APPENDIX B: TOPOLOGICAL DERIVATION OF FRACTIONAL CHARGE

(The $\mathbb{Z}_3$ Fundamental Group of the Void)

B.1 The Geometric Origin of Gauge Fields

Standard physics assumes charge quantization is an axiomatic property of the $U(1)$ gauge group. In SBF, we derive charge as the Berry Phase accumulated by a topological defect traversing the void network.

$$q = \frac{e}{2\pi} \oint \mathcal{A} \cdot d\mathbf{l} = \frac{e}{2\pi} \gamma$$

where $\gamma$ is the geometric Holonomy of the path.

B.2 Tetrahedral Voids (The Quark Sector)

The tetrahedral void (4 grains) possesses symmetry group $T_d$. We analyze the configuration space $M$ of a defect winding around this void.

• The Fundamental Group: Because the void has 3-fold rotational symmetry axes, a path $\mathcal{C}$ that winds once corresponds to a rotation of $2\pi/3$. Three windings ($\mathcal{C}^3$) are topologically homotopic to a trivial loop (no winding).

• The Result: This implies the fundamental group of the path space is the cyclic group of order 3:
  $$\pi_1(M) \cong \mathbb{Z}_3$$

• Quantization: The Holonomy $\rho$ maps this group to the $U(1)$ gauge phase. Since $\rho([\mathcal{C}]^3) = 1$, the phase must be a third root of unity:
  $$e^{i\gamma} = e^{2\pi i m / 3}, \quad m \in \{0, 1, 2\}$$

• Charge Spectrum: This forces the electric charge to be quantized in exact thirds:
  $$q = m \cdot \frac{e}{3}$$
  This rigorously derives the quark charge spectrum ($+2/3, -1/3$) from the topology of the 4-grain void, removing the need for heuristic solid-angle arguments.

B.3 Octahedral Voids (The Lepton Sector)

The octahedral void (6 grains) has point inversion symmetry ($O_h$). The fundamental group of its path space allows for integer windings ($\mathbb{Z}$), yielding integer charges ($q = n \cdot e$), corresponding to the Lepton sector.

B.4 Confinement (Anomaly Cancellation)

This topology also explains Confinement. An isolated quark ($m=1$) has a non-trivial holonomy ($e^{i 2\pi/3} \neq 1$), creating a topological obstruction string that connects it to the void.

However, combining three quarks (a Baryon) sums the phases:

$$\gamma_{total} = 3 \times \frac{2\pi}{3} = 2\pi \equiv 0 \pmod{2\pi}$$

The total phase vanishes, allowing the composite particle to detach from the void lattice and propagate freely. Confinement is thus the requirement for Topological Neutrality in the void network.