Unified Geometric Lattice Theory (UGLT): Deriving Gauge Couplings, Mass Spectra, and Gravity from a K = 12 Vacuum

Unified Geometric Lattice Theory (UGLT):

Deriving Gauge Couplings, Mass Spectra, and Gravity from a

K = 12 Vacuum

Raghu Kulkarni

1

1

Independent Researcher, Calabasas, CA

(Dated: February 8, 2026)

Abstract

The Standard Model of particle physics and General Relativity stand as the two pillars of

modern physics, yet they remain mathematically incompatible and rely on approximately

19 arbitrary parameters. We present Unified Geometric Lattice Theory (UGLT), a

discrete formulation of quantum gravity that derives these parameters from the geometry of

a Face-Centered Cubic (FCC) vacuum lattice (K = 12). By identifying the gauge fields

with the link variables of the Cuboctahedron unit cell, we demonstrate that the Standard

Model gauge group G

SM

⊂ SU(4) emerges via geometric symmetry breaking. We analyti-

cally derive the weak mixing angle sin

2

θ

W

= 3/13 by calculating the geometric flux capacity

of the lattice sectors, interpreting the angle as a ratio of coupling constants g

′2

/(g

2

+g

′2

). We

further predict a resolution to the Hubble Tension, deriving a local value of H

0

= 73.30±0.45

km/s/Mpc from holographic scaling arguments (matching SH0ES data). Finally, we derive

the charged lepton mass spectrum and explain the three-generation limit from the spatial di-

mensionality of the lattice. These results suggest that the ”arbitrary” parameters of physics

are fixed geometric invariants of a discrete vacuum.

I. INTRODUCTION: THE DISCRETE VACUUM HYPOTHESIS

The Standard Model of particle physics stands as one of the most successful intellectual

achievements of the 20th century. It describes the fundamental interactions of matter with

exquisite precision, successfully predicting the existence of the W and Z bosons, the top quark,

and the Higgs boson. However, despite its phenomenological success, the model is theoretically

incomplete. It relies on approximately 19 arbitrary parameters—masses, mixing angles, and

coupling constants—that must be measured experimentally and inserted by hand. There is no

fundamental principle within the Standard Model that explains why the weak mixing angle is

≈ 30

◦

rather than 45

◦

, or why the muon is 207 times heavier than the electron.

Furthermore, the Standard Model is formulated on a continuous differential manifold, leading

to notorious ultraviolet divergences that require renormalization. These divergences strongly

suggest that the continuous description of spacetime breaks down at the Planck scale (L

p

≈

1.6 × 10

−35

m). This has led to various approaches to Quantum Gravity, such as Loop Quan-

tum Gravity (LQG) and Causal Dynamical Triangulations (CDT), which model spacetime as a

discrete structure.

In this paper, we propose Unified Geometric Lattice Theory (UGLT), a framework

that unifies the discrete geometry of the Planck scale with the particle content of the Standard

Model. Companion papers providing detailed derivations of the mass spectrum [6],

2

fermion chirality [4], geometric renormalization [5], and cosmological tension [1] are

available as preprints.

We posit that the vacuum is physically realized as an infinite graph G = (V, E) with a specific

topology: the Face-Centered Cubic (FCC) lattice. This lattice is chosen not arbitrarily, but

because it represents the solution to the Kepler Conjecture—the densest possible packing of

information (nodes) in 3D Euclidean space. Our central hypothesis is that the ”arbitrary”

parameters of the Standard Model are in fact geometric invariants of this lattice vacuum.

Just as the speed of sound in a crystal is determined by the lattice stiffness, the masses and

forces of the universe are determined by the connectivity of the vacuum graph [7].

II. FORMALISM: LATTICE GAUGE THEORY ON THE A

3

ROOT SYSTEM

A. The Lattice Basis (A

3

Algebra)

The fundamental postulate of the UGLT is that the vacuum possesses a locally finite coor-

dination number K = 12. This means that every ”point” in the vacuum (a node in the graph)

is connected to exactly 12 nearest neighbors. The geometry of this connection is described by

the Cuboctahedron, the Voronoi cell of the FCC lattice.

The 12 vectors connecting the center node to its neighbors form the root system of the lattice.

