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 a collider recast of CMS’s dijet angular distribution approaches the same scale from below, so the two calculations cross.
One ladder, three rungs. Expanding the one-loop effective potential about the symmetric point produces a single identity read at p = 5, 3, 1: the value, the curvature and the fourth moment are the same formula at three values of one index. At p = 1 the antiperiodic halves cancel their pole into a finite ln 2 while the periodic pole survives as the x4ln x branch point — which is why the answer cannot be a polynomial, where the finite-temperature theory at an even index has Gross–Pisarski–Yaffe’s exact quartic.
The hierarchy in closed form. Solving stationarity gives αmin in a Lambert-W form in three moments, x² = −d/W(−de−b) — not an algebraic root, since it carries ln x, but solved rather than iterated, with the two branches the minimum and the maximum and the fold at zero Higgs mass. Median error 0.13 % over 272 contents against numerical minimisation of the same potential. The logarithm and the third moment cancel identically at the stationary point, so both columns of their Table 1 become functions of two moments with no minimisation anywhere.
The arithmetic of the content. The moments are quantised: 8D ≡ 2A4 + 3 (mod 6), because c4 ≡ c² (mod 3). What holds whatever the gauge sector contributes is the matter half — every bulk multiplet moves 8D − 2A4 by a multiple of six — while the residue belongs to the gauge base point. Whether the same argument also makes 8D odd is a separate question, and it is the one open input of this paper.
The ceiling. Maximising the compactification scale is an integer program over a rational cone, closed by an exact two-variable dual whose vertices are enumerated in rational arithmetic: 1/R5 ≤ 10.03 TeV for arbitrary bulk content. Writing the third moment in the basis {1, ln 2, ln 3} lifts the content to a four-dimensional lattice, where the relaxation’s vertex turns out empty and the bound sharpens to 10.01 TeV attained. Put on the exact potential, however, that content is a false vacuum. Requiring the electroweak point to be the true minimum is the orbifold stability criterion of Cacciapaglia, Cossu and Deandrea, whose arithmetic is theirs; summed over a content it is a fourth linear functional, and what is new is only that a linear functional fits inside the integer program. With it imposed the bound is 9.22 TeV on the three-rung algebraic surface, and 9.09 TeV at the Higgs mass the world actually has.
Five coordinates, and they are complete. Two bulk contents have the same one-loop potential, as a function of the phase, if and only if they agree on five integer invariants. That is proved by two kernels meeting, computed by routes sharing no arithmetic, and it explains a gap the earlier work had left open between a certificate and an enumeration.
Which state a collider would bound. 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 at the fixed point where that same paper localises the quarks — which answers, for the gauge channel, the rapid proton decay their conclusions flag and leave to future work. What survives at the brane acts through a colour-octet operator in the dijet angular distribution, so published contact-interaction limits, stated for a colour-singlet operator, do not directly apply to it; the data behind them do constrain it, which is what the recast does. Carried through CMS’s unfolded distribution, 77×77 covariance and NNLO baseline gives a nominal sensitivity of 1/R5 > 10.86 TeV. It is 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 — aimed at their own published limit the machinery returns 17.5 against 17 TeV and 29.5 against 37.
The one condition, and it is not hidden. The chain passes through one coefficient this paper does not derive from the product-orbifold geometry: the gauge weight of eq. (68) of arXiv:2503.04090. Theorem 1 carries its hypothesis in its own statement, and a red note on the front page says so. A parity-resolved covariant count of the same orbit, done here, gives a different split — shown to be independent of the gauge parameter, with the ghosts’ boundary conditions derived from the orbifold action on the gauge parameter rather than assumed. So the fork is not symmetric, and the paper says so: its own machinery points at the branch where that hypothesis is not met. Both branches are costed in a table, and no number is quoted without saying which branch it lives on. Between the two seeds considered, and on the three-rung surface, 9.22 TeV remains the conservative bound: the ambiguity does not open a route to larger scales, it closes one.
Reproducibility. Every displayed number regenerates from the ancillary scripts, whose standard output is archived beside them; the ledger 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. Fifteen gates guard the two editions, and one of them executes every cited script and diffs it against its archive: 70 of 70 executable scripts reproduce byte for byte. Two claims the earlier drafts made are corrected in place and left visible rather than removed.
