Structural Correspondence between the Standard Model and Vacuum
Geometry: SU (3) × SU (2) × U (1) from the Cuboctahedron
Raghu Kulkarni
1, ∗
1
Independent Researcher, Calabasas, CA
(Dated: February 6, 2026)
The Standard Model is defined by its gauge group SU(3)×SU(2)×U (1). We propose that
this structure corresponds to the face topology of a saturated vacuum lattice. Identifying the
vacuum unit cell as the Cuboctahedron (the Voronoi cell of the Face-Centered Cubic lattice,
K = 12), we identify a structural mapping between its faces and the Standard Model forces.
The 8 non-bipartite triangular faces support geometric frustration, corresponding to the 8
generators of the confining SU (3) sector. The 6 bipartite square faces support flux screening,
corresponding to the electroweak SU(2)×U(1) sector. We further propose geometric ansatzes
for the fundamental constants, obtaining a Weinberg angle sin
2
θ
W
= 3/13 and a Fine
Structure inverse α
−1
≈ 137. Finally, we argue that the non-bipartite lattice topology
dynamically suppresses instanton formation, potentially resolving the Strong CP problem
via geometry.
I. INTRODUCTION
Narrative: The gauge group of the Standard Model, G
SM
= SU(3) × SU(2) × U (1), is the
foundational input of modern particle physics. While typically viewed as an abstract symmetry
chosen by nature, we propose it arises from the constructive geometry of a discrete vacuum. Unlike
Grand Unified Theories (GUTs) which embed G
SM
in a larger group like SU(5) [1], we suggest
G
SM
is the natural gauge structure of a lattice saturated at the maximum packing limit.
Formalism: We utilize a lattice gauge framework where the vacuum is modeled as a tensor
network saturated at the Kepler packing limit (K = 12) [2]. The local geometry is the **Cuboc-
tahedron**, a polyhedron with 14 faces (8 triangles, 6 squares). We demonstrate a **Structural
Correspondence** between the topology of these faces and the forces of the Standard Model,
segregated into confining (triangular) and screening (square) sectors.
∗
raghu@idrive.com
2
II. GAUGE TOPOLOGY FROM THE CUBOCTAHEDRON
Narrative: Why does the Standard Model have two distinct types of forces: one that confines
(Strong) and one that screens (Electroweak)? The vacuum lattice has two distinct types of faces:
triangles and squares. A triangle is a ”frustrated” loop. You cannot alternate charges (+/-) around
a triangle without conflict. This topological frustration prevents the field from relaxing, leading
to confinement. A square is a ”relaxed” loop. You can alternate charges perfectly (+/-+/-). This
allows the field to form dipoles and screen itself. We map the frustrated faces to the Strong force
and the relaxed faces to the Electroweak force.
Formalism: In Lattice Gauge Theory [3], gauge fields are defined by holonomies around ele-
mentary plaquettes.
1. **The Strong Sector (8 Triangles):** A triangle is a graph of odd cycle length (L = 3),
making it **non-bipartite**. On a non-bipartite lattice, antiferromagnetic ordering is im-
possible, leading to **Geometric Frustration**. This corresponds to a confining phase. The
minimal unitary group acting on the 3-state frustration basis is SU(3). We identify the 8
triangular faces with the 8 generators of the SU(3) algebra.
2. **The Electroweak Sector (6 Squares):** A square is a graph of even cycle length (L =
4), making it **bipartite**. Bipartite graphs support perfect charge alternation, allowing
for flux screening (Debye shielding). The 6 squares form 3 orthogonal pairs, defining the
fundamental spatial axes. For a lattice supporting chiral fermions, the rotation group is the
double cover SU(2). The scalar trace provides the U(1) generator.
III. THE WEINBERG ANGLE
Narrative: The Weinberg angle θ
W
measures how much the Electroweak force mixes the
”spatial” geometry with the ”internal” geometry. Electromagnetism is a vector force—it propagates
through the 3 dimensions of space. The Weak force is a topological force—it changes the identity
(flavor) of the particle itself. To do this, it must act on the entire ”cluster” that defines the particle
(neighbors + center). The mixing angle is simply the ratio of the spatial dimensions (3) to the
total cluster dimensions (13).
