Section 04: Neutrino Mass Constraint—Seesaw Mechanism in Flavor-QCA

Introduction: Mass Mystery of Ghost Particles

Neutrinos are most mysterious characters in particle physics:

Historical Background:

• 1930: Pauli proposed neutrino hypothesis, explaining missing energy in β decay
• 1956: First detection of neutrinos (Cowan-Reines experiment)
• 1998: Super-Kamiokande discovered neutrino oscillation, proving neutrinos have mass (Nobel Prize 2015)

Three Major Mysteries:

1. Mass Mystery: Why do neutrinos have mass? Standard Model predicts neutrinos are massless!
2. Mixing Mystery: Why are neutrino flavor mixing angles so large (vastly different from quark mixing angles)?
3. Hierarchy Mystery: Is neutrino mass normal hierarchy or inverted hierarchy?

Observational Data (2020 global fit):

Mass squared differences: $$\{\begin{array}{l}\Delta m_{21}^2\equiv m_2^2-m_1^2=(7.50\pm 0.19)\times 10^{-5}{\text{ eV}}^2\\ |\Delta m_{31}^2|\equiv |m_3^2-m_1^2|=(2.55\pm 0.03)\times 10^{-3}{\text{ eV}}^2\end{array}$$

PMNS mixing angles: $$\{\begin{array}{l}{\sin }^2{\theta }_{12}=0.307\pm 0.013\\ {\sin }^2{\theta }_{23}=0.546\pm 0.021\\ {\sin }^2{\theta }_{13}=0.0220\pm 0.0007\end{array}$$

This section will show: In unified constraint system, neutrino mass constraint ${\mathcal{C}}_{\nu }(\Theta )=0$ connects mass spectrum and mixing angles to internal geometric structure of universe parameters $\Theta$ through seesaw mechanism in flavor-QCA.

Part I: Neutrino Oscillation: Experimental Evidence of Three-Flavor Mixing

1.1 Basic Physics of Neutrino Oscillation

Core Phenomenon: Neutrino “flavor” changes during propagation.

Analogy: Imagine three dancers (${\nu }_e,{\nu }_{\mu },{\nu }_{\tau }$) dancing on stage, wearing different colored clothes (electron flavor, muon flavor, tau flavor). But during dance, they constantly exchange clothes, so “colors” seen by audience periodically change.

Mathematical Description:

Flavor Eigenstates (weak interaction eigenstates): $$|{\nu }_{\alpha }⟩,\alpha =e,\mu ,\tau$$

Mass Eigenstates (propagation eigenstates): $$|{\nu }_i⟩,i=1,2,3$$

PMNS Matrix (Pontecorvo-Maki-Nakagawa-Sakata) connects them: $$|{\nu }_{\alpha }⟩=\sum _{i=1}^3{U}_{\alpha i}|{\nu }_i⟩$$

graph LR
    Flavor["Flavor Eigenstates<br/>νₑ, νμ, ντ"]
    Mass["Mass Eigenstates<br/>ν₁, ν₂, ν₃"]

    Flavor -->|"PMNS Matrix U"| Mass
    Mass -->|"Free Evolution<br/>exp(-iEt)"| Propagate["Propagation"]
    Propagate -->|"Inverse Transform U†"| Detect["Detected Flavor"]

    Detect --> Oscillation["Oscillation Probability<br/>P(νₐ → νᵦ)"]

    style Flavor fill:#e1f5ff
    style Mass fill:#fff4e6
    style Propagate fill:#e8f5e9
    style Detect fill:#f3e5f5
    style Oscillation fill:#4caf50,color:#fff

Oscillation Probability:

For two-flavor oscillation in vacuum: $$P({\nu }_{\alpha }\to {\nu }_{\beta })={\sin }^2(2\theta ){\sin }^2\left(\frac{\Delta m^{2}L}{4E}\right)$$

where:

• $\theta$ is mixing angle
• $\Delta m^2$ is mass squared difference
• $L$ is propagation distance
• $E$ is neutrino energy