2026-08-27noteaffects the record: no
A verdict can have no subject β the true-vacuum flag was a boolean where three answers were needed
- what
A second outside audit read the corrected source and found the correction's own edge. The
verdict had become true: symmetricOK && deepest !== false β right whenever there was
something to test, and wrong whenever there was not. deepest is null on every content that
never reaches the minimiser, and in this language null !== false is true. So a content with
no electroweak breaking and W > 0 exported alpha_min: no electroweak breaking next to
vacuum.true: true. The auditor built it from this page's own coordinates: 2 Γ 7(+,+),
where the gauge seed gives 8D = β27, 2W = β3 and each 7(+,+) adds 8D = β6, 2W = +2, so
8D = β39 < 0 and 2W = +1 > 0. Reproduced here to every digit, and the Python engine of Part VII
now produces that row into tests/reference_models.json so a reader can check the null against
an implementation that shares no line of code with the page. The screen was never wrong β it
said "But D β€ 0, so there is no interior minimum for it to be the vacuum of" β but the screen
is not what a third party reads. vacuum.true is now one of {true, false, null} beside a
named state: true-vacuum, false-vacuum, no-electroweak-breaking, no-branch-located,
undetermined. Null is not a verdict, and it no longer pretends to be one.
The same audit asked for the globality half to stop resting on a positional tolerance β
it decided the branch was the deepest point if |Ξ±_global β Ξ±_closed| < 0.02 or the gap in F
was negligible. The closed form is an expansion, accurate to 0.71 % under the largest Ξ± of
their Table 1 and to 20 % out at Ξ± = 0.229, so a fixed window in Ξ± is a guess about how wide a
basin is. It is gone. The closed form now only locates the basin: localMin walks downhill
from it to the minimum it is about, numericMin finds the deepest point with no bracket at
all, and the verdict is F against F at two numerically refined minima. On their five rows the
two points coincide to better than 10β»βΆ; on the 2026-08-26 counterexample they are 0.0839 and
0.5660, and F is lower at the second by 1.07.
- why
The first is the honesty floor read at the wrong layer. The instrument had learned that a
necessary condition is not a sufficient one, wrote the two halves, and then joined them with an
operator that cannot tell "no" from "not asked" β a three-valued question answered in a
two-valued type. The second is the same instinct as the first correction, applied one level
deeper: a tolerance in Ξ± is a claim about the potential's shape, made where a comparison of two
computed depths was available for the same cost.
- so
No printed number moves and the frozen record is untouched: every published row has a branch,
and on all five the refined branch minimum is the global one. What moves is what the
instrument exports about contents that break nothing β and the standard the globality half is
held to. Both are pinned: _test_hierarchy.mjs fails on the old boolean and on a localMin
that returns its input, and the shipped tests/run.mjs now checks that a reference row with no
electroweak point makes the page claim neither vacuum. Fired both on purpose before believing
them.
- check
- load 2 Γ 7(+,+) in the hierarchy section β 8D = β39 < 0 with 2W = +1 > 0, so there is no electroweak breaking at all; the exported card now reads
state: no-electroweak-breaking and true: null where it used to read true: true. tests/run.mjs, which ships with the page, pins it against the Python engine, and the globality half no longer rests on any positional tolerance
2026-08-26noteaffects the record: no
A true vacuum needs more than W > 0 β and three smaller things an outside audit found
- what
An outside audit of the deployed source found four things, and every one of them is real.
First and largest: the instrument wrote true: W > 0 and labelled it a theorem about "the
true vacuum". [8]'s criterion β and Part VII's own sentence, "W > 0 sufficient for the
electroweak point to be the deeper one" β says only that Ξ± = 0 is deeper than the *other
symmetric point*, Ξ± = 1. It says nothing about a deeper minimum elsewhere in (0, 1). The
auditor built a counterexample from this page's own tables: 7(+,+) + 48(+,β) + 84(+,+) has
D = 5/8, W = 5/2, a small-Ξ± branch at 0.0848 with F = β0.626 β and F = β1.698 at Ξ± = 0.566.
The page called it a true vacuum. Reproduced here to every digit. The verdict now has two
halves with their own standing: the symmetric-point half stays the theorem it is (necessary,
never a tie); the deepest-point half is measured, by minimising the same F directly on
(0, 1] on every render β which is exactly what the sweep already did and counted as notGlobal,
without the single-model verdict ever consulting it. vacuum.true is now the conjunction.
