/schur-nonidentity-o4 · Zenodo 10.5281/zenodo.21463000
• Part V — What the Higgs Potential Cannot See → github.com/karlesmarin/higgs-blind-class · Zenodo 10.5281/zenodo.21727094
• Part VI — Proton Decay in SU(7) Grand Gauge-Higgs Unification → github.com/karlesmarin/su7-proton-row · Zenodo 10.5281/zenodo.22033302
• Part VII — An Upper Bound on the Compactification Scale of SU(7) GHU (this repo)
• Part VIII — A Certified 2.68 TeV Gap in the Closed-Form Map of the Compactification Scale → github.com/karlesmarin/su7-certified-gap · Zenodo 10.5281/zenodo.22159036
• Part IX-A — The Alphabet of Orbifold Boundary Conditions → github.com/karlesmarin/orbifold-alphabet · Zenodo 10.5281/zenodo.22254861
• Part IX-B — An Affine Semigroup from Orbifold Boundary Conditions → github.com/karlesmarin/orbifold-semigroup · Zenodo 10.5281/zenodo.22254863

In gauge-Higgs unification the Higgs is the Wilson-line phase, so the electroweak hierarchy is a minimum of a potential rather than a ratio of scales. Part VI computed the curvature of that vacuum for the SU(7) model of Komori and Maru (arXiv:2503.04090) and reached a sign. This paper computes the vacuum itself, and then asks how large the compactification scale can be over arbitrary multiplicities of the representations they put in the bulk.

The answer is finite, and the second half of the title is why that matters: a collider recast of CMS's dijet angular distribution approaches the same scale from below, so the two calculations cross.

🙏 With thanks

This work exists because Komori and Maru wrote their model out fully — every parity matrix, every mode expansion, the whole one-loop potential — so that a reader outside their group can rebuild it from its own equations rather than transcribe it. That is not the norm and it is what made this paper possible.

Fifty-nine quoted equations from eleven sources carry the load, and it is not spread evenly: twenty-five from Komori–Maru, fourteen from Cacciapaglia, Cossu and Deandrea on orbifold vacuum stability, seven from Haba, Hosotani and Kawamura and five from Haba and Yamashita — the two papers that made Wilson-line potentials on orbifolds something one computes rather than models. §12 of the paper says which piece is whose, one by one, and a gate keeps that table honest.

🧭 What is in it

The ladder	Expanding the one-loop potential about the symmetric point gives one identity read at three rungs, p = 5, 3, 1. The value, the curvature and the fourth moment are the same formula. At p = 1 the antiperiodic half cancels its pole into a finite ln 2 while the periodic pole survives as the x⁴ln x branch point — which is why no polynomial exists.
The closed form	α_min in Lambert-W, x² = −d/W(−d e^{−b}), in three moments. Not an algebraic root — it carries ln x — but solved rather than iterated, with W₋₁ the minimum and W₀ the maximum, proved rather than assigned. Median error 0.13 % over 272 contents.
Both columns for free	The logarithm and G cancel identically at the stationary point, so α_min and m_h both follow from two moments with no minimisation anywhere.
The arithmetic	8D ≡ 2A₄ + 3 (mod 6), because c⁴ ≡ c² (mod 3). The matter half holds whatever the gauge sector does; the residue is the gauge base point's, and both seeds considered give the same 9.
The ceiling	An integer program over a rational cone, closed by an exact two-variable dual: 1/R₅ ≤ 10.03 TeV for arbitrary bulk content, sharpened to 10.01 attained once the lattice is lifted to four dimensions, and to 9.22 TeV once the vacuum is required to be the true one — on the three-rung algebraic surface. At the measured Higgs mass, 9.09 TeV.
Which state a collider bounds	Read off their three parity matrices and nothing else. Every colour-changing gauge boson is odd under the small-circle parity, so its tower starts at 1/R₆ and its wavefunction vanishes on the quark brane — which answers, for the gauge channel, the rapid proton decay their conclusions flag and leave to future work.
The recast	The surviving coloured tower acts through a colour-octet operator in the dijet angular distribution, so published contact-interaction limits, stated for a singlet, do not directly apply. Carried through CMS's unfolded distribution, 77×77 covariance and NNLO baseline: a nominal sensitivity of 1/R₅ > 10.86 TeV. Deliberately not called an exclusion — a Gaussian χ² over unfolded data is not a detector-level CLₛ, and the paper measures that gap rather than waving at it.
Where it stops	One inherited coefficient this paper does not compute: the gauge weight of their eq. (68). Everything needing it to be odd is marked conditional, in the abstract, on the front page, in the theorem's own signature and in the ledger.

⚠️ The one condition, stated plainly

Theorem 1 carries its hypothesis in its own statement: if the gauge sector's contribution to 8D is odd, then 8D is an odd integer for every bulk content. On the coefficients Komori and Maru print, it is. On a parity-resolved covariant count it would not be — and this paper does part of that count, finding the split ξ-independent and equal to 3 + 1, which is the candidate.

So the fork is not symmetric, and the paper says so: its own machinery points twice, by different routes, at the branch where that hypothesis fails. What is not settled is whether the ghosts' boundary conditions are the assumed ones, and whether their spectrum construction hides a cancellation. Both branches are costed in a table, and no number is quoted without saying which branch it lives on.

🔍 Reproducing it

Every displayed number regenerates from the ancillary scripts, whose standard output is archived beside them in outputs/. The ledger in the paper cites the script for each entry rather than the number alone, so a reader who disagrees with a claim can re-run the thing that made it.

python check_reproduces.py     # runs every cited script and diffs it against its archived output

Fifteen gates guard the two editions — attribution, branch conditionality, figures, formulas, inline maths, framed boxes, layout, narrative, numbers, parity, prose, references, scripts, EN/ES synchrony, and reproduction. comprobar.py decides which of them a given change can break.

Three scripts need CIJET's output, whose licence permits academic use and forbids redistribution; they are named, with the reason, rather than silently skipped. Five .sage files need the container.

📁 Layout

paper/     the two editions, their sources, figures, and the fifteen gates
scripts/   every cited computation
outputs/   the archived standard output of each, byte for byte

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Carles Marín · independent researcher · ORCID 0009-0007-5637-9688 Written with Claude (Anthropic) as a research instrument.