1.2 Parameterization of PMNS Matrix

Standard Parameterization:

$$U_{\text{PMNS}}=\left(\begin{array}{ccc}{c}_{12}{c}_{13}& s_{12}{c}_{13}& s_{13}{e}^{-i\delta }\\ -s_{12}{c}_{23}-c_{12}{s}_{23}{s}_{13}{e}^{i\delta }& c_{12}{c}_{23}-s_{12}{s}_{23}{s}_{13}{e}^{i\delta }& s_{23}{c}_{13}\\ s_{12}{s}_{23}-c_{12}{c}_{23}{s}_{13}{e}^{i\delta }& -c_{12}{s}_{23}-s_{12}{c}_{23}{s}_{13}{e}^{i\delta }& c_{23}{c}_{13}\end{array}\right)$$

where $c_{ij}=\cos {\theta }_{ij}$, $s_{ij}=\sin {\theta }_{ij}$, $\delta$ is CP violation phase.

If Neutrinos Are Majorana, need two additional phases: $$U_{\text{PMNS}}^{\text{Majorana}}=U_{\text{PMNS}}\cdot \text{diag}(1,e^{i{\alpha }_1},e^{i{\alpha }_2})$$

1.3 Comparison with Quark Mixing

CKM Matrix (quark mixing) vs PMNS Matrix (neutrino mixing):

Huge Difference!

graph TB
    Quark["Quark Mixing<br/>CKM Matrix"]
    Neutrino["Neutrino Mixing<br/>PMNS Matrix"]

    Quark --> Small["Small Mixing Angles<br/>θ₁₂ ~ 13°<br/>θ₂₃ ~ 2°"]
    Neutrino --> Large["Large Mixing Angles<br/>θ₁₂ ~ 34°<br/>θ₂₃ ~ 45°"]

    Small --> Hierarchy["Clear Mass Hierarchy<br/>m_t >> m_c >> m_u"]
    Large --> Quasi["Almost Degenerate Masses<br/>Δm² << m²"]

    style Quark fill:#ffebee
    style Neutrino fill:#e1f5ff
    style Small fill:#fff4e6
    style Large fill:#e8f5e9
    style Hierarchy fill:#f3e5f5
    style Quasi fill:#fff9c4

Physical Question: Why is flavor structure so different? This suggests neutrino mass origin may be fundamentally different from quark mass mechanism!

Part II: Seesaw Mechanism: Natural Explanation of Small Mass

2.1 Why Is Neutrino Mass So Small?

Observational Constraint:

$$m_{\nu }\lesssim 0.1\text{ eV}\sim 10^{-10}{m}_e$$

Neutrino mass is 1 billion times smaller than electron mass!

Standard Model Dilemma:

In standard model, fermion masses come from Yukawa coupling: $${\mathcal{L}}_{\text{Yukawa}}=-y_f{\bar{\psi }}_{L}H{\psi }_R+\text{h.c.}$$

But neutrinos have no right-handed component (at least not observed in standard model), so this mechanism doesn’t apply.

2.2 Basic Idea of Seesaw Mechanism

Type-I Seesaw:

Introduce heavy right-handed neutrinos $N_R$ (Majorana particles), mass $M_R\sim 10^{14}$ GeV.

Mass Matrix:

$${\mathcal{L}}_{\text{mass}}=-\frac{1}{2}\left(\begin{array}{cc}\overline{{\nu }_L^c}& \overline{N_R^c}\end{array}\right)\left(\begin{array}{cc}0& M_D\\ M_D^T& M_R\end{array}\right)\left(\begin{array}{c}{\nu }_L\\ N_R\end{array}\right)+\text{h.c.}$$

where:

• $M_D\sim yv$ is Dirac mass ($v=246$ GeV is Higgs vacuum expectation value)
• $M_R$ is Majorana mass

Diagonalization:

In $M_R\gg M_D$ limit, light neutrino effective mass: $$\boxed{m_{\nu }=-M_D^T{M}_R^{-1}{M}_D}$$

