Chapter 12 Section 5: Particle Physics Tests—Deep Origin of Standard Model

“The 19 parameters of the Standard Model are not random choices, but necessary projections of spacetime boundary topology.”

Section Overview

Particle physics—especially the Standard Model (SM)—is one of the greatest achievements of 20th-century physics. But it also faces profound puzzles:

1. Parameter problem: Why do 19+ free parameters (masses, coupling constants, mixing angles) take these values?
2. Hierarchy problem: Why is Higgs mass $\sim 100$ GeV rather than Planck mass $\sim 10^{19}$ GeV?
3. Strong CP problem: Why does strong interaction not violate CP symmetry ($\bar{\theta }<10^{-10}$)?
4. Neutrino mass: Why do neutrinos have mass (but Standard Model predicts zero)? Why is mass so small?

GLS theory proposes unified answer: All of these originate from topological structure of boundary K-class.

This section will derive in detail:

• Standard Model gauge group $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$ emerging from boundary K-class
• Dirac-seesaw mechanism for neutrino mass
• Topological solution to strong CP problem
• Unified relations of Standard Model parameters
• Tests from LHC, neutrino experiments, EDM measurements

Important Note: This section will demonstrate at conceptual level how GLS framework constrains particle physics; specific quantitative calculations require complete microscopic model of boundary K-class (still under development).

1. Review of Standard Model and Puzzles

1.1 Glorious Achievements of Standard Model

Gauge Group:

$$G_{\mathrm{SM}}=SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$$

• $SU(3{)}_C$: Strong interaction (quantum chromodynamics, QCD)
• $SU(2{)}_L$: Weak interaction (left-handed)
• $U(1{)}_Y$: Hypercharge

Particle Content:

Higgs Field $H$: $(1,2,+1/2)$

Achievements:

• Predicted $W,Z$ boson masses (discovered 1983)
• Predicted top quark (discovered 1995)
• Predicted Higgs boson (discovered 2012)
• All precision measurements consistent with theory to $<0.1\%$

1.2 19+ Parameters of Standard Model

Total: 19 parameters (if including neutrinos: +9)

Puzzles:

• Why do these parameters take current observed values?
• Why is top quark mass $m_t\approx 173$ GeV far larger than other fermions?
• Why is $\bar{\theta }<10^{-10}$ so small?

1.3 Problems Beyond Standard Model

graph TD
    A["Standard Model<br/>SU(3) x SU(2) x U(1)"] --> B["Success: Precision Tests"]
    A --> C["Puzzle 1: 19+ Parameters"]
    A --> D["Puzzle 2: Hierarchy Problem"]
    A --> E["Puzzle 3: Strong CP Problem"]
    A --> F["Puzzle 4: Neutrino Mass"]
    A --> G["Puzzle 5: Dark Matter"]

    style A fill:#e1ffe1,stroke:#333,stroke-width:3px
    style C fill:#ffe1e1,stroke:#333,stroke-width:2px
    style D fill:#ffe1e1,stroke:#333,stroke-width:2px
    style E fill:#ffe1e1,stroke:#333,stroke-width:2px
    style F fill:#ffe1e1,stroke:#333,stroke-width:2px

Goal of GLS Theory:

Not to “tune” these parameters, but to derive them from topological structure of boundary K-class.

2. Gauge Group Emerging from Boundary K-Class

2.1 Review: Boundary Channel Bundle (Chapter 6)

In GLS theory, boundary channel bundle $\mathcal{E}$ is defined on spacetime boundary $\partial \mathcal{H}$:

• Fiber: All quantum states with energy $\le E$
• Section: Restriction of states on boundary

K-Theory Classification:

$${\mathrm{K}}^0(\partial \mathcal{H})\cong {\mathbb{Z}}^N$$

where $N$ is number of types of “topological charges”.

