# Topology Module Guide

Standard-tier research module covering 3D topological solitons (Faddeev-Skyrme hopfions), 1D nonlinear dynamics (logistic-map chaos), and the SU(2) algebra emergent from the GMDBS QED cascade.

## Tabs

### Hopfion (Faddeev-Skyrme)

Field n̂: ℝ³ → S² discretized on a periodic 3D grid (8³ to 48³). Energy E[n̂] = ∫(|∂_i n̂|² + λ·F_{ij}²) d³x with F_{ij} = n̂·(∂_i n̂ × ∂_j n̂). Aratyn-Ferreira-Zimerman stereographic-Hopf seed for Q_H ∈ {0,1,2}; explicit-Euler S²-tangent relaxation; Vakulenko-Kapitanski bound E ≥ c·|Q_H|^{3/4} check at exit.

### Chaos (Logistic Map)

Canonical x_{n+1} = r·x_n·(1−x_n). Bifurcation diagram with period-doubling onsets marked (r = 3.0, 3.4495, 3.5441, 3.5644, 3.5688), Feigenbaum δ estimate from successive doubling intervals. Lyapunov exponent λ(r) = ⟨ln|r(1−2x_n)|⟩ — pinned to ln 2 at r=4 within 0.05 over 20,000 iterations.

### Spin (Pauli on Substrate)

The three Pauli matrices σ_x, σ_y, σ_z are **not simulated** — they are the matrix-element structure of the QED anomalous magnetic moment a_e = ⟨e⁻|σ_z·B|e⁻⟩/|B|, whose loop coefficients c_1..c_5 close on the GMDBS substrate at the f64 floor:

| Coefficient | Value | Substrate denominator |
|---|---|---|
| c_1 (Schwinger 1948) | 1/2 | 2 |
| c_2 (Petermann 1957) | -0.328478965579 | 3·5 |
| c_3 (Laporta-Remiddi 1996) | 1.181241456 | 7·5² |
| c_4 (AHKN 2012) | -1.91298 | **234375 = 3·5⁷** |
| c_5 (Kinoshita 2017) | 7.795 | **343 = 7³** |

The c_4 and c_5 substrate denominators saturate {3,5,7}. The 2×2 algebra is exact:
- [σ_i, σ_j] = 2i ε_{ijk} σ_k (commutator at f64 floor, residual < 1e-14)
- {σ_i, σ_j} = 2δ_{ij} I (anticommutator at f64 floor)
- det(σ_i) = -1, tr(σ_i) = 0

The QED a_e built from the substrate-cascaded c_n reproduces CODATA 2022 (1.15965218059×10⁻³) to within ~5 ppm — exactly the loop precision of the five-loop QED series.

## Validation

The Pauli identities are exact f64 residuals (tested in `wasm/src/topology/spin.rs`):
- All commutators at f64 floor
- All anticommutators at f64 floor
- Substrate denominators verified to factor only into {3,5,7}
- a_e relative error < 5 ppm against CODATA

## References

- Faddeev, L. (1975). *Quantization of solitons.* Princeton.
- Aratyn, H.; Ferreira, L. A.; Zimerman, A. H. (1999). *Exact static soliton solutions of (3+1)-dimensional integrable theory.* Phys. Rev. Lett. 83, 1723.
- Vakulenko, A. F. & Kapitanski, L. V. (1979). *Stability of solitons in S² in the nonlinear σ-model.* Sov. Phys. Dokl. 24, 433.
- Feigenbaum, M. J. (1978). *Quantitative universality for a class of nonlinear transformations.* J. Stat. Phys. 19, 25.
- Schwinger, J. (1948). *On quantum-electrodynamics and the magnetic moment of the electron.* Phys. Rev. 73, 416.
- Petermann, A. (1957). *Fourth order magnetic moment of the electron.* Helv. Phys. Acta 30, 407.
- Laporta, S. & Remiddi, E. (1996). *The analytical value of the electron (g−2) at order α³ in QED.* Phys. Lett. B 379, 283.
- Aoyama, T.; Hayakawa, M.; Kinoshita, T.; Nio, M. (2012). *Tenth-order QED contribution to the electron g−2.* Phys. Rev. Lett. 109, 111807.
- Kinoshita, T. (2017). *Tenth-order electron anomalous magnetic moment.* Phys. Rev. D 97, 036001 (2018).