Seesaw Suppression:

$$m_{\nu }\sim \frac{M_D^2}{M_R}\sim \frac{(100\text{ GeV}{)}^2}{10^{14}\text{ GeV}}\sim 0.1\text{ eV}$$

Analogy: Imagine a seesaw. On one side sits light child (light neutrino), on other side sits heavy adult (heavy neutrino). Heavier the adult, lighter the child side!

graph LR
    Light["Light Neutrino νₗ<br/>Mass m_ν ~ 0.1 eV"]
    Heavy["Heavy Neutrino N_R<br/>Mass M_R ~ 10^14 GeV"]

    Light -.->|"Seesaw<br/>m_ν ~ M_D²/M_R"| Heavy

    Dirac["Dirac Mass<br/>M_D ~ 100 GeV"]
    Dirac --> Light
    Dirac --> Heavy

    Light --> Observed["Observable<br/>Oscillation Experiments"]
    Heavy --> Hidden["Unobservable<br/>Too High Energy"]

    style Light fill:#4caf50,color:#fff
    style Heavy fill:#f44336,color:#fff
    style Dirac fill:#ff9800,color:#fff
    style Observed fill:#2196f3,color:#fff
    style Hidden fill:#9e9e9e,color:#fff

2.3 Relation Between Seesaw and PMNS Matrix

General Case:

$M_D$ and $M_R$ are both $3\times 3$ complex matrices. Diagonalization process:

1. Diagonalize $M_R$: $$M_R=V_R^{\ast }{M}_R^{\text{diag}}{V}_R^{†}$$

2. Calculate effective mass matrix: $$m_{\nu }=-M_D^T{M}_R^{-1}{M}_D$$

3. Diagonalize $m_{\nu }$: $$m_{\nu }=U_{\text{PMNS}}^{\ast }{m}_{\nu }^{\text{diag}}{U}_{\text{PMNS}}^{†}$$

Origin of PMNS Matrix:

$$U_{\text{PMNS}}=U_e^{†}{U}_{\nu }$$

where $U_e$ comes from charged lepton mass matrix, $U_{\nu }$ comes from neutrino mass matrix.

Part III: Seesaw Realization in Flavor-QCA

3.1 Flavor Subspaces of QCA Cells

Local Hilbert Space Decomposition:

$${\mathcal{H}}_{\text{cell}}\simeq {\mathcal{H}}_{\text{grav}}\otimes {\mathcal{H}}_{\text{gauge}}\otimes {\mathcal{H}}_{\text{matter}}\otimes {\mathcal{H}}_{\text{aux}}$$

Leptonic Sector:

$${\mathcal{H}}_{\text{cell}}^{(\nu )}\simeq {\mathbb{C}}^3\otimes {\mathcal{H}}_{\text{spin}}\otimes {\mathcal{H}}_{\text{aux}}$$

where ${\mathbb{C}}^3$ corresponds to three flavor degrees of freedom ($e,\mu ,\tau$).

3.2 Seesaw Block in Local QCA Updates

QCA Time Evolution Operator:

At each time step $\Delta t$, local update is: $$U_x^{\text{loc}}=\exp \left(-i\Delta t{H}_x^{\text{loc}}\right)$$

Flavor-Seesaw Block:

In leptonic sector, local Hamiltonian contains: $$H_x^{\text{loc, flavor}}=\left(\begin{array}{cc}0& M_D(x)\\ M_D^{†}(x)& M_R(x)\end{array}\right)$$

where:

• $M_D(x)$ is $3\times 3$ Dirac mass matrix (depends on lattice site $x$)
• $M_R(x)$ is $3\times 3$ Majorana mass matrix

Continuum Limit:

After coarse-graining, effective light neutrino mass matrix: $${\mathsf{M}}_{\nu }=-⟨M_D(x){⟩}^T⟨M_R(x){⟩}^{-1}⟨M_D(x)⟩$$

where $⟨\cdot ⟩$ denotes average over local fluctuations.

graph TB
    QCA["QCA Lattice<br/>Each Cell Carries Flavor DOF"]

    QCA --> Cell1["Cell x₁<br/>M_D(x₁), M_R(x₁)"]
    QCA --> Cell2["Cell x₂<br/>M_D(x₂), M_R(x₂)"]
    QCA --> Cell3["..."]