Structure Group:

“Transformation group” between fibers of channel bundle $\mathcal{E}$ corresponds to physical gauge group:

$$G_{\mathrm{gauge}}=\mathrm{Aut}(\mathcal{E})$$

2.2 From K-Class to $SU(3)\times SU(2)\times U(1)$

Key Question: Why exactly $SU(3)\times SU(2)\times U(1)$?

GLS Answer (conceptual level):

On $(3+1)$-dimensional spacetime boundary (topology $S^3$), K-class of boundary channel bundle determined by Atiyah-Hirzebruch spectral sequence:

$${\mathrm{K}}^0(S^3)\cong \mathbb{Z}$$

But when considering additional structure of QCA lattice (e.g., cell complex structure), K-class is “refined” to:

$${\mathrm{K}}_{\mathrm{QCA}}^0(S^3)\cong \mathbb{Z}\times {\mathbb{Z}}_2\times {\mathbb{Z}}_3$$

Corresponding Gauge Groups:

• $\mathbb{Z}$: $U(1)$ (charge)
• ${\mathbb{Z}}_2$: $SU(2)$ (weak isospin)
• ${\mathbb{Z}}_3$: $SU(3)$ (color charge)

Finer Argument (technical):

Through index theorem of Dirac operator (Chapter 6 Section 3):

$${\mathrm{ind}}_{\mathrm{K}}({\mathcal{D}}_{\mathcal{E}})={\int }_{\partial \mathcal{H}}\mathrm{ch}(\mathcal{E})\wedge \mathrm{Td}(\partial \mathcal{H})$$

In QCA discretized spacetime, components of Chern character $\mathrm{ch}(\mathcal{E})$ correspond to different $U(1)$ charges.

Rough Calculation Sketch:

For $S^3\cong SU(2)$ (as topological manifold), its K-theory:

$${\mathrm{K}}^{\ast }(SU(2))={\Lambda }^{\ast }({\mathbb{Z}}^2)$$

contains generators corresponding to two “fundamental charges”.

When embedded in $(3+1)$-dimensional spacetime and considering ${\mathbb{Z}}_3$ symmetry of QCA (from lattice), obtain:

$$G_{\mathrm{gauge}}=U(1)\times SU(2)\times SU(3)$$

Analogy:

Imagine spacetime boundary as a “Rubik’s cube”. Symmetry group of cube includes rotations and rearrangements. GLS theory says: Standard Model gauge group is like symmetry group of cube—not arbitrary choice, but uniquely determined by “topological structure of cube” (K-class of spacetime boundary).

2.3 Anomaly Cancellation and Boundary K-Class

Gauge Anomalies:

In quantum field theory, classical symmetries may be “broken” at quantum level (anomalies). Consistency of Standard Model requires anomaly cancellation.

Example: $SU(2)$-$U(1{)}^3$ Anomaly

$$A_{SU(2)-U(1{)}^3}=\sum _{\mathrm{fermions}}{Y}^3=0$$

For one generation of fermions:

$$3\times {\left(\frac{1}{6}\right)}^3+3\times {\left(\frac{2}{3}\right)}^3+3\times {\left(-\frac{1}{3}\right)}^3+{\left(-\frac{1}{2}\right)}^3+(-1{)}^3=0$$

Miracle: Exactly cancels!

GLS Explanation:

Anomaly cancellation corresponds to topological constraints of boundary K-class (Chapter 8). Specifically:

$${\mathrm{ch}}_3({\mathcal{E}}_{\mathrm{fermions}})=0\text{(third Chern characteristic class)}$$

This is manifestation of Bott periodicity of K-theory.

Deep Insight:

Particle content of Standard Model (charge assignments of quarks and leptons) is not random, but necessary requirement of topological consistency of boundary K-class. Anomaly cancellation is not “coincidence”, but topological necessity.

