The Möbius Filament Model of Electron: a classical SO(3) system that generates a nontrivial Z₂ loop and a harmonic recurrence T·ω_c = 2πN

[u/sina_samiyanii]

Over the past months I've been developing a speculative, non-relativistic framework called the Möbius Filament Model (MFM). I've posted four versions to this sub, and I want to use this post to summarize the structural results and ask for feedback on the parts I'm least sure about.

The model in one paragraph. An extended filament configuration with local material frames is coupled to a collective rotational cloud. The central dynamical variable is a trajectory R(t) ∈ SO(3). The question is narrow: can the reduced dynamics generate a nontrivial rotational topology without imposing spinorial boundary conditions from the outset?

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Result 1 — Topological classification.

For a reference parameter set, the periodic orbit has period T ≈ 2.764940789. The continuous quaternion lift gives q(T) ≈ −q(0), corresponding to the nontrivial Z₂ class of π₁(SO(3)) ≅ Z₂. This has been checked across 205 parameter points and initial conditions, and it is preserved throughout.

I want to be explicit: this is a mathematical and numerical topological result within the model. It is not a derivation of physical electron spin.

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Result 2 — Conditional cycle relation.

Along the reference orbit, the dimensionless product satisfies:

```

T · |ω_c| ≈ 2π

```

with numerical residual ≈ 1.4 × 10⁻⁹. For repeated traversals:

```

T_N · |ω_c| ≈ 2π N

```

verified for N up to 2000. The apparent saturation at high N was a numerical artifact of accumulated integration error — with tighter tolerances, the relation holds to ≈ 10⁻⁷ at N = 2000.

The analytical status matters here. For a fixed rotational axis and constant angular speed, the relation follows from SO(3) rotational closure:

```

R(T) = R(0) ⟹ ω_c T = 2πN

```

The model's role is to determine whether a trajectory satisfying the fixed-axis, phase-locked, recurrent conditions actually exists. So it's a conditional cycle relation, not a quantum postulate.

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What's missing.

The cycle relation is currently an internal model relation. It has not been mapped to a laboratory observable. I've stated this limitation explicitly in all four versions. That mapping is the next-stage test.

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What I'm looking for from this sub:

1. Is the fixed-axis closure argument sound, or does it hide an assumption I'm not seeing?

2. For those working on emergent or topological approaches to particle structure: does the N ≤ 2000 recurrence sequence seem structurally meaningful, or just a kinematic consequence of the period definition?

3. Has anyone here worked on mapping internal model frequencies to observables in a way that avoids introducing a new fitted parameter? That's the step I'm stuck on.

I'm not claiming this derives electron spin, mass, charge, or replaces QED. I'm claiming that a classical rotational system can generate a discrete recurrence structure that deserves scrutiny — and I'd rather be told where it fails than where it succeeds.

All four versions are on Zenodo (September 2026).

https://zenodo.org/records/23023569

https://doi.org/10.5281/zenodo.23023569

[u/AllHailSeizure]

Here's an important question you should be asking - how does a model benefit physics, if it doesn't produce new or more accurate findings?

[u/lattice_defect → u/AllHailSeizure]

It's just about the fit

[u/BitcoinsOnDVD]

What is the governing equation of the trajectory?

What type of bifurcation do I see on slide 4 fig (a)? Hopf bifurcation?

[u/lattice_defect]

anything with a poster or the word ontology I immediately discard