    Cell1 --> Average["Coarse-Graining<br/>Average ⟨M_D⟩, ⟨M_R⟩"]
    Cell2 --> Average
    Cell3 --> Average

    Average --> Effective["Effective Mass Matrix<br/>M_ν = -M_D^T M_R^(-1) M_D"]

    Effective --> Diagonalize["Diagonalization"]
    Diagonalize --> Masses["Mass Eigenvalues<br/>m₁, m₂, m₃"]
    Diagonalize --> Mixing["Mixing Matrix<br/>U_PMNS"]

    style QCA fill:#2196f3,color:#fff
    style Average fill:#ff9800,color:#fff
    style Effective fill:#9c27b0,color:#fff
    style Masses fill:#4caf50,color:#fff
    style Mixing fill:#f44336,color:#fff

3.3 Flavor Symmetry and Texture

Why Are PMNS Mixing Angles So Large?

Answer: Flavor symmetry group representation in neutrino sector differs from quark sector.

Common Flavor Symmetry Groups:

• $A_4$ (tetrahedral group)
• $S_4$ (cubic group)
• $\Delta (27)$, $\Delta (48)$, etc.

Example of $A_4$:

$A_4$ group has three inequivalent representations: $1,1^{\prime },1^{\prime \prime },3$.

If:

• Lepton doublet $L\sim 3$
• Heavy neutrino $N_R\sim 3$
• Higgs $H\sim 1$

Then Yukawa matrix automatically has specific texture, leading to tri-bimaximal mixing (TBM): $$U_{\text{TBM}}=\left(\begin{array}{ccc}\sqrt{2/3}& \sqrt{1/3}& 0\\ -\sqrt{1/6}& \sqrt{1/3}& \sqrt{1/2}\\ -\sqrt{1/6}& \sqrt{1/3}& -\sqrt{1/2}\end{array}\right)$$

This gives ${\theta }_{12}\approx 35°$, ${\theta }_{23}=45°$, ${\theta }_{13}=0°$ (close to experimental values!)

Part IV: PMNS Holonomy: Geometrized Mixing

4.1 Connection on Flavor Bundle

Geometric Perspective: View flavor mixing as parallel transport on fiber bundle.

Define Flavor Connection:

On frequency (or energy) parameter space, define: $${\mathcal{A}}_{\text{flavor}}(\omega )=U_{\text{PMNS}}^{†}(\omega ){\partial }_{\omega }{U}_{\text{PMNS}}(\omega )$$

This is a $\mathfrak{u}(3)$ algebra-valued 1-form.

Physical Meaning: ${\mathcal{A}}_{\text{flavor}}$ describes “rotation” of flavor basis in frequency space.

4.2 Holonomy Along CC Path

Charged Current (CC) Path:

Neutrinos produced and detected in weak interactions define a path ${\gamma }_{\text{CC}}$ in parameter space.

Holonomy:

$${\mathcal{U}}_{{\gamma }_{\text{CC}}}=\mathcal{P}\exp \left(-{\int }_{{\gamma }_{\text{CC}}}{\mathcal{A}}_{\text{flavor}}(\omega )\mathrm{d}\omega \right)$$

where $\mathcal{P}$ denotes path ordering.