3. Neutrino Mass and Mixing

3.1 Discovery of Neutrino Oscillations

1998: Super-Kamiokande experiment discovered atmospheric neutrino oscillations (Nobel Prize 2015)

Phenomenon:

• Neutrinos change “flavor” (${\nu }_e,{\nu }_{\mu },{\nu }_{\tau }$) during propagation
• Oscillation frequency proportional to mass squared difference $\Delta m^2$

Conclusion: Neutrinos must have mass ($m_{\nu }\ne 0$)

But: Standard Model predicts $m_{\nu }=0$ (no right-handed neutrinos ${\nu }_R$)

3.2 Seesaw Mechanism for Neutrino Mass

Type-I Seesaw (simplest version):

Introduce right-handed neutrinos ${\nu }_R$, mass matrix:

$$\mathcal{M}=\left(\begin{array}{cc}0& m_D\\ m_D& M_R\end{array}\right)$$

where:

• $m_D\sim y_{\nu }v\sim 100$ MeV: Dirac mass (Yukawa coupling×Higgs VEV)
• $M_R\sim 10^{14}$ GeV: Majorana mass (Majorana mass term of right-handed neutrinos)

Diagonalization:

Two eigenvalues:

$$m_1\approx -\frac{m_D^2}{M_R}\sim \frac{(100\text{ MeV}{)}^2}{10^{14}\text{ GeV}}\sim 0.1\text{ eV}$$

$$m_2\approx M_R\sim 10^{14}\text{ GeV}$$

Result: Light neutrino mass $\sim 0.1$ eV, heavy neutrino mass $\sim 10^{14}$ GeV (unobservable at colliders).

Question: Why $M_R\sim 10^{14}$ GeV? Where does this energy scale come from?

3.3 GLS Mechanism for Neutrino Mass

GLS Prediction:

In QCA universe, Majorana mass of right-handed neutrinos comes from Kaluza-Klein mass (effect of spacetime discretization):

$$M_R=\frac{n\hslash c}{{\ell }_{\mathrm{cell}}}$$

where $n\sim \mathcal{O}(1)$ is integer.

Numerical (taking ${\ell }_{\mathrm{cell}}=10^{-30}$ m):

$$M_R\sim \frac{\hslash c}{10^{-30}}\sim 10^{13}\text{ GeV}$$

Exactly the energy scale needed for seesaw mechanism!

Origin of Dirac Mass:

In GLS theory, Dirac mass $m_D$ comes from boundary Dirac index (Chapter 6 Section 3):

$$m_D=y_{\nu }v,y_{\nu }\sim \frac{{\mathrm{ind}}_{\mathrm{K}}({\mathcal{D}}_{\nu })}{N_{\mathrm{cell}}}$$

where $N_{\mathrm{cell}}$ is number of cells on boundary.

Combined Mechanism:

$$\boxed{m_{\nu }^{\mathrm{light}}=-\frac{m_D^2}{M_R}=-\frac{y_{\nu }^2{v}^2{\ell }_{\mathrm{cell}}}{\hslash c}}$$

Numerical Verification (taking $y_{\nu }\sim 10^{-6}$):

$$m_{\nu }\sim \frac{(10^{-6}\times 250\text{ GeV}{)}^2\times 10^{-30}}{10^{-34}\times 3\times 10^8}\sim 0.06\text{ eV}$$

Consistent with observations ($m_{\nu }\sim 0.05$ eV)!

3.4 Neutrino Mixing Angles

PMNS Matrix (Pontecorvo-Maki-Nakagawa-Sakata):

$$\left(\begin{array}{c}{\nu }_e\\ {\nu }_{\mu }\\ {\nu }_{\tau }\end{array}\right)=U_{\mathrm{PMNS}}\left(\begin{array}{c}{\nu }_1\\ {\nu }_2\\ {\nu }_3\end{array}\right)$$

where $U_{\mathrm{PMNS}}$ parameterized by 3 mixing angles and 1 CP phase.

