Meta Theory of the Zeckendorf-Hilbert Universe

17.4 Geometric Interpretation of Neutrino Oscillation: Flavor Mixing Mechanism on Discrete Spacetime Lattices

In Section 17.3, we solved the strong CP problem through the axion mechanism, showing how vacuum topological structure restores symmetry through dynamical relaxation. The final section of this chapter turns to another puzzling phenomenon in the Standard Model of particle physics—Neutrino Oscillation.

Discovery of neutrino oscillation proved that neutrinos have non-zero mass, and mixing exists between different “flavors” (Flavor, i.e., electron, muon, tau neutrinos). This phenomenon is usually phenomenologically described in the Standard Model by introducing the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. However, why are neutrino masses so tiny ($m_{\nu }\ll m_e$)? Why are mixing angles so large? These questions lack geometric explanations in continuous field theory frameworks.

This section will use QCA discrete ontology to propose a geometric topological interpretation of neutrino oscillation. We will prove that on discrete spacetime lattices, “flavor” is not an intrinsic label of particles, but projections of different topological excitation modes in QCA networks onto eigenbases of propagation operators. Neutrino oscillation is essentially geometric phase interference of these topological modes during discrete evolution.

17.4.1 Geometric Definition of Flavor: Misalignment of Internal Registers and Mass Basis

In QCA universe models (see Sections 3.1 and 16.1), each spacetime cell $x$ carries an internal Hilbert space ${\mathcal{H}}_{int}$. For the lepton sector, this internal space encodes “generation” or “flavor” information.

Definition 17.4.1 (Geometric Duality of Flavor Basis and Mass Basis)

On internal fiber bundles of QCA networks, there exist two natural orthogonal bases:

1. Flavor Basis $\{|{\nu }_{\alpha }⟩\}$ ($\alpha =e,\mu ,\tau$): This is the basis aligned with interaction vertices. In weak interaction processes (such as $\beta$ decay), local update rules of QCA (weak charge current operator $J_W^{\mu }$) are diagonal in this basis. Geometrically, this is the “interaction frame” defined on local fibers.

2. Mass Basis $\{|{\nu }_i⟩\}$ ($i=1,2,3$): This is the basis aligned with free propagation operator $U_{free}$. When freely propagating in vacuum, evolution operator $U$ of QCA is diagonal in this basis. Geometrically, this is the “inertial frame” defined on global tangent bundle.

Geometric Misalignment:

The root of neutrino oscillation lies in that interaction frames and inertial frames do not coincide in total space geometry. This misalignment is described by a unitary rotation matrix $U_{PMNS}$:

$$|{\nu }_{\alpha }⟩=\sum _i(U_{PMNS}{)}_{\alpha i}^{\ast }|{\nu }_i⟩$$

In QCA discrete ontology, this is no longer an artificial parameter, but non-trivial holonomy of total space unified connection $\mathbb{A}$ in flavor space components. When a neutrino propagates from its production point (interaction region), it is forced to “project” from flavor basis to mass basis for transport.

17.4.2 Microscopic Mechanism of Oscillation: Phase Beating on Discrete Lattices

Consider a neutrino with energy $E$ propagating on a QCA one-dimensional chain.

According to the quantum walk model in Section 4.2, eigenstates of different masses $m_i$ correspond to different microscopic rotation angles ${\theta }_i\approx m_{i}a$ (where $a$ is lattice spacing). This causes them to have different phase and group velocities.

Theorem 17.4.2 (Discrete Oscillation Formula)

Let a flavor eigenstate $|{\nu }_{\alpha }⟩={\sum }_i{U}_{\alpha i}^{\ast }|{\nu }_i⟩$ be produced at initial time $t=0$.

After $N$ discrete time steps (physical time $t=N\tau$), the state evolves to:

$$|\nu (t)⟩=\sum _i{U}_{\alpha i}^{\ast }{e}^{-i{E}_{i}t}|{\nu }_i⟩$$

In QCA, energy eigenvalues $E_i$ are determined by dispersion relation $\cos (E_i\tau )=\cos (pa)\cos ({\theta }_i)$. For extremely relativistic neutrinos ($p\gg m$), phase difference approximates to:

$$\Delta {\Phi }_{ij}=(E_i-E_j)t\approx \frac{m_i^2-m_j^2}{2E}t$$

(Note: This is the standard result, but in QCA, when $E$ approaches Planck energy scale $1/a$, there will be higher-order corrections).

Probability of detecting flavor $\beta$ at time $t$ is:

$$P({\nu }_{\alpha }\to {\nu }_{\beta })=|⟨{\nu }_{\beta }|\nu (t)⟩{|}^2={\left|\sum _i{U}_{\beta i}{e}^{-i\frac{m_i^2}{2E}t}{U}_{\alpha i}^{\ast }\right|}^2$$

This is the standard neutrino oscillation formula.