Formalism: We define the **Bulk Coordination Number** N
bulk
as the total number of neigh-
3
bors plus the central site:
N
bulk
= K + 1 = 12 + 1 = 13 (1)
The geometric mixing angle is the projection of the spatial manifold (D = 3) onto the full cluster
topology (N = 13):
sin
2
θ
W
=
D
spatial
N
bulk
=
3
13
≈ 0.23077 (2)
This agrees with the measured value at the Z-pole (sin
2
θ
W
(M
Z
) = 0.23122 [4]) to within 0.2%.
IV. GEOMETRIC ANSATZES FOR COUPLINGS
Narrative: Can we estimate the strength of the forces from geometry? The strength of the
electromagnetic force (α) is determined by how likely a photon is to couple to a vertex. This
probability is diluted by three factors: 1. Geometry: Only half the faces at a vertex are squares
(Electroweak). 2. Algebra: The photon only uses the vector part of the spacetime algebra. 3.
Scale: The interaction must cross the full width of the lattice unit cell. The Strong force is
stronger because it is confining—it doesn’t get diluted by the background screening factors.
Formalism: We propose the following combinatorial relations:
1. **Fine Structure Constant (α
−1
):** We model the inverse coupling as a product of dilution
factors plus a unitary self-interaction term (+1). The coherence length is the lattice diagonal
λ = ⌈K
√
2⌉ = 17. The topological fraction is f
topo
= 1/2. The algebraic fraction (Vector vs.
Clifford dimension) is f
alg
= 4/16 = 1/4 [5].
α
−1
≈
1
f
topo
f
alg
f
λ
+ 1 = (2 × 4 × 17) + 1 = 137 (3)
2. **Strong Coupling (α
s
):** The confining force bypasses the Clifford dilution. The ratio of
couplings is set by the ratio of the full algebra (N = 16) to the effective EM background
(α
−1
≈ 137):
α
s
≈
16
137
≈ 0.117 (4)
This matches α
s
(M
Z
) ≈ 0.118.
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V. RESOLUTION OF THE STRONG CP PROBLEM
Narrative: Why doesn’t the Strong force violate CP symmetry? In standard physics, this
is a mystery (the ”Strong CP Problem”) often solved by inventing a new particle, the Axion.
We propose the lattice solves it. CP violation requires the vacuum to support ”twisted” field
configurations called instantons. In a grid made of squares (bipartite), these twists are easy to
make. But our vacuum is made of triangles (non-bipartite). The built-in frustration of the triangles
makes it energetically impossible to form the global twists needed for CP violation.
Formalism: The Strong CP problem asks why θ
QCD
≈ 0 [6]. Standard hypercubic lattices are
bipartite, allowing for global instanton windings. The SSM lattice is **non-bipartite** due to the
triangular faces. This geometric frustration raises the action of instanton configurations. Unlike
the smooth windings possible on a bipartite grid, topological charge on a frustrated lattice incurs
a significant energy penalty. This **Dynamical Suppression** implies:
Z
θ
∼ e
−S
f rust
→ 0 (5)
This forces θ
eff
→ 0, resolving the problem geometrically without an axion.
VI. CONCLUSION
Narrative: We have shown that the ”arbitrary” structure of the Standard Model—its groups,
its mixing angles, and its lack of CP violation—maps directly to the geometry of a simple K = 12
sphere packing. The forces are not random; they are the inevitable result of filling space with
information.
Formalism: The Cuboctahedral geometry of the K = 12 vacuum establishes a structural
correspondence for the Standard Model. The separation into frustrated (triangular) and screened
(square) sectors maps to the SU (3) and SU(2)×U(1) gauge groups. The Weinberg angle sin
2
θ
W
=
3/13 and the suppression of Strong CP violation follow from the topological constraints of the
lattice.
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[2] T. Hales, Annals of Math. 162, 1065 (2005).
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[4] Particle Data Group, Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
5
[5] P. Lounesto, Clifford Algebras and Spinors (Cambridge Univ. Press, 2001).
[6] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).
[7] Super-Kamiokande Collaboration, Phys. Rev. D 95, 012004 (2017).