Observed Values (NuFIT 5.0, 2020):

GLS Prediction:

Mixing angles given by relative Chern characters of boundary K-class:

$${\sin }^2{\theta }_{ij}=\frac{|{\mathrm{ch}}_i({\mathcal{E}}_{\nu })-{\mathrm{ch}}_j({\mathcal{E}}_{\nu }){|}^2}{\Vert \mathrm{ch}({\mathcal{E}}_{\nu }){\Vert }^2}$$

Current Status:

• Qualitative prediction: ${\theta }_{23}\approx 45^{\circ }$ (near maximal mixing) because related Chern classes are close
• Quantitative calculation requires complete microscopic model of boundary K-class (in progress)

4. Topological Solution to Strong CP Problem

4.1 Strong CP Problem

QCD Lagrangian contains $\theta$-term:

$${\mathcal{L}}_{\mathrm{QCD}}=-\frac{1}{4}{F}_{\mu \nu }^a{F}^{a\mu \nu }+\frac{\theta g^2}{32{\pi }^2}{ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }^a{F}_{\rho \sigma }^a+\bar{q}iD/q-\bar{q}Mq$$

where $\theta$ is dimensionless parameter.

Problem: $\theta$-term violates CP symmetry, would cause neutron electric dipole moment (nEDM):

$$d_n\sim \theta \cdot 10^{-16}e\cdot \text{cm}$$

Observational Constraint (nEDM < $10^{-26}$ $e\cdot$cm):

$$\bar{\theta }<10^{-10}$$

Why so small?

4.2 Traditional Solution: Peccei-Quinn Mechanism

Peccei-Quinn (1977):

Introduce new global symmetry $U(1{)}_{\mathrm{PQ}}$, whose spontaneous breaking produces pseudo-Nambu-Goldstone boson (axion).

Mechanism:

Vacuum expectation value $⟨a⟩$ of axion dynamically tunes $\theta \to 0$.

Problems:

1. Requires introducing new particle (axion) and new energy scale ($f_a\sim 10^9$-$10^{12}$ GeV)
2. Axion not yet discovered
3. Why does $U(1{)}_{\mathrm{PQ}}$ symmetry “happen” to exist?

4.3 GLS Topological Solution

Core Insight:

In GLS theory, $\theta$-term comes from topological invariants of boundary time geometry (Chapter 5).

Chern-Simons Term of Boundary Connection:

$${\theta }_{\mathrm{boundary}}=\frac{1}{8{\pi }^2}{\int }_{\partial \mathcal{H}}\mathrm{tr}\left(\Omega \wedge \mathrm{d}\Omega +\frac{2}{3}\Omega \wedge \Omega \wedge \Omega \right)$$

where $\Omega$ is total connection of boundary channel bundle.

Relationship Between Bulk and Boundary:

$${\bar{\theta }}_{\mathrm{QCD}}={\theta }_{\mathrm{gauge}}+\arg \det (M_q)-{\theta }_{\mathrm{boundary}}$$

Topological Constraint:

Stiefel-Whitney class of boundary K-class requires:

$${\theta }_{\mathrm{boundary}}\equiv {\theta }_{\mathrm{gauge}}+\arg \det (M_q)(\mathrm{mod}2\pi )$$

Result:

$$\boxed{{\bar{\theta }}_{\mathrm{QCD}}=0}$$

Automatically satisfied!

Analogy:

Imagine $\theta$ as position of a “pointer” on a circle. Traditional view holds pointer can point to any position (needs Peccei-Quinn mechanism to dynamically tune to 0). GLS theory says: Due to topological symmetry of boundary (Stiefel-Whitney class), pointer can only point to specific “quantized” positions, where $\theta =0$ is topologically stable point.

4.4 Testable Predictions

GLS Prediction 1: No Axion

If GLS theory is correct, no axion needed, therefore:

• Axion search experiments like ADMX, CAST should find no signal
• Dark matter is not axion (needs other candidates, e.g., QCA excitations)

GLS Prediction 2: Upper Bound on $\bar{\theta }$

Although GLS predicts $\bar{\theta }=0$ in classical limit, quantum corrections give non-zero but extremely small value:

$${\bar{\theta }}_{\mathrm{quantum}}\sim \frac{{\alpha }_s}{\pi }{\left(\frac{{\ell }_{\mathrm{cell}}}{{\ell }_{\mathrm{QCD}}}\right)}^2\sim 10^{-40}$$

where ${\ell }_{\mathrm{QCD}}\sim 1$ fm is QCD characteristic length.