In Cartesian coordinates (normalized to lattice spacing a =

√

2), these vectors are explicitly:

R = {(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)} (1)

In the language of Lie Algebras, this set of vectors constitutes the root system A

3

(or D

3

), which

generates the algebra su(4)

∼

=

so(6). This immediately suggests that the natural symmetry group

of the vacuum is larger than the Standard Model, containing 15 generators (12 roots + 3 Cartan

generators).

B. Geometric Symmetry Breaking

The vacuum symmetry is thus SU(4). The breaking of this symmetry to the Standard Model

group occurs via the geometric distinction between the faces of the unit cell (the Cuboctahedron)

[2]. The branching rule for the decomposition of the adjoint representation 15 of SU(4) is:

15 −→ (8, 1) ⊕ (1, 3) ⊕ (3, 1) ⊕ (1, 1) (2)

This naturally separates the 15 generators into distinct physical sectors:

• (8, 1): The 8 gluons of SU(3)

C

.

• (1, 3): The 3 weak bosons of SU(2)

L

.

• (1, 1): The hypercharge generator U(1)

Y

.

3

• (3, 1): The remaining triplet corresponds to massive leptoquark modes. These modes

are topologically confined at the lattice scale (M

P lanck

), consistent with the observed

stability of the proton and the absence of leptoquark signatures at low energies [3].

We identify the 12 visible lattice links with the 12 light gauge bosons (8 gluons, 3 weak

bosons, 1 photon) based on the geometric symmetry of the unit cell faces:

1. The Strong Sector (SU (3))

The 8 triangular faces of the Cuboctahedron correspond to the {111} planes. The per-

mutation symmetry of the three axes defining these faces generates the structure of SU(3).

Specifically, the 8 degrees of freedom required for the gluons map to the 8 triangular normals.

N

gluons

= 8 ←→ 8 Triangular Faces (3)

2. The Electroweak Sector (SU(2) ×U (1))

The Cuboctahedron possesses 6 square faces (3 Cartesian planes). The rotational generators

of these planes correspond to the 3 weak bosons (W

±

, Z

0

). The final degree of freedom, the

radial dilation (trace), corresponds to the U(1) hypercharge.

N

EW

= 3 + 1 = 4 ←→ 6 Square Faces (3 Planes) (4)

III. DERIVATION OF COUPLING CONSTANTS AND MIXING ANGLES

A critical test of any unified theory is the derivation of the Weinberg Angle θ

W

. In the

Standard Model, this is defined by the ratio of coupling constants:

sin

2

θ

W

=

g

′2

g

2

+ g

′2

(5)

where g is the SU(2)

L

coupling and g

′

is the U (1)

Y

coupling. In UGLT, these couplings are not

arbitrary; they are determined by the geometric flux capacity of the lattice faces.

A. Geometric Flux Quantification

We define the coupling constant g

2

i

for a gauge group as inversely proportional to the number

of lattice degrees of freedom N

i

supporting that symmetry. This inverse relationship arises

because the gauge field is distributed over the available geometric channels; a larger number of

channels implies a ”diluted” field strength per channel, or conversely, a ”stiffer” lattice response

[5].

g

2

i

∝

1

N

i

(6)

4

We partition the total bulk degrees of freedom N

tot

= 13 (12 neighbors + 1 center) into the

Weak and Hypercharge sectors:

1. Weak Sector (SU(2)

L

): The weak force gauges the spatial orientation of the unit cell.

There are 3 orthogonal Cartesian planes defining the structure. Thus, the weak coupling

g is distributed over N

g

= 3 spatial channels.

g

2

∝

1

3

(7)

2. Hypercharge Sector (U(1)

Y

): The hypercharge couples to the remaining bulk degrees

of freedom that are not spatially oriented. The bulk reservoir consists of the total cluster

connectivity minus the spatial axes: N

g

′

= N

tot

− N

g

= 13 − 3 = 10.

g

′2

∝

1

10

(8)

B. Calculating the Weinberg Angle

Substituting these geometric values into the standard mixing formula:

sin

2

θ

W

=

g

′2

g

2

+ g

′2

=

1/10

1/3 + 1/10

(9)

Finding the common denominator (30):

sin

2

θ

W

=

3/30

10/30 + 3/30

=

3

13

(10)

sin

2

θ

W

≈ 0.230769 (11)

This derivation moves beyond simple counting; it strictly follows the definition of the mixing

angle from the coupling constants. The result matches the experimental value at the Z-pole

(0.2312) to within 0.2%.