Unique QCA Perspective: Beat Frequency

From the perspective of discrete time crystals (DTC) (Chapter 10), each mass eigenstate $|{\nu }_i⟩$ corresponds to a specific detuning mode of universe fundamental frequency ${\omega }_{univ}$.

• Production of flavor $\alpha$ is a coherent excitation, simultaneously striking three different “tuning forks” ($|{\nu }_1⟩,|{\nu }_2⟩,|{\nu }_3⟩$).

• Oscillation phenomenon is essentially beating between frequencies of these three tuning forks.

• PMNS matrix elements $U_{\alpha i}$ determine amplitude weights of exciting each tuning fork per strike (interaction).

17.4.3 Mass Hierarchy and Geometrization of Seesaw Mechanism

Why are neutrino masses so tiny? In the Standard Model, right-handed neutrinos at grand unification energy scale are usually introduced and explained through Seesaw mechanism: $m_{\nu }\approx m_D^2/M_R$.

In QCA discrete ontology, this mechanism acquires topological interpretation.

Definition 17.4.3 (Topological Mass Generation)

Recalling Section 17.1, fermion mass arises from coupling of left-right chirality components (Zitterbewegung).

For charged fermions (such as electrons), this coupling is local, achieved through direct scattering with Higgs field (vacuum condensation), hence larger mass.

For neutrinos, due to their electrical neutrality, they are almost “transparent” in QCA networks. Their left-right chirality components (if right-handed neutrinos exist) are located in topologically isolated sectors.

• Left-handed component ${\nu }_L$: Coupled to “shallow” layer of spacetime geometry.

• Right-handed component ${\nu }_R$: Coupled to “deep” layer of spacetime geometry or “extra dimensions” (corresponding to high-frequency modes in KK theory).

Theorem 17.4.4 (Geometric Seesaw Theorem)

In QCA networks, chirality flip ${\nu }_L\leftrightarrow {\nu }_R$ of neutrinos requires an extremely rare non-local topological tunneling process. This process involves traversing the entire internal manifold or through higher-order quantum corrections.

Its effective coupling strength (i.e., mass $m_{\nu }$) is exponentially suppressed by topological barrier:

$$m_{\nu }\sim M_{Planck}\cdot e^{-S_{tunnel}}$$

This explains why neutrino masses are far smaller than other particles: they acquire mass through quantum tunneling effects of spacetime foam, not direct Higgs coupling.

17.4.4 Pontecorvo Geometric Phase and CP Violation

Possible CP violation (${\delta }_{CP}$ phase) in neutrino oscillation is key to explaining matter-antimatter asymmetry in the universe.

In Section 16.3, we proved that gauge field strength corresponds to total space curvature.

CP-violating phase ${\delta }_{CP}$ in PMNS matrix, geometrically corresponds to Berry flux in flavor space.

Corollary 17.4.5 (Aharonov-Bohm Effect in Flavor Space)

When three neutrino mass eigenstates evolve in flavor space and form closed loops (e.g., cyclic conversion in early thermal bath), they accumulate an ineliminable geometric phase ${\Phi }_{geom}\propto {\delta }_{CP}$.

This phase causes different oscillation probabilities for matter (neutrinos) and antimatter (antineutrinos):

$$P({\nu }_{\alpha }\to {\nu }_{\beta })\ne P({\bar{\nu }}_{\alpha }\to {\bar{\nu }}_{\beta })$$

This is a direct manifestation of chirality of spacetime geometry in flavor space. QCA universes “prefer” certain matter forms at the most fundamental geometric structure level.

Summary

This section completed the final argument for topological origin of matter.

1. Oscillation as Interference: Neutrino oscillation is phase beating of different topological mass modes propagating on discrete lattices.

2. Mixing as Misalignment: PMNS matrix describes rotational misalignment of interaction frames and inertial frames in total space geometry.

3. Mass as Tunneling: Tiny neutrino masses arise from topologically protected chirality flip barriers.

At this point, all discussions in Part IX Chapter 17 on “Topological Origin of Matter” conclude. We have proved: fermions, spin, strong CP axions, and neutrino oscillation are all natural emergences of topological structures (knots, holonomy, tunneling) in QCA discrete networks.

In the upcoming Volume IV, we will enter the most exciting and challenging field of the entire book—Physics of Agency. We will explore: in this universe composed of bits and geometry, how are observers born? Is consciousness also some topological structure?

(End of Volume III Part IX, continuing to Volume IV Part X: Algebraic Structure of Observers)