Orbifold Model Calculator Six-dimensional SU(4) gauge–Higgs unification on T²/Z₂. Type a matter content; read the one-loop Wilson-line potential, its vacuum, the Higgs mass matrix and what the boundary condition can and cannot do. No Kaluza–Klein spectrum is enumerated.

theorem · Lean 4 — blindness & census verified — Part IV closed form measured vs AHMN self-check: running…
  1. 1Build a content. Add multiplets from the catalogue, set how many of each, matter or gauge, and the boundary sign η. Or press load AHMN for a published model.
  2. 2Read what comes out. The vacuum, the two Higgs masses, the mixing and the electroweak verdict — each labelled theorem, verified or measured.
  3. 3Interrogate the landscape. Drag any of the three panels: the reading bar reports at your cursor. Double-click drops it there and it rolls downhill.
  4. 4Flip η. On a content of blind multiplets nothing may move — that is the theorem, checked in front of you.

1 · The catalogue i

(a,b,c) dim box (p,q,r) i ζ i η i

η is the boundary-condition sign η = η₀η₁. A multiplet marked blind has Dλ ≡ 0: the boundary condition has no consequence any measurement of this Higgs sector can see. That classification is a theorem, machine-checked in Lean 4.

2 · Your matter content i

multipletdimn ηrole
Pick multiplets from the catalogue, or load AHMN's content.
V(α) = Σk≠0 |k|⁻⁶ Σq [Aq + Bq(−1)^k₂] cos(2πq·kα) Aq+Bq = Σλ Aq−Bq = Dλ ← the two halves

3 · What comes out i

4 · The landscape i

plan · the torus of vacua
relief · the same V, turned

drag move α, anywhere double-click drop it and watch it roll turn view / shift-drag rotate the relief wheel raise the relief

reading 
cut · V along α₂ — drag

The curve is the potential along the line through the cursor (α₂ varying, α₁ fixed at the cursor’s value): the vertical slice of the map above. The well is the vacuum. If it has no well, this content does not break the symmetry.

— V — ✕ vacuum ○ cursor — drag any panel

Left, the one-loop potential over the Wilson-line torus in plan (α₁ horizontal, α₂ vertical, one period each), with its isolines; right, the same V as a relief you can turn. Drag any panel to move the cursor — the reading above is taken there, not at the vacuum — and double-click to drop it and watch it roll downhill. On the relief, drag turns the view and the wheel raises it. Flip η below and watch: on a content of blind multiplets nothing moves — that is Theorem 1 on the screen.

5 · Before you compute anything

Glossary — every symbol on this page, in one place

α = (α₁, α₂)
The two Wilson-line phases. In gauge–Higgs unification the Higgs is this phase, so the potential over α is the Higgs potential, and its minimum is the vacuum.
Σλ and Dλ
The two halves. Σ is a graded dimension: its coefficients are non-negative and it can never cancel. D is an index: differences of multiplicities, so it can vanish. Even windings carry Σ, odd windings carry D.
η = η₀η₁
The boundary-condition sign a multiplet carries, ±1. It multiplies the coset half and nothing else — which is why some multiplets cannot feel it at all.
blind
Dλ ≡ 0. The boundary condition has no consequence any measurement of this Higgs sector can see. Classified and counted in closed form, machine-checked in Lean 4.
box (p,q,r) and ζ
Part IV's closed form: Dλ = ζ χpχqχr, the three indices read off the 2-quotient of the partition. This is what makes the potential computable without enumerating a Kaluza–Klein spectrum.
matter / gauge
How the multiplet enters the loop: matter with a plus sign, the gauge and ghost sector with a minus and with the antiperiodic twist that exchanges the two mode counts.
Aq, Bq
Mode counts at charge q of each orbifold parity. A+B = Σ and A−B = D; the field writes the same pair as {A + B(−1)^k₂}.

What each label means. theorem proved, and machine-checked in Lean 4: which multiplets are blind to η, the count of them, and the census of boundary-condition classes. verified Part IV's closed form for Dλ, on which the mode counts rest — exhaustively checked, not proved. measured the vacuum, the masses and the mixing: the engine reproduces AHMN's published vacuum exactly and their published mass matrix to 0.1 %, and that single published case is the whole of its validation.

What this does not do. It says what a model predicts, not what nature does — in this model the Higgs comes out light, as AHMN themselves report. A closed form for the potential is not a closed form for the mass: the mass lives at the true minimum, found here numerically. The sign of the off-diagonal mass, and hence of the mixing angle, is a convention — the potential is invariant under α → −α, so only invariants are reported. Everything assumes boundary conditions that can be simultaneously diagonalised.

Papers. Part III 10.5281/zenodo.21438226 · Part IV 10.5281/zenodo.21463000 · Part V 10.5281/zenodo.21727094 · code higgs-blind-class. Carles Marín Muñoz, independent researcher, with Claude (Anthropic) as AI research assistant. Apache 2.0. Nothing you type leaves this page.