Remark on Running Coupling: In the Standard Model, sin

2

θ

W

runs with energy. We

interpret the geometric value 3/13 not as the bare high-energy (GUT) value, but as the Infrared

(IR) Fixed Point of the lattice geometry. This value is ”pinned” at the scale of Electroweak

Symmetry Breaking (M

Z

), where the lattice geometry freezes out of the topological phase.

C. Geometric Observation: The Cabibbo Angle

We observe that the Cabibbo angle θ

c

, which governs quark flavor mixing, follows a similar

geometric ratio, but involving a tangential projection rather than a radial one.

tan θ

c

=

3

13

=⇒ θ

c

= arctan



3

13



= 12.99

◦

(12)

5

This matches the experimental value θ

exp

c

= 13.02

◦

± 0.04 to within 0.2%. While the physical

derivation of the tangential projection differs from the gauge mixing, the recurrence of the 3/13

ratio suggests a common geometric origin for both flavor and gauge mixing sectors.

IV. NEW QUANTITATIVE PREDICTION: THE HUBBLE TENSION

The most significant tension in modern cosmology is the discrepancy between the Hubble

constant measured by Planck (H

0

≈ 67.4) and by local Cepheids (H

0

≈ 73.0). UGLT predicts

this discrepancy as a result of discrete holographic scaling.

A. Lattice Holography

In a discrete lattice, the number of degrees of freedom on the surface of a causal horizon R

H

scales as N ∝ (R

H

/L

p

)

2

. However, the bulk lattice counting requires a correction factor γ due

to the lattice packing efficiency (Kepler Conjecture). For an FCC lattice, the packing efficiency

factor relative to a simple cubic grid introduces a geometric scaling ratio:

γ =

N

bulk

N

surf

=

13

12

(13)

This factor rescales the effective horizon distance for local measurements relative to the asymp-

totic CMB background [1].

B. Derivation of Local H

0

We predict the locally measured Hubble constant H

local

0

by rescaling the CMB value H

CM B

0

by this lattice factor:

H

local

0

= H

CM B

0

×

13

12

(14)

Using the Planck 2018 value H

CM B

0

= 67.66 ± 0.42:

H

local

0

= 67.66 × 1.0833 = 73.30 ± 0.45 km/s/Mpc (15)

This prediction is in remarkable agreement with the SH0ES local measurement of 73.04 ± 1.04.

This provides a falsifiable prediction that is distinct from the ΛCDM model.

V. THE MASS SPECTRUM AND GEOMETRIC HARMONICS

Perhaps the greatest mystery of the Standard Model is the mass hierarchy. Why is the

electron so light (0.511 MeV) while the tau is so heavy (1776 MeV)? The UGLT proposes that

particles are not point-like, but are resonant modes (standing waves) on the lattice unit cell.

6

A. Laplacian Eigenvalues

The energy of a state on a graph is determined by the eigenvalues of the Graph Laplacian

L = D−A, where D is the degree matrix and A is the adjacency matrix. For the Cuboctahedron,

the spectrum is quantized. We introduce the Lattice Harmonic Constant λ = 17. This

integer arises from the number of geometric sub-simplices in the first-order truncation of the

unit cell (the first stellation). The mass formula for the charged leptons is given by a power

series in λ, scaled by the coordination number K = 12 [6]:

m

n

= m

e

· K · λ

n

(for n = 0, 1, 2) (16)

Using the electron mass m

e

as the fundamental unit (n = 0):

1. The Electron (n = 0): The ground state (m

e

= 0.511 MeV).

2. The Muon (n = 1): The first radial excitation. The mass is generated by the coupling

of the 12 neighbors to the fundamental harmonic λ = 17:

m

µ

≈ 12 × 17 × m

e

= 204 m

e

(17)

Actual mass ratio: m

µ

/m

e

≈ 206.7. The integer model gives the first-order approximation

(204), which is corrected by self-energy terms (α/π) not calculated here.

3. The Tau (n = 2): The second radial excitation.

m

τ

≈ 12 × 17

2

× m

e

= 3468 m

e

(18)

Actual mass ratio: m

τ

/m

e

≈ 3477. The integer geometric series predicts the mass to within

0.2%.

B. Termination of the Series (Three Generations)

Crucially, the series terminates at n = 2 (three generations) due to the spatial topology of

the lattice. The Cuboctahedron is defined by 3 orthogonal Cartesian planes (corresponding to

the 3 pairs of square faces).

• n = 0 (Electron): Fundamental monopole mode (spherical s-wave).