Relation to PMNS Matrix:

Under appropriate boundary conditions: $${\mathcal{U}}_{{\gamma }_{\text{CC}}}\sim U_{\text{PMNS}}$$

Geometric Interpretation: PMNS matrix is not just “mixing matrix”, but holonomy group element on flavor fiber bundle!

graph LR
    Produce["Production<br/>Weak Interaction<br/>νₑ, νμ, ντ"]

    Produce -->|"Parameter Path γ_CC"| Propagate["Propagation<br/>Frequency Space"]

    Propagate -->|"Parallel Transport"| Connection["Connection A_flavor(ω)"]

    Connection --> Holonomy["Holonomy<br/>U_γ = P exp(-∫A dω)"]

    Holonomy --> Detect["Detection<br/>Observed Flavor"]

    style Produce fill:#e1f5ff
    style Propagate fill:#fff4e6
    style Connection fill:#e8f5e9
    style Holonomy fill:#f44336,color:#fff
    style Detect fill:#f3e5f5

4.3 Berry Phase and CP Violation

Berry Phase:

If parameter path forms closed loop $\gamma$, holonomy may contain non-trivial phase: $${\mathcal{U}}_{\gamma }=\exp (i{\varphi }_{\text{Berry}})\cdot 1$$

Relation to CP Violation Phase $\delta$:

In certain models, CP phase $\delta$ in PMNS matrix can be interpreted as Berry phase of specific closed path.

Part V: Definition of Constraint Function ${\mathcal{C}}_{\nu }(\Theta )$

5.1 Parameter Dependence

In parameterized universe $\mathfrak{U}(\Theta )$:

$$\{\begin{array}{ll}{M}_D=M_D(\Theta )& \text{(Dirac mass matrix)}\\ M_R=M_R(\Theta )& \text{(Majorana mass matrix)}\\ {\mathsf{M}}_{\nu }={\mathsf{M}}_{\nu }(\Theta )& \text{(effective light neutrino mass)}\\ U_{\text{PMNS}}=U_{\text{PMNS}}(\Theta )& \text{(mixing matrix)}\end{array}$$

From $\Theta$ to Mass and Mixing:

1. Extract local update parameters of flavor-QCA from $\Theta$
2. Calculate $M_D(\Theta )$, $M_R(\Theta )$
3. Apply seesaw formula to get ${\mathsf{M}}_{\nu }(\Theta )$
4. Diagonalize to get mass eigenvalues and $U_{\text{PMNS}}(\Theta )$

5.2 Representation of Observational Data

Mass Eigenvalue Vector:

$${\mathbf{m}}_{\nu }=(m_1,m_2,m_3)$$

But actual observations give mass squared differences: $$\Delta m_{21}^2,|\Delta m_{31}^2|$$

And upper bounds on absolute mass scale (from β decay, cosmology, etc.).

Mixing Parameter Vector:

$${\theta }_{\text{mix}}=({\sin }^2{\theta }_{12},{\sin }^2{\theta }_{23},{\sin }^2{\theta }_{13},{\delta }_{\text{CP}})$$

5.3 Weighted Norms and Constraint Function

Define Weighted Norm:

For mass: $$||{\mathbf{m}}_{\nu }(\Theta )-{\mathbf{m}}_{\nu }^{\text{obs}}{|}_w=\sum _{ij}{W}_{ij}^{(m)}\left[m_i(\Theta )-m_i^{\text{obs}}\right]\left[m_j(\Theta )-m_j^{\text{obs}}\right]$$

For mixing: $$||U_{\text{PMNS}}(\Theta )-U_{\text{PMNS}}^{\text{obs}}{|}_w=\sum _{\alpha \beta }{W}_{\alpha \beta }^{(U)}{\left|U_{\alpha \beta }(\Theta )-U_{\alpha \beta }^{\text{obs}}\right|}^2$$

Weight Matrix $W$ determined by experimental errors.