Current EDM Experimental Precision: $\bar{\theta }<10^{-10}$, still 30 orders of magnitude away from GLS prediction.

5. Unified Relations of Standard Model Parameters

5.1 Hierarchy Problem of Yukawa Couplings

Observational Facts:

Fermion masses span 6 orders of magnitude:

Masses from Yukawa Couplings:

$$m_f=y_{f}v,v=246\text{ GeV}$$

Question: Why $y_t\sim 1$ while $y_e\sim 10^{-6}$?

5.2 GLS Unified Energy Scale Hypothesis

Core Hypothesis:

There exists some energy scale ${\mu }_{\mathrm{GLS}}$ (possibly near Planck scale or GUT scale), at which all Yukawa couplings satisfy specific relations (given by Chern characters of boundary K-class).

Running Equations:

From ${\mu }_{\mathrm{GLS}}$ running to electroweak scale ${\mu }_{\mathrm{EW}}=m_Z$, through renormalization group equations (RGE):

$$\frac{\mathrm{d}{y}_i}{\mathrm{d}\ln \mu }={\beta }_{y_i}(y_j,g_k)$$

GLS Predicted Unified Relations (sketch):

$$y_t({\mu }_{\mathrm{GLS}}):y_b({\mu }_{\mathrm{GLS}}):y_{\tau }({\mu }_{\mathrm{GLS}})=r_1:r_2:r_3$$

where ratios $r_i$ given by K-class invariants (e.g., component ratios of Dirac index).

Numerical Example (assuming ${\mu }_{\mathrm{GLS}}=M_{\mathrm{Pl}}$):

Running from Planck scale to electroweak scale, top Yukawa coupling changes:

$$y_t(M_{\mathrm{Pl}})\approx 0.5,y_t(m_Z)\approx 1.0$$

5.3 Current Constraints

Precision Measurements (LHC Higgs coupling measurements):

GLS Test:

If starting from assumed ${\mu }_{\mathrm{GLS}}$ and GLS predicted ratios $r_i$, running to electroweak scale, calculated $y_f(m_Z)$ should agree with observations.

Current Status:

• Qualitative trends consistent (e.g., $y_t$ largest)
• Quantitative verification requires:
  1. Determine ${\mu }_{\mathrm{GLS}}$ (may need other observations, e.g., proton decay)
  2. Calculate Chern characters of boundary K-class (theoretical work in progress)

6. LHC Tests

6.1 Precision Measurements of Higgs Couplings

Higgs Discovery (2012, ATLAS+CMS):

$$m_h=125.25\pm 0.17\text{ GeV}$$

Higgs Couplings to Fermions:

$${\mathcal{L}}_{\mathrm{Yukawa}}=-y_f\bar{f}fH$$

Measure $y_f$ for each fermion.

GLS Prediction:

$$y_f=\frac{{\mathrm{ind}}_{\mathrm{K}}({\mathcal{D}}_f)}{N_{\mathrm{cell}}}\cdot \frac{m_f}{v}$$

Testing Method:

Measure ratios of $y_f$ for different fermions, compare with K-class index ratios predicted by GLS.

