Part VI · Proton Decay in SU(7) Grand Gauge-Higgs Unification: An Obstruction, Its Minimal Escapes, and the One Row of Their Table 1 That Can Pay for Them

Komori and Maru leave proton decay to future work in one sentence. Taking that sentence as the question, their own anomaly conditions answer half of it: at one generation the extra U(1) their SU(7) model carries cannot forbid proton decay — and the reason is structural rather than an oversight, in a paper that sets the question aside explicitly and says so.

The obstruction. All six anomaly channels are evaluated on their published content with their own conjugate-pairing prescription. Three of them are linear in the brane-quark charge and share one root, X_Q = −1/6 — which is exactly the point at which the four dimension-6 |ΔB| = 1 operators become neutral. There U(1)′ = T_3L + Y − (B − L) identically, and every operator in that list has Y = B − L = 0: any U(1) in that span is blind to them, so the failure is structural rather than a coincidence of a fitted charge. Two further channels cannot cancel at all. On the charges their own representations supply, the pair of Diophantine conditions leaves the one-parameter family e = −1 − 4w, u = 5w, in which the net number of 28 singlets is never zero: a right-handed neutrino in a 28 is necessary and not merely sufficient, and only two rows of their Table 1 carry one at the right parity. That requirement is not stated in their paper; it is reported here as a consequence of their own content, not as a defect in it.

The escapes, classified inside their own group. The literature’s cure for exactly this disease — a family-dependent abelian charge — fits inside their SU(7) as a ladder extra(L) = (1 − k)/2 indexed by the number k of index-7 boxes, together with the identity [SU(2)_L]²U(1)′ = ½∑_jA_j, which makes anomaly cancellation and proton protection the same equation at one generation and different equations at three. Of the fourteen assignments that cancel all six channels, exactly two fit inside their own tensors, and both need that same charge −1 singlet. The rung that hosts a third, inequivalent lepton generation sits in an 84^(+,+) that every row of their Table 1 already contains. The 48 cannot host it, and the reason is a three-line proposition rather than a box count: under diagonal ±1 parities a real representation furnishes no chiral generation, the Z₃ projection that cures this in the literature being unavailable here.

The bill, in eighths. By their own argument a lepton-hosting bulk multiplet is massive and leaves the Higgs potential, so the escape is charged against their table. Expanding their potential at the symmetric point gives V″(0) = −π²ζ(3)D with D an exact rational per multiplet — a 48^(+,+) is worth exactly 0 and an 84^(+,+) exactly 5/4. Donating one takes case (3) from 9/8 to −1/8 and electroweak symmetry breaking with it. Case (2) is the only one of the five published rows that satisfies both conditions at once — within the minimal extension, at one-loop curvature, and it is also the row with the lowest compactification scale. That is a benchmark selected, not a model delivered: the modified potential is not recomputed here.

The selection rule is a half-line, not a tuning. The one datum their paper leaves unstated — the charge of the scalar that breaks U(1)′ — turns out to be decidable in closed form: protection fails iff q_φ = |A_j|/n, a countable set with a maximum. And q_φ = 3/2, the charge of the 84’s pure index-7 component, forbids all four operators to all orders in ⟨φ⟩/M for every half-integer brane-quark charge, by a mod-3 argument — with its preconditions written in, including that any second condensing scalar destroys exactly this protection.

Over a third of the paper checks our own instrument rather than theirs. Their §3 is rebuilt from their own equations instead of assumed, and everything of theirs that was checked came out right: 13/13 structure constants, the omitted one exactly zero, the Kaluza-Klein spectrum, all four branchings, their eq. (67) proved rather than sampled, and their eq. (68) derived from the adjoint’s own charge-and-parity table with the coefficient 7 split as 4 + 3. Their eq. (81), imported from a different (five-dimensional) model, is shown to be a theorem of their own SU(7). Their Table 1 α_min column is not reproduced from their published formulas, and that is recorded as an open question rather than as a defect: the k ⋙ 1 truncation, the transcription (verified three independent ways), the gauge sector and every multiplicative or single-channel repair are excluded at one loop, each by a control that re-runs the whole search on scrambled data. One by-product is a test rather than a result: K = m_hα_min/√F″(α_min) is independent of the row and of R₅, R₆ and their ratio, so any table of this kind must return one number on every line. Theirs does on three rows of five.

Why the open question does not reach the verdict. Treating the residual as the most general per-representation reweighting, D is exactly linear in the weights and D(48) vanishes identically — so w(48) = 5.59, the largest repair the α_min column asks for, is invisible to the conclusion. What is left is a wedge in two weights containing w = 1 and every repair ever fitted, a factor 1.77 wide. What the wedge does not absorb is loop order, and that exposure is reported at its true size and on the wrong side: with the published fermionic half included, the transplanted two-loop estimate sits at 1.033 times the margin at the nominal coupling, crossing the ceiling at g₄ = 0.6205. An earlier draft answered this with the α_min column; that answer is withdrawn inside the paper rather than removed from it, because five row-by-row fits scattering over 20 % measure a discrepancy and not a weight, and a two-loop term is under no obligation to be a uniform rescaling of anything. So two-loop safety is left open rather than defused: the one-loop verdict is robust over every multiplicative deformation studied, its margin is quoted exactly, and the boundary of validity is located rather than argued away.

Included: the paper in English and Spanish (32 and 33 pp); every script that regenerates a quoted number, with its archived standard output; and the ten gates the artifact must pass before publication — among them one that checks every printed number against an archived run in both editions, one that compares the display formulas of the two editions body by body, one that checks the page map of the letter sent to the authors against the PDF they were actually sent, and one that re-runs every cited script and diffs it against the archive, because the others believe the archive. That last one earned its place on its first run: it caught two scripts that had stopped running at all, their archived numbers unchanged.

Honesty ledger. No defect in anyone’s paper is claimed anywhere in this work. Three transcription controls pass, so a residual we cannot place is more likely ours than theirs. What is reported about their 28 being Sym²(7̄) and about π₅ = π₇ are unstated data with computed consequences, not errors. The mechanisms are other people’s and a scope section says whose, one by one — including the one this paper proves and does not own: that matter in the adjoint is vector-like is stated as well known in Gogoladze–Mimura–Nandi (2004), and what is ours there is the specialisation to their parities and the counting. Not claimed: that case (2) works; any α_min, 1/R₅ or m_h for modified content; any vacuum expectation value, scale or proton lifetime; and operators above dimension 6, whose charges were not enumerated.

With thanks. This work exists because Komori and Maru wrote their model out fully enough that a reader can rebuild its section 3 from its own equations, and because they said plainly what they had not done. Almost nothing here is a new mechanism: the ones used cost other people a great deal of time — Gogoladze–Mimura–Nandi for the vector-like adjoint and its Z₃ cure, Lee–Ma for the family-dependent charge, Costa–Dobrescu–Fox for the general U(1) anomaly solution, Adachi–Maru for reading the curvature at the origin as a selection rule, Dumitru–Guo–Korthals Altes and Guo–Du for the two-loop halves, and Hosotani–Maru–Takenaga–Yamashita for the only two-loop Wilson-line potential in this framework. The paper says which piece is whose, one by one, and that table is the part we would most like a reader to check.