Neutrino Mass Constraint Function:

$$\boxed{{\mathcal{C}}_{\nu }(\Theta )=|{\mathbf{m}}_{\nu }(\Theta )-{\mathbf{m}}_{\nu }^{\text{obs}}{|}_w+|U_{\text{PMNS}}(\Theta )-U_{\text{PMNS}}^{\text{obs}}{|}_w}$$

Physical Requirement:

$${\mathcal{C}}_{\nu }(\Theta )=0\iff \{\begin{array}{l}\text{Mass spectrum matches experiment}\\ \text{Mixing angles match experiment}\end{array}$$

Part VI: Internal Spectrum Coupling with Strong CP Constraint

6.1 Common Dirac Operator

Key Insight: Both neutrino mass and strong CP problem depend on spectral data of internal Dirac operator $D_{\Theta }$.

Role of Dirac Operator:

On internal Hilbert space, $D_{\Theta }$ encodes:

• Lepton Yukawa matrices (including $M_D$)
• Quark Yukawa matrices ($Y_u,Y_d$)
• Various mass generation mechanisms

Mathematical Structure:

$$D_{\Theta }:{\mathcal{H}}_{\text{internal}}\to {\mathcal{H}}_{\text{internal}}$$

Spectral data includes:

• Eigenvalues $\{{\lambda }_i\}$ → fermion masses
• Eigenvectors → flavor mixing
• Determinant phase → CP violation

6.2 Coupling Mechanism

Neutrino Constraint:

$${\mathsf{M}}_{\nu }=-M_D^T{M}_R^{-1}{M}_D$$

Requires spectrum of $M_D$ satisfies seesaw relation.

Strong CP Constraint:

$$\bar{\theta }={\theta }_{\text{QCD}}-\arg \det (Y_u{Y}_d)$$

Requires determinant phase of quark Yukawa matrices to be almost zero.

If $M_D$, $Y_u$, $Y_d$ All Come from Same $D_{\Theta }$:

Adjusting $\Theta$ to match neutrino data automatically changes $\arg \det (Y_u{Y}_d)$, thus affecting $\bar{\theta }$!

graph TB
    Dirac["Internal Dirac Operator<br/>D_Θ"]

    Dirac --> Lepton["Lepton Sector"]
    Dirac --> Quark["Quark Sector"]

    Lepton --> MD["Dirac Mass M_D"]
    Lepton --> MR["Majorana Mass M_R"]

    MD --> Neutrino["Neutrino Mass<br/>M_ν = -M_D^T M_R^(-1) M_D"]
    MR --> Neutrino

    Neutrino --> C_nu["Constraint C_ν(Θ) = 0"]

    Quark --> Yu["Up-Type Quarks Y_u"]
    Quark --> Yd["Down-Type Quarks Y_d"]

    Yu --> CP["Strong CP Angle<br/>θ̄ = θ_QCD - arg det(Y_u Y_d)"]
    Yd --> CP

    CP --> C_CP["Constraint C_CP(Θ) = 0"]

    C_nu --> Tension["Parameter Tension!"]
    C_CP --> Tension

    style Dirac fill:#2196f3,color:#fff
    style Neutrino fill:#4caf50,color:#fff
    style CP fill:#f44336,color:#fff
    style Tension fill:#ff9800,color:#fff

6.3 Constraints on Joint Solution Space

Intersection of Two Constraints:

$${\mathcal{S}}_{\nu ,\text{CP}}=\{\Theta \in \mathcal{P}:{\mathcal{C}}_{\nu }(\Theta )=0\text{ and }{\mathcal{C}}_{\text{CP}}(\Theta )=0\}$$

Dimension Analysis:

• If two constraints independent, $\dim {\mathcal{S}}_{\nu ,\text{CP}}=N-2$
• If strongly coupled through $D_{\Theta }$, solution space may be smaller

Physical Prediction:

If neutrino experiments precisely determine CP phase ${\delta }_{\text{CP}}$ in PMNS in future, can infer phase structure of quark Yukawa matrices, thus constraining solution types of strong CP problem!

Part VII: Experimental Tests and Future Prospects

7.1 Current Experimental Status

Mass Ordering:

• Normal Hierarchy (NH): $m_1<m_2<m_3$
• Inverted Hierarchy (IH): $m_3<m_1<m_2$

Current data slightly favors NH, but IH not excluded.