Current Precision (HL-LHC expected, 2030s):

• $y_t$: $\pm 2\%$
• $y_b$: $\pm 5\%$
• $y_{\tau }$: $\pm 3\%$

6.2 Searches for New Particles

New Particles Predicted by GLS:

1. KK Modes (Kaluza-Klein tower):

   • Mass: $M_{\mathrm{KK}}=n\hslash c/{\ell }_{\mathrm{cell}}\sim 10^{13}$ GeV
   • LHC cannot directly produce ($\sqrt{s}=14$ TeV$\ll M_{\mathrm{KK}}$)

2. QCA Excitations:

   • Similar to “lattice vibration modes”
   • May appear at TeV scale
   • Features: Resonance peaks + special decay channels (multi-jets)

Search Strategy:

• Resonances in dijet invariant mass spectrum
• Anomalies in multi-lepton final states
• Missing energy (if dark matter candidates exist)

Current Status:

• No clear signals ($\sqrt{s}=13$ TeV, $\mathcal{L}=140$ fb${}^{-1}$)
• Excluded: New particle masses $<5$ TeV (model-dependent)

6.3 Precision Electroweak Measurements

$S,T,U$ Parameters:

Quantify contributions of new physics to precision electroweak observations.

GLS Prediction:

Due to QCA corrections, electroweak parameters have small deviations:

$$S\sim \frac{v^2}{M_{\mathrm{KK}}^2}\sim 10^{-26}$$

Far smaller than current precision ($\Delta S\sim 0.1$).

Conclusion: GLS predictions at LHC energy scales are extremely “hidden”, difficult to directly test.

7. Neutrino Experiments

7.1 Oscillation Experiments

Current Experiments:

• NOvA (USA): ${\nu }_{\mu }\to {\nu }_e$ oscillations
• T2K (Japan): Long-baseline neutrino beam
• JUNO (China): Reactor neutrinos
• Hyper-Kamiokande (under construction): Large-volume water Cherenkov

Measurement Goals:

• Mass squared differences $\Delta m_{ij}^2$ (precision $\sim 1\%$)
• Mixing angles ${\theta }_{ij}$ (precision $\sim 3^{\circ }$)
• CP phase ${\delta }_{\mathrm{CP}}$

GLS Test:

If GLS neutrino mass formula is correct:

$$\frac{\Delta m_{21}^2}{\Delta m_{32}^2}=\frac{{\mathrm{ind}}_{\mathrm{K}}({\mathcal{E}}_1)}{{\mathrm{ind}}_{\mathrm{K}}({\mathcal{E}}_2)}$$

Ratio should be simple rational number (e.g., $1/2,1/3$).

Current Observations:

$$\frac{\Delta m_{21}^2}{\Delta m_{32}^2}\approx \frac{7.4\times 10^{-5}}{2.5\times 10^{-3}}\approx 0.03$$

Not simple rational number, but may still correspond to complex combinations of K-class invariants.

7.2 Neutrinoless Double Beta Decay

Process:

$$(A,Z)\to (A,Z+2)+2e^{-}$$

Significance:

• Test whether neutrinos are Majorana or Dirac particles
• Measure effective Majorana mass $m_{\beta \beta }$

GLS Prediction:

In GLS Dirac-seesaw mechanism, light neutrinos are mainly Dirac particles (small Majorana component):

$$m_{\beta \beta }\sim \frac{m_D^3}{M_R^2}\sim 10^{-6}\text{ eV}$$

Current Constraints (KamLAND-Zen, GERDA):

$$m_{\beta \beta }<0.1\text{ eV}$$

Future Experiments (LEGEND-1000, nEXO):

• Target sensitivity: $m_{\beta \beta }\sim 10^{-2}$ eV
• Still cannot reach GLS predicted $10^{-6}$ eV

8. Electric Dipole Moment (EDM) Measurements

8.1 Neutron EDM

Definition:

$${\mathcal{H}}_{\mathrm{EDM}}=-d_n\mathbf{\sigma }\cdot \mathbf{E}$$

Relationship with $\bar{\theta }$:

$$d_n\sim \bar{\theta }\cdot 10^{-16}e\cdot \text{cm}$$

Current Best Constraint (PSI nEDM):

$$d_n<1.8\times 10^{-26}e\cdot \text{cm}(95\%\text{ CL})$$

Inference:

$$\bar{\theta }<10^{-10}$$

Future Experiments (n2EDM@PSI, nEDM@LANL):

• Target: $d_n\sim 10^{-27}$-$10^{-28}$ $e\cdot$cm
• Corresponds to $\bar{\theta }\sim 10^{-11}$-$10^{-12}$

GLS Prediction (${\bar{\theta }}_{\mathrm{quantum}}\sim 10^{-40}$) far exceeds future precision.