Second: the sweep's "largest Ξ± their Table 1 reaches" was 0.0836 β our Ξ± on their rows β
when their printed maximum is 0.081. The regime boundary stays on our axis, because that
is the axis the closed form is evaluated on, but both numbers are now carried and the sentence
says which is which. Third: the SU(4) calculator minimised over the whole torus and then
reported searched: [0, Β½] whenever the selection rule allowed the halving β a true answer
with a false provenance. minimise() now receives the licensed domain, and what is reported is
what ran. Fourth: validate() accepted a parity list of any length; termTable reads two.
It now demands exactly two.
- why
The first is the class of error this whole instrument exists to refuse β a necessary condition
shipped as a sufficient one, under the wrong chip. The page knew the distinction (the sweep's
notGlobal count is the same fact) and did not apply it to itself: two instruments that agree
on a definition and disagree on where it is used. The other three are provenance drifting from
computation β a label naming the wrong number, a report naming the wrong domain, a check that
did not check.
- so
The frozen record is untouched: the paper's sentence is the modest one, its true-vacuum level
9.22 TeV is a maximum over a class its witness was checked to belong to on the exact potential,
and no printed number moves. What moves is the instrument's verdict on contents like the
counterexample, the wording of the regime boundary, the provenance of the SU(4) minimum, and
one validation. Each has a harness check that fails on the old behaviour. The audit that found
them read the source, not the page β which is the second time today that reading the artifact
found what the artifact's own tests had not.
- check
- load 7(+,+) + 48(+,β) + 84(+,+) in the hierarchy section β W = 5/2 > 0 and the verdict now reads "a false vacuum W alone cannot see", with the deeper minimum at Ξ± = 0.566 named; _test_hierarchy.mjs pins it, the printed 0.081, the searched domain and the parity check
2026-08-26noteaffects the record: no
The instrument stood on the draft; it now stands on the published paper
- what
The hierarchy and anomalies sections were closed on the 9 August draft of Part VII, whose
ceiling was the relaxation's 10.03 TeV, called certified and drawn as one point. The paper
as deposited on 24 August keeps that number as the bound the dual proves and adds three levels
below it: 10.01 TeV attained at (Aβ, 8D) = (212, 1) once the lattice is lifted β the
relaxation's own vertex (215, 1) is empty β 9.22 TeV once the electroweak point must be the
true vacuum, and 9.09 TeV at the Higgs mass the Higgs actually has. The content that attains
10.01 sits in a false vacuum: the potential is deeper at the other symmetric point by 316. The
instrument had no way to see that, because it did not compute W.
It now does. Every content carries W = Ξ£_{c odd} m(βs), F(1) β F(0) = (31/16) ΞΆ(5) W exactly,
and a verdict β the numbers at a stationary point in a false vacuum are still shown, labelled.
The ceiling a content is measured against is 9.22 TeV, and the four levels are a table with what
each one bounds. Theorem 1 (8D odd) is now stated as the paper states it, conditional on the
gauge seed: a switch lets the model stand on the candidate parity-resolved split of Β§13, under
which 8D is even, Aβ half-integral, the relaxation ceiling drops to 7.38 TeV at 8D = 2, and
Theorem 2 and the 2W theorem survive untouched. The five complete invariants (Aβ, 8D, 2U, V, 2W)
are computed in the kernel and checked, generator by generator, against lattice_lift.py.
- why
A page that says "certified ceiling 10.03 TeV" over a content in a false vacuum is showing a
number the paper it advertises has since qualified. Publishing a part did not update the
instrument built on its draft β the same failure that left the series pages saying "not yet
deposited" for five days after Part VI was deposited, and it was found by auditing the whole
site against the papers rather than the part that had just moved.
- so
The record is unaffected: nothing in the frozen Part VII depends on this page. Two things in the
instrument were wrong and are corrected β the single ceiling, and Theorem 1 stated without its
hypothesis β and one thing was missing, the stability functional, which is [8]'s criterion and
not ours. And one bug of the shell surfaced while photographing the seed switch: a page opened
with a permalink rendered nothing at all, so every deep link the tool had ever handed out was
dead. Fixed in the same commit; the harness now opens one.
- check
- open the hierarchy section β the four levels of the ceiling, the other-symmetric-point panel, and the gauge-seed switch; _test_hierarchy.mjs checks each against the archived runs of the paper