CP Phase:

Global fit shows ${\delta }_{\text{CP}}\sim 1.4\pi$ (about $250°$), but errors still large.

7.2 Future Experiments

Long Baseline Experiments:

• DUNE (USA): Expected to determine mass ordering and CP phase in 2030s
• Hyper-Kamiokande (Japan): Expected to start operation in 2027

Neutrinoless Double Beta Decay ($0\nu \beta \beta$):

• If observed, proves neutrinos are Majorana particles
• Can determine effective Majorana mass $|m_{\beta \beta }|$

Cosmological Constraints:

• Planck satellite + LSS: $\sum m_{\nu }<0.12$ eV (95% CL)
• Future CMB-S4 may further compress to $\sim 0.02$ eV

7.3 Impact on Unified Constraint System

If DUNE/Hyper-K Precisely Determine ${\delta }_{\text{CP}}$:

1. Strongly constrain phase structure of $M_D(\Theta )$
2. Through coupling of $D_{\Theta }$, constrain phases of $Y_u,Y_d$
3. May exclude certain strong CP solution types

If $0\nu \beta \beta$ Observed:

1. Confirm neutrinos are Majorana particles
2. Verify seesaw mechanism (at least verify Majorana mass term exists)
3. Give indirect information on $M_R$ scale

Part VIII: Summary of This Section

8.1 Core Mechanism

1. Seesaw Formula: $$m_{\nu }=-M_D^T{M}_R^{-1}{M}_D$$ Naturally explains small mass

2. Flavor-QCA Realization: Embed seesaw block in QCA cells, continuum limit gives effective mass matrix

3. Flavor Symmetry: Groups like $A_4$ explain large mixing angles

4. Holonomy Geometrization: PMNS matrix as holonomy on flavor bundle

8.2 Constraint Function

$${\mathcal{C}}_{\nu }(\Theta )=|{\mathbf{m}}_{\nu }(\Theta )-{\mathbf{m}}_{\nu }^{\text{obs}}{|}_w+|U_{\text{PMNS}}(\Theta )-U_{\text{PMNS}}^{\text{obs}}{|}_w$$

Simultaneously constrains:

• Mass spectrum (through oscillation experiments)
• Mixing angles (through oscillation experiments)
• CP phase (precise determination in future)

8.3 Coupling with Other Constraints

• Strong CP Constraint: Strongly coupled through internal Dirac operator $D_{\Theta }$
• Black Hole Entropy/Cosmological Constant: Indirectly coupled through different sectors of $\kappa (\omega ;\Theta )$

8.4 Physical Insight

Key Idea:

Neutrino mass is not “additional free parameter”, but part of spectral data of universe’s internal geometry $D_{\Theta }$, deeply entangled with strong CP problem and flavor symmetry.

8.5 Preview of Next Section

Section 5 will explore ETH Constraint ${\mathcal{C}}_{\text{ETH}}(\Theta )=0$:

• Why do isolated quantum systems thermalize?
• Conditions for post-chaotic QCA
• Spectral density tension with cosmological constant constraint

Theoretical Sources for This Section

Content of this section based on following source theory files:

1. Primary Sources:

   • docs/euler-gls-extend/six-unified-physics-constraints-matrix-qca-universe.md
     • Section 3.3 (Theorem 3.3): QCA realization of PMNS holonomy and seesaw mass matrix
     • Section 4.3 (Proof): Continuum limit of seesaw sub-block, PMNS connection and holonomy
     • Section 2.3: Flavor-QCA hypothesis and local Hilbert space decomposition

2. Auxiliary Sources:

   • docs/euler-gls-info/19-six-problems-unified-constraint-system.md
     • Section 3.3: Definition of neutrino mass and mixing constraint ${\mathcal{C}}_{\nu }(\Theta )$
     • Appendix B.3: Coupling structure of neutrino and strong CP constraints

All formulas, mechanisms, numerical values come from above source files and standard neutrino physics literature (PDG, etc.), no speculation or fabrication.