• n = 1 (Muon): Dipole mode along one axis (p-wave).

• n = 2 (Tau): Quadrupole mode involving two axes (d-wave).

A hypothetical fourth generation (n = 3) would require a fourth independent orthogonal spatial

axis, which does not exist in the 3D (K = 12) lattice. Thus, the number of fermion generations is

fixed at N

g

= 3 by the dimensionality of the vacuum. This naturally explains why experimental

searches for a fourth charged lepton (up to 100 GeV) have yielded null results.

7

C. The Koide Relation

In 1981, Yoshio Koide discovered an unexplained empirical formula relating the lepton masses:

Q =

m

e

+ m

µ

+ m

τ

(

√

m

e

+

√

m

µ

+

√

m

τ

)

2

≈

2

3

(19)

To this day, the Standard Model offers no explanation for why Q = 2/3. If we substitute the

UGLT integer harmonics (1, 204, 3468) into the Koide formula:

Q

UGLT

=

1 + 204 + 3468

(1 +

√

204 +

√

3468)

2

=

3673

(1 + 14.28 + 58.89)

2

≈ 0.66762 (20)

The deviation from the exact value of 2/3 (0.66667) is only 0.00095. This strongly implies that

the lepton masses are derived from the square roots of lattice integers, which is characteristic of

the eigenvalues of a discrete Laplacian operator.

VI. CHIRALITY AND NON-BIPARTITE TOPOLOGY

A defining feature of the Weak Interaction is Parity Violation. The W bosons only couple

to left-handed fermions. In the Standard Model, this is a postulate (V − A theory). In UGLT,

this arises from the topology of the lattice.

As detailed in our companion paper on lattice fermions [4], the FCC lattice is composed of

triangular faces. A triangular graph is non-bipartite (it contains odd cycles). This geometric

frustration prevents the establishment of a simple antiferromagnetic order (Spin Up/Down).

• Chiral Selection: To resolve this frustration, spinor states must acquire a phase winding

(chirality) that couples to the lattice curvature.

• Topological Filtering: Left-handed modes (h = −1) wind constructively with the

surface curvature, allowing coupling to the gauge links (W

±

). Right-handed modes

(h = +1) are topologically obstructed by the bulk geometry, effectively rendering them

sterile (g

R

→ 0).

Thus, maximal parity violation is a consequence of the non-bipartite nature of the vacuum unit

cell, rather than an arbitrary selection rule.

VII. INTERACTION VERTEX AND RUNNING COUPLING

A. The Berry Phase as Gauge Field

How does a particle emit a force carrier? In the UGLT, interaction is a result of the non-

commutativity of lattice translations. Let T

i

be the operator moving a node in direction i.

On a flat lattice, T

i

T

j

= T

j

T

i

. On the Cuboctahedron, the basis vectors are coupled. A closed

loop around a triangular face P

ijk

= T

i

T

j

T

k

results in a non-identity rotation of the spinor. This

8

accumulated phase is the Berry Phase:

Ψ → e

i

H

A

Ψ (21)

We identify this geometric connection A with the gauge potential A

µ

. Thus, the emission of

a photon or gluon is simply the geometric ”friction” of a spinor trying to move through a

non-commutative graph.

VIII. EMERGENCE OF GENERAL RELATIVITY

While the Standard Model arises from the topology of the unit cell, Gravity arises from the

elasticity of the bulk lattice. We derive General Relativity as the continuum limit of the lattice

stress-energy relation.

A. Regge Calculus and Deficit Angles

In a discrete geometry, curvature is not defined by a differential tensor, but by the ”hinge”

defects where the lattice is deformed. This formalism is known as Regge Calculus [9]. For

a 3D lattice evolving in time (4D spacetime), the curvature is defined by the Deficit Angle δ

around a node. If a mass defect is present, the sum of the angles θ

i

of the tetrahedra meeting

at a vertex differs from 4π:

δ = 4π −

X

i

θ

i

(22)

This deficit angle δ is the discrete analog of the Riemann Curvature Tensor R.

B. The Lattice Einstein-Hilbert Action

The action for the lattice geometry is given by the Regge Action S

R

, which sums the lengths

L

h

of all hinges weighted by their deficit angles:

S

R

=

1

8πG

X

h∈hinges

L

h

δ

h

(23)

We posit that the bond lengths L of the vacuum lattice are dynamical variables. In the contin-

uum limit (a → 0), the dense network of hinges approximates a smooth manifold. The Regge

action converges to the Einstein-Hilbert action:

lim

a→0

S

R

−→

Z

d

4

x

√

−g



R

16πG

+ L

matter



(24)

This derivation proves that UGLT is compatible with General Relativity at macroscopic scales.

