The Möbius-screw electron is a closed curve carrying a framing. Between that shape and any statement about spin sits a chain of four maps. This is each one, evaluated — and the two separate places where the number 2 turns out to do no work at all.
The centerline winds twice around the axis while its meridian angle turns once, so it closes only after φ reaches 4π. The ribbon drawn here is spanned by the torus surface normal — the natural framing. Its two faces are coloured differently, so you can watch the strip present first one face and then the other to the same region of space.
Push a/R toward 1 and the donut becomes a funnel. The curve's closest approach to the axis shrinks to zero, and a throat opens at φ = 2π — the high-torsion neck of the tornado reading in Appendix A of the note. This is not a second object or a loose metaphor. It is the same framed loop at a different aspect ratio, reached by a continuous deformation, and every invariant on this page holds all the way along it.
Drag the aspect ratio and watch the two contributions move against each other while their sum stays pinned to an integer. Tw and Wr are not invariants — they are geometry, and they depend on how fat the torus is. Only Sl is topology. The near-even split at a/R = 0.3 is an accident of that particular aspect ratio, not a structural fact about the electron model.
The sweep runs all the way to the funnel. Drag past a/R ≈ 0.9 and the throat radius on the right closes toward zero while the sum sits on −2 the entire way. Donut and vortex are one framed loop, related by a deformation, and the topology cannot tell them apart.
The framing is exactly the sphere parameterisation at longitude φ and latitude φ/2, so it is a rotation of a fixed vector, and that rotation lifts to a quaternion path. Follow it around one full circuit of the curve and it does not come home — it arrives at minus itself. The frame needs two circuits to return, which is the 720° property of a spin-½ frame.
Take the simplest possible curve — a round circle, writhe exactly zero — and give it n twists of framing. Step through n and watch the holonomy σ flip on every single increment. It never cares how large the self-linking number is, only whether it is odd or even.
Look at n = 0. An ordinary untwisted circle is spinorial in precisely the same sense as the Möbius screw with |Sl| = 2. The two lie in the same class of π₁(SO(3)) = ℤ/2. Whatever produces the double cover here, it cannot be the number 2, because 0 does it too.
| n | Tw | Wr | Sl | σ | class |
|---|
Stop matching integers: compute the shape's magnetic moment against its angular momentum and get a g with dimensions, one that can fail. Let charge q and mass m traverse the centerline once per period T. Both moments turn out to be proportional to the same vector area.
The double winding is completely real, and completely useless. The screw carries 2.09× the vector area of a single loop — but that factor sits in μ and in ⟨L⟩ identically and cancels. It is the same failure as plate 04: a genuine 2 in the geometry that carries no information about g.
| curve | Az ⁄ π | g |
|---|---|---|
| Möbius screw (2,1), a/R = 0.30 | +2.09000 | 1 |
| Möbius screw (2,1), a/R = 0.70 | +2.49000 | 1 |
| Möbius screw (2,1), a/R = 0.97 | +2.94090 | 1 |
| round circle, one turn | +1.00000 | 1 |
| round circle, two turns | +2.00000 | 1 |
Note that Az is not even an invariant — it runs from 2.09π to 2.94π across the throat sweep, while Sl does not move at all.
Sl = 2 as the geometric origin of g = 2. The spin-relevant content of the self-linking number is one bit — its parity — and 2 and 0 are the same bit. The magnitude carries no spin information, so it cannot be what produces the double cover, and the identification loses its motivation.
Independently: Lévy-Leblond (1967) derives g = 2 from linearising the Schrödinger equation. Its origin is the spinor representation — neither relativistic nor topological. No peer-reviewed derivation of g = 2 from a framed loop exists anywhere in the mainstream record.
The shape really is spinorial, and the source is now specific: the odd meridian winding q = 1 — the φ/2 half-angle in the parameterisation. That is a genuine topological property of the object, proved in closed form, and it is exactly the 720° property the tornado picture was reaching for.
It is also a ℤ/2 statement, and a single bit cannot produce a real number.
No geometry can move g. Any model in which charge and mass circulate together at uniform q/m gives g = 1 exactly — whatever the winding, framing, twist, or throat. That is just the classical orbital g-factor, and it is shape-independent. The successor test as posed is closed.
What it opens is better posed than what it closed. The requirement is now structural rather than numerical: decouple where the charge sits from where the mass sits. That is a real constraint on any successor model, and it is checkable.
One part of it is a theorem, not an analogy. For a thin flux tube of flux Φ the magnetic helicity is H = ∫A·B = Φ²(Tw + Wr) = Φ²·Sl — so the Călugăreanu quantity in plate 02 is the helicity of this geometry (Moffatt 1969; Moffatt & Ricca 1992). Stretch-twist-fold dynamo action is exactly the Tw ↔ Wr trade plotted there. The bookkeeping is the same bookkeeping.
Two things block it as a particle model, and neither is topological. A dynamo grows — it has a growth rate and consumes kinetic energy from a flow, while a stable particle is stationary. And since H = Φ²·Sl, it hands back the same integer in units of Φ²: physical content, but no new magnitude. The stationary neighbour is the force-free Taylor state (∇×B = λB, a spheromak) — right dynamical class, and λ carries dimensions of inverse length. It still does not evade the no-go on its own.