5.4 Origin of Fermion Statistics: Riccati Square Root and ${\mathbb{Z}}_2$ Phase

After establishing the picture that “mass is topological knots,” we face the most profound question of this chapter: Why must these massive “knots” be fermions?

In standard physics, the connection between spin and statistics (spin-statistics theorem) is derived through axioms in relativistic quantum field theory. It shows that particles with half-integer spin must follow Fermi-Dirac statistics (wave function changes sign upon exchange), while integer spin particles follow Bose-Einstein statistics (unchanged upon exchange). But in our discrete ontology, we do not presuppose continuous spacetime symmetry groups, nor spinor fields.

This section will prove a startling conclusion: Fermion statistics is not an arbitrary rule, but a topological fingerprint that any self-referential structure (massive particle) capable of maintaining its existence in discrete networks necessarily carries. This fingerprint originates from the inherent square root structure of Riccati equations describing feedback loops.

5.4.1 Input-Output Relations of Self-Referential Systems

A massive particle is essentially a “self-maintaining” information loop. We can abstract it as a black box that receives input information ${\psi }_{in}$ and produces output information ${\psi }_{out}$, while part of the output is fed back to the input to maintain its own state.

In QCA’s transmission line model, the physical quantity describing this input-output relationship is impedance $Z$ (or reflection coefficient $r$).

$$Z=\frac{V}{I}(\text{voltage/current})$$

In quantum mechanical correspondence, $Z$ corresponds to logarithmic derivative of wave function at boundary $Z\sim {\psi }^{\prime }/\psi$.

A stable particle means its internal impedance structure $Z(x)$ must satisfy some self-consistent equation. For discrete iterative systems, impedance evolution follows Riccati equation (discrete form is Möbius transformation):

$$Z_{n+1}=\frac{A{Z}_n+B}{C{Z}_n+D}$$

where $A,B,C,D$ are determined by QCA’s local evolution operator $\hat{U}$.

5.4.2 Fixed Points and Square Root Branches

A particle as a stable topological soliton means it is not only localized in space, but also a fixed point in time evolution. That is, after one period of internal evolution, its impedance structure should restore:

$$Z^{\ast }=\frac{A{Z}^{\ast }+B}{C{Z}^{\ast }+D}$$

Solving this equation, we get a quadratic equation:

$$C(Z^{\ast }{)}^2+(D-A)Z^{\ast }-B=0$$

Its solution is:

$$Z^{\ast }=\frac{(A-D)\pm \sqrt{(A-D{)}^2+4BC}}{2C}$$

Here appears a decisive mathematical feature: Square Root.

This means, any self-consistent, non-trivial stable particle state has its physical parameters (impedance $Z$) defined on a double-sheeted Riemann surface.

5.4.3 Rotation, Exchange, and ${\mathbb{Z}}_2$ Phase

Now, we consider particle’s exchange statistics.

In $3+1$ dimensional spacetime, exchanging two identical particles $1$ and $2$ is topologically equivalent to rotating one particle around the other by $360^{\circ }$ (or rotating the particle itself by $360^{\circ }$ in center-of-mass frame).

In parameter space, this rotation operation corresponds to evolving along a closed path on the Riemann surface for one cycle.

Since $Z^{\ast }$ contains square root $\sqrt{\Delta }$ (where $\Delta$ is a complex function of evolution parameters), when parameters rotate $2\pi$ around origin, square root function changes sign:

$$\sqrt{e^{i2\pi }}=e^{i\pi }=-1$$

This is the origin of ${\mathbb{Z}}_2$ topological phase.

• For Bosons (photons): They have no internal self-referential feedback loops ($v_{int}=0$), so they don’t need to solve Riccati fixed points. Their states are directly defined on single-valued plane. Rotating $360^{\circ }$ returns to origin, phase unchanged ($+1$).

• For Fermions (massive particles): They must maintain a self-referential dead knot ($v_{int}>0$). Mathematical solution of this knot lies on branch cut of square root. Rotating $360^{\circ }$ causes system to slide to another sheet of Riemann surface, wave function acquires $-1$ phase.

Theorem 5.4 (Mass-Statistics Theorem):

In local unitary QCA networks, any stable excitation maintained by nonlinear self-referential feedback (i.e., massive particles) necessarily has double-valued wave functions, thus exhibiting fermion statistics under exchange operations.

5.4.4 Ontological Status of Spinors

This discovery completely changes our understanding of spinors.

In traditional geometry, spinors are defined as “geometric objects that change sign upon $360^{\circ }$ rotation,” usually viewed as abstract mathematical constructs.

But in our theory, spinors are “square roots of scalars.”

• Physical observables (such as energy, charge, impedance) are single-valued (scalars or vectors).

• Underlying probability amplitudes (wave functions) must be square roots of these observables to satisfy self-consistent equations.

Therefore, fermions are not strange, special particles. Fermions are the normal state of matter existence. Any information structure attempting to “stop” and maintain its identity in spacetime networks must anchor itself on topological structure through “square root” operations, thus inevitably becoming fermions.

5.4.5 Summary: The Trio of Matter

At this point, we have completed Part III “The Emergence of Matter.” Starting from Axiom $\Omega$, we constructed the complete microscopic picture of matter:

1. Photons: Massless, non-self-referential, single-valued phase translation modes.

2. Mass: Topological dead knots of information flow, resisting external forces (inertia) through internal oscillation ($v_{int}$).

3. Fermions: Self-referential logic required to maintain such dead knots necessarily introduces square root structure, leading to $-1$ exchange phase.

This picture not only explains “what,” but also “why.” It unifies mass, spin, and statistical properties under a simple geometric-logical framework.

In the next Part IV, we will explore the most mysterious component of this cosmic machine—observers. We will see how, when these fermion knots form sufficiently complex networks, they begin to “see” themselves.