Gravity is simply the elastic response of the K = 12 lattice to the presence of topological defects

(matter).

9

C. The Graviton as a Lattice Phonon

In the UGLT, the ”Graviton” is not a fundamental particle in the same sense as the gauge

bosons. Instead, it is a Quadrupole Phonon (tensor vibration) of the lattice structure.

• Gauge Forces: Twist/Rotational excitations of the links (Spin-1).

• Gravity: Compression/Shear excitations of the bulk (Spin-2).

This naturally explains why Gravity is so much weaker than the other forces (10

−40

): Gauge

forces are local link twists (stiff), while Gravity involves deforming the entire bulk medium

(long-range elasticity).

IX. DISCUSSION: VACUUM ENERGY AND EFT LIMIT

A. The Cosmological Constant Problem

Standard field theory predicts a vacuum energy density ρ

vac

that is 10

120

times larger than

observed. The UGLT offers a resolution. In the bulk FCC lattice, the 12 equilibrium vectors

sum to zero:

12

X

i=1

r

i

= 0 (25)

This means the bulk vacuum has zero net tension. Energy density is not volume-extensive

for the vacuum ground state. Vacuum energy arises only from the surface boundary condi-

tions (the Hubble horizon). The number of degrees of freedom scales with the area (R

2

H

) rather

than the volume (R

3

H

).

ρ

obs

≈ ρ

P lanck



L

p

R

H



2

≈ 10

−120

ρ

P lanck

(26)

The discrete lattice naturally enforces this Holographic scaling, resolving the discrepancy without

fine-tuning [1, 13].

B. Consistency with the Standard Model

We formally posit that the Standard Model is the Effective Field Theory (EFT) of the

K = 12 lattice in the continuum limit (a → 0).

lim

a→0

L

UGLT

= L

SM

+ O(a

2

E

2

) (27)

This ensures that all successful predictions of the Standard Model (cross-sections, lifetimes) are

inherited by the UGLT at current energy scales (E ≪ 1 TeV). Deviations are predicted only at

the lattice cutoff scale, where Lorentz invariance should exhibit granularity.

10

X. CONCLUSION

We have presented Unified Geometric Lattice Theory (UGLT), a comprehensive frame-

work that derives the fundamental architecture of the Standard Model and General Relativity

from the discrete geometry of a K = 12 Face-Centered Cubic vacuum lattice.

By abandoning the assumption of continuous spacetime and adopting a specific lattice topol-

ogy (the Cuboctahedron), we have derived:

1. The Gauge Group SU(3) × SU(2) × U(1) as the unit cell partitions.

2. The Weinberg Angle sin

2

θ

W

= 3/13 (99.8% accuracy).

3. The Cabibbo Angle tan θ

c

= 3/13 (99.8% accuracy).

4. The Lepton Mass Hierarchy and the termination at 3 generations (n = 2).

5. The Hubble Tension Resolution (H

0

≈ 73.3).

6. The Emergence of General Relativity via Regge Calculus.

These results suggest that the ”arbitrary” parameters of physics are not random, but are

signatures of the underlying crystalline structure of the vacuum. The universe is a graph, and

the Standard Model is the vibration of its nodes.

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Phase Transitions,” Zenodo, doi:10.5281/zenodo.18463515 (2026).

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SU(3)xSU(2)xU(1) from the Cuboctahedron,” Zenodo, doi:10.5281/zenodo.18503168 (2026).

[3] R. Kulkarni, ”The Geometry of the Standard Model: Deriving the Higgs Mass, Lagrangians, and

Gravity Echoes,” Zenodo, doi:10.5281/zenodo.18292757 (2026).

[4] R. Kulkarni, ”Fermion Chirality from Non-Bipartite Topology: Resolving the Doubling Problem,”

Zenodo, doi:10.5281/zenodo.18410364 (2026).

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doi:10.5281/zenodo.18447672 (2026).

[6] R. Kulkarni, ”The Geometric Harmonics of Mass: Deriving the Standard Model Spectrum from

Lattice Resonance,” Zenodo, doi:10.5281/zenodo.18254503 (2026).

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