The arithmetic of the moments A₄, D, W in SU(7) gauge-Higgs unification

The one-loop Wilson-line potential of the six-dimensional SU(7) model of Komori and Maru has moments A₄, D, W that are sums over the bulk multiplets plus a gauge-sector contribution. The gauge-sector contribution is read from eq. (68) of Komori–Maru and is a physical input, not a logical step; two values ("seeds") are considered: the one printed (gaugePublished) and a candidate with the ghost subtraction (gaugeCandidate). This file proves the statements that do not depend on the seed, and states the one that does with its hypothesis explicit.

The coordinates are (2 A₄, 8 D, 2 W): with the candidate seed A₄ is a half-integer, so 2 A₄ is the integer coordinate, and the proof of the mod-6 law only uses the combination 8 D − 2 A₄.

Main results

• matter_key_mod6, matter_w2_even, matter_d_even: the eight bulk multiplets (7, 28, 48, 84, each with two parities) have 8D − 2A₄ ≡ 0 (mod 6), 2W even and 8D even.
• key_seed_independent: both seeds give 8D − 2A₄ = 9.
• mod6_law: for every content with natural multiplicities, 8D ≡ 2A₄ + 3 (mod 6).
• twoW_odd: 2W is odd for every content.
• oddEighths: if the gauge contribution to 8D is odd, then 8D is odd for every content (so D ≠ 0); oddEighths_published: the hypothesis holds for the printed seed; candidate_hypothesis_fails: it fails for the candidate seed.

References

• Y. Komori, N. Maru, SU(7) grand gauge-Higgs unification, arXiv:2503.04090, eq. (68).

def GaugeHiggsUnification.OddEighths.matter : List (ℤ × ℤ × ℤ)

The eight bulk multiplets in coordinates (2 A₄, 8 D, 2 W). A₄ is an integer for matter, so the first coordinate is its double.

Equations

• GaugeHiggsUnification.OddEighths.matter = [(0, -6, 2), (2, 8, -2), (34, 16, 6), (8, 2, -6), (36, 0, 12), (16, 28, -12), (136, 10, 18), (218, 80, -18)]

def GaugeHiggsUnification.OddEighths.gaugePublished : ℤ × ℤ × ℤ

The gauge sector as printed in eq. (68).

Equations

• GaugeHiggsUnification.OddEighths.gaugePublished = (-36, -27, -3)

def GaugeHiggsUnification.OddEighths.gaugeCandidate : ℤ × ℤ × ℤ

The gauge sector with the ghost subtraction.

Equations

• GaugeHiggsUnification.OddEighths.gaugeCandidate = (-27, -18, -3)

def GaugeHiggsUnification.OddEighths.key (v : ℤ × ℤ × ℤ) : ℤ

The combination 8 D − 2 A₄.

Equations

• GaugeHiggsUnification.OddEighths.key v = v.2.1 - v.1

def GaugeHiggsUnification.OddEighths.w2 (v : ℤ × ℤ × ℤ) : ℤ

The third coordinate, 2 W.

Equations

• GaugeHiggsUnification.OddEighths.w2 v = v.2.2

The matter multiplets

theorem GaugeHiggsUnification.OddEighths.matter_key_mod6 (v : ℤ × ℤ × ℤ) : v ∈ matter → 6 ∣ key v

Every matter multiplet has 8 D − 2 A₄ ≡ 0 (mod 6).

theorem GaugeHiggsUnification.OddEighths.matter_w2_even (v : ℤ × ℤ × ℤ) : v ∈ matter → 2 ∣ w2 v

Every matter multiplet has 2 W even.

The gauge base point: the same 9 for both seeds

theorem GaugeHiggsUnification.OddEighths.key_published : key gaugePublished = 9

The printed seed gives 8 D − 2 A₄ = 9.

theorem GaugeHiggsUnification.OddEighths.key_candidate : key gaugeCandidate = 9

The candidate seed gives the same 9.

theorem GaugeHiggsUnification.OddEighths.key_seed_independent : key gaugePublished = key gaugeCandidate

The two seeds agree on 8 D − 2 A₄.

theorem GaugeHiggsUnification.OddEighths.gauge_w2_odd : ¬2 ∣ w2 gaugePublished ∧ ¬2 ∣ w2 gaugeCandidate

Both seeds have 2 W odd.

Linearity: arbitrary content

def GaugeHiggsUnification.OddEighths.combo (ns : List ℕ) (xs : List ℤ) : ℤ

A content is a list of natural multiplicities against the generators; combo ns xs is the weighted sum of one coordinate.

Equations

• GaugeHiggsUnification.OddEighths.combo ns xs = (List.zipWith (fun (k : ℕ) (x : ℤ) => ↑k * x) ns xs).sum

theorem GaugeHiggsUnification.OddEighths.dvd_combo {d : ℤ} (ns : List ℕ) (xs : List ℤ) (h : ∀ (x : ℤ), x ∈ xs → d ∣ x) : d ∣ combo ns xs

If d divides every generator, it divides every combination with natural multiplicities.

The three statements

theorem GaugeHiggsUnification.OddEighths.mod6_law (g : ℤ × ℤ × ℤ) (hg : key g = 9) (ns : List ℕ) : (key g + combo ns (List.map key matter)) % 6 = 3

The mod-6 law, independent of the seed. For every gauge base point with 8 D − 2 A₄ = 9 (both seeds, by key_published and key_candidate) and every content, 8 D − 2 A₄ ≡ 3 (mod 6).

theorem GaugeHiggsUnification.OddEighths.twoW_odd (g : ℤ × ℤ × ℤ) (hg : w2 g = -3) (ns : List ℕ) : ¬2 ∣ w2 g + combo ns (List.map w2 matter)

2 W is odd, independently of the seed. The gauge sector contributes −3 with both seeds and matter contributes an even number.

theorem GaugeHiggsUnification.OddEighths.oddEighths (dGauge : ℤ) (hodd : ¬2 ∣ dGauge) (ns : List ℕ) (hmatter : 2 ∣ combo ns (List.map (fun (v : ℤ × ℤ × ℤ) => v.2.1) matter)) : ¬2 ∣ dGauge + combo ns (List.map (fun (v : ℤ × ℤ × ℤ) => v.2.1) matter)

Odd eighths, conditional. If the gauge contribution to 8 D is odd and the matter contribution is even, then 8 D is odd, hence non-zero. The hypothesis on the gauge sector is an input from eq. (68).

theorem GaugeHiggsUnification.OddEighths.matter_d_even (v : ℤ × ℤ × ℤ) : v ∈ matter → 2 ∣ v.2.1

Every matter multiplet contributes an even 8 D: the matter hypothesis of oddEighths.

theorem GaugeHiggsUnification.OddEighths.oddEighths_published (ns : List ℕ) : ¬2 ∣ -27 + combo ns (List.map (fun (v : ℤ × ℤ × ℤ) => v.2.1) matter)

For the printed seed (8 D = −27, odd), 8 D is odd for every content.

theorem GaugeHiggsUnification.OddEighths.candidate_hypothesis_fails : 2 ∣ gaugeCandidate.2.1

For the candidate seed the hypothesis of oddEighths fails: its 8 D is −18, even.