Quantised vortex windings realise U(1) holonomies; SU(2) matrix toy shows how non-Abelian factors could be composed (full SM not claimed).

Purpose

Provide a reviewer-auditable **toy** linking quantised vortex windings to U(1) holonomy phases and showing how non-Abelian factors could be composed — without claiming the full Standard Model gauge group from 2D GPE alone.

What it proves

On a four-vortex (+1,+1,−1,−1) cell, circular loops yield integer windings matching the enclosed vortex charge; the discrete holonomy phase ΣΔφ matches 2πQ to numerical tolerance; a standalone SU(2) product of Pauli exponentials is unitary with det = 1 to machine precision.

Relation to current theory

Addresses the 'where is SU(3)×SU(2)×U(1)?' criticism at the level of a **numerical existence proof** for holonomy arithmetic; extending to three generations and anomaly cancellation remains open (see Open issues).

Key equation

W_{\rm U(1)} \;=\; e^{i\oint \nabla\phi\cdot d\ell} \;=\; e^{2\pi i Q}

Interactive visualization

SVT predictionQuantised vortex windings realise U(1) holonomies; SU(2) matrix toy shows how non-Abelian factors could be composed (full SM not claimed).

Plots

stdout tail

  enc +1 at (6,-2): Q_exp=1  wind=1  ΣΔφ=6.2832  wind_PASS=True  holo_PASS=True
  enc both lower +1: Q_exp=2  wind=2  ΣΔφ=12.5664  wind_PASS=True  holo_PASS=True
  enc global (net 0): Q_exp=0  wind=0  ΣΔφ=0.0000  wind_PASS=True  holo_PASS=True
  SU(2) toy: unitarity max err = 2.22e-16, |det-1| = 0.00e+00  -> PASS

=================== VALIDATION ===================
  All integer windings match      : PASS
  U(1) holonomy phases (tolerance): PASS
  SU(2) composition toy           : PASS
  Overall                         : PASS

  SVT Prediction: Quantised vortex windings realise U(1) holonomies; SU(2) matrix toy shows how non-Abelian factors could be composed (full SM not claimed).
==================================================

  Matches data:   YES — Validated

/home/kruiserx/code/SVT2.0/sim_56_braid_holonomy_gauge_toy_2026.py:151: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown
  plt.show()
/home/kruiserx/code/SVT2.0/sim_56_braid_holonomy_gauge_toy_2026.py:177: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown
  plt.show()