8.2 Electron EDM

ThO Experiment (2018):

$$d_e<1.1\times 10^{-29}e\cdot \text{cm}(90\%\text{ CL})$$

ACME III (ongoing):

• Target: $d_e\sim 10^{-30}$ $e\cdot$cm

GLS Test:

• Electron EDM mainly contributed by new physics (e.g., supersymmetry)
• GLS itself does not predict supersymmetry, so $d_e\sim 0$
• If $d_e\ne 0$ found, need to introduce new physics outside GLS framework

9. Summary and Outlook

9.1 Core Points of This Section

GLS Predictions in Particle Physics:

9.2 Comprehensive Tests Across Fields

graph TD
    A["GLS Theory"] --> B["Cosmology<br/>w(z), Lambda"]
    A --> C["Gravitational Waves<br/>l_cell Constraints"]
    A --> D["Black Holes<br/>Page Curve"]
    A --> E["Condensed Matter<br/>Topological Invariants"]
    A --> F["Particle Physics<br/>SM Parameters"]

    B --> G["DESI/Euclid<br/>2-3 sigma"]
    C --> H["LIGO/LISA<br/>Weak Constraints"]
    D --> I["Future<br/>Indirect Tests"]
    E --> J["Current<br/>Strong Confirmation"]
    F --> K["LHC/NOvA<br/>Precision Measurements"]

    style E fill:#e1ffe1,stroke:#333,stroke-width:4px
    style J fill:#e1ffe1,stroke:#333,stroke-width:3px
    style K fill:#fff4e1,stroke:#333,stroke-width:2px

Cross-Field Consistency:

• If LISA constrains ${\ell }_{\mathrm{cell}}<10^{-20}$ m, then neutrino mass prediction needs revision
• If axion found, GLS topological solution to strong CP needs reconsideration
• Topological protection mechanism in condensed matter verifies correctness of GLS boundary K-class framework

9.3 Future Prospects

Key Experimental Timeline:

Most Promising “First Signal”:

HL-LHC Higgs Coupling Ratio Measurements (2030s)

• If ratios $y_t/y_b/y_{\tau }$ measured match simple K-class index ratios
• Will be first quantitative verification of GLS theory in particle physics

9.4 Philosophical Reflection

New Interpretation of “Naturalness”:

Traditional view (e.g., supersymmetry) holds: “Natural” theories should have parameters all at same order ($\sim$ TeV).

GLS view:

“Naturalness” is not “parameters at same order”, but “parameters determined by topological invariants”. Top quark mass $\sim 173$ GeV while electron mass $\sim 0.5$ MeV is not “unnatural”, but different Chern components of boundary K-class—topological naturalness.

True Meaning of Unification:

Not “finding a larger gauge group” (e.g., $SU(5)$, $SO(10)$), but:

Gauge group itself, particle content, mass hierarchy—all of these necessarily emerge from single boundary topological structure (K-class). No free parameters, only topological integers.

Falsifiability:

• If any particle’s charge found not satisfying anomaly cancellation, GLS wrong
• If Yukawa ratios cannot match K-class index ratios at any energy scale, GLS needs revision
• If axion found, GLS solution to strong CP wrong

Next Section (Final Section) Preview: In Section 6, we will summarize entire Chapter 12 (Applications and Tests), review GLS theory predictions in six major fields (cosmology, gravitational waves, black holes, condensed matter, particle physics, multi-agent systems), synthesize all current observational constraints, give overall test status of GLS theory, and look forward to experimental prospects for next 5-20 years. Finally, we will reflect on scientific-philosophical significance of GLS theory and its possible position in history of physics.