Research Note: Geometric Origin of the PMNS Matrix
Deriving Large Neutrino Mixing Angles from the FCC Bulk
Lattice
Raghu Kulkarni
1
1
Independent Researcher, Calabasas, CA
∗
(Dated: February 8, 2026)
Abstract
The Standard Model of particle physics offers no explanation for the striking disparity be-
tween the Quark Mixing Matrix (CKM), which is characterized by small mixing angles
(θ
C
≈ 13
◦
), and the Neutrino Mixing Matrix (PMNS), which exhibits large, near-maximal
angles. We present a rigorous derivation of these values within the Unified Geometric
Lattice Theory (UGLT) [1]. By identifying neutrinos as unanchored bulk defects in a
Face-Centered Cubic (FCC) vacuum lattice, we demonstrate that their mixing angles are
determined by the fundamental symmetry vectors of the Cuboctahedron unit cell. We ana-
lytically derive the Atmospheric angle θ
23
= 45
◦
from the diagonal symmetry of the lattice
faces, and the Solar angle sin
2
θ
12
= 1/3 (θ
12
≈ 35.3
◦
) from the projection of the bulk Body
Diagonal onto the interaction Face Diagonal. We resolve the tension between this geomet-
ric prediction (0.333) and the global fit (0.307) by applying a unitarity constraint with the
measured reactor angle θ
13
as an input, which corrects the solar angle to sin
2
θ
12
≈ 0.318.
This framework resolves the ”Flavor Puzzle” as a consequence of topological dimensionality:
quarks are confined to 2D surface geometries, while neutrinos traverse the 3D bulk.
I. INTRODUCTION: THE FLAVOR PUZZLE
In the Standard Model, the mixing between mass and flavor eigenstates for quarks and
leptons is described by unitary matrices: the CKM matrix for quarks and the PMNS matrix
for neutrinos. Despite their mathematical similarity, their physical parameters are drastically
different:
• Quarks (CKM): Small mixing angles. The dominant Cabibbo angle is θ
C
≈ 13
◦
.
• Neutrinos (PMNS): Large mixing angles. The Atmospheric angle is near-maximal
(≈ 49
◦
), and the Solar angle is large (≈ 34
◦
).
Standard approaches often utilize discrete non-Abelian symmetries (e.g., A
4
, S
4
) to enforce these
values, specifically the ”Tri-bimaximal” mixing pattern (θ
23
= 45
◦
, sin
2
θ
12
= 1/3). However,
these symmetries are typically imposed ad hoc without a dynamical origin.
In this note, we derive the PMNS parameters naturally from the Unified Geometric Lat-
tice Theory (UGLT) [1], specifically the geometry of the K = 12 Face-Centered Cubic (FCC)
∗
raghu@idrive.com
2
vacuum lattice. We propose that the disparity between quark and neutrino mixing is topologi-
cal: Charged fermions are ”anchored” surface defects governed by tangential projections (3/13),
while neutrinos are ”unanchored” bulk defects governed by 3D volumetric projections (1/3).
II. THE GEOMETRY OF THE VACUUM
We posit that the vacuum is a discrete graph with the topology of a Cuboctahedron, the
Voronoi cell of the FCC lattice. This structure dictates the allowed propagation modes for
fermions.
A. Surface vs. Bulk Anchoring
1. Surface Anchoring (Quarks/Charged Leptons): Particles carrying gauge charges
(Color/Electric) couple to the link variables on the surface of the unit cell. Their effective
geometry is 2D. As derived in previous work, their mixing angle is determined by the ratio
of the radial to tangential components of the hexagonal face, yielding tan θ
C
≈ 3/13 [3].
2. Bulk Propagation (Neutrinos): Neutrinos carry neither electric nor color charge, and
thus do not couple to the surface gauge links. Instead, they propagate as rotational defects
(microrotations) through the **bulk** of the lattice [2]. Their mixing is governed by the
angle between the bulk propagation vector (Mass Eigenstate) and the surface interaction
vector (Flavor Eigenstate).
III. DERIVATION OF THE MIXING ANGLES
We define the mixing angle θ
ij
as the geometric angle between the Mass Basis (
M, propa-
gation direction) and the Flavor Basis (
F , interaction plane normal).
A. Defining the Basis Vectors
In the FCC lattice, the fundamental vectors are:
• Flavor Basis (
F ): The weak interaction is mediated by W
±
bosons, which live on the
square faces of the unit cell . The normal vector to a square face (or its diagonal in the
dual basis) defines the flavor axis.
F = (1, 1, 0) (Face Diagonal) (1)
• Mass Basis (
M): The mass eigenstate corresponds to the freest, longest coherent path
through the bulk lattice. In an FCC crystal, the longest inter-nodal distance that main-
tains symmetry is the **Body Diagonal**.
M = (1, 1, 1) (Body Diagonal) (2)
3
B. The Atmospheric Angle (θ
23
)
The Atmospheric angle θ
23
represents the mixing between ν
µ
and ν
τ
. Geometrically, this
corresponds to the orientation of the flavor basis relative to the principal Cartesian axes of the
lattice. The flavor vector
F = (1, 1, 0) bisects the x and y axes. The angle between the principal
axis (1, 0, 0) and the flavor vector is:
cos θ
23
=
(1, 0, 0) · (1, 1, 0)
1 ·
√
2
=
1
√
2
(3)
θ
23
= 45
◦
(4)
Prediction: UGLT predicts exact maximal mixing (θ
23
= 45
◦
). While current global fits prefer
θ
23
≈ 49
◦
, the maximal value is consistent within experimental systematics and remains a key
falsifiable prediction for future long-baseline experiments like DUNE.
C. The Solar Angle (θ
12
)
The Solar angle θ
12
is the projection of the Mass eigenstate onto the Flavor eigenstate.
It represents the probability amplitude of a bulk neutrino being detected in a specific flavor
interaction plane.
We calculate the angle α between the Mass vector
M = (1, 1, 1) and the Flavor vector
F = (1, 1, 0):
cos α =
M ·
F
|
M||
F |
=
(1)(1) + (1)(1) + (1)(0)
√
3 ·
√
2
=
2
√
6
=
r
2
3
(5)
The mixing probability is the sine-squared of this projection angle:
sin
2
θ
12
= 1 − cos
2
α = 1 −
2
3
=
1
3
(6)
sin
2
θ
12
≈ 0.333 =⇒ θ
12
≈ 35.26
◦
(7)
Observation: This geometric value (0.333) reproduces the Tri-bimaximal prediction exactly.
However, it is in tension with the current global fit value of 0.307 ± 0.013 (≈ 2σ tension).
IV. REACTOR ANGLE CORRECTION VIA UNITARITY
The tension in θ
12
and the non-zero value of the reactor angle θ
13
(sin
2
θ
13
≈ 0.022) sug-
gest that the Tri-bimaximal geometry is perturbed. In UGLT, we attribute this to the lattice
**packing defect** (the geometric gap angle required to close the unit cell).
Rather than applying an arbitrary perturbation, we posit that the dominant flavor component
|U
e1
|
2
is **topologically protected** at its geometric value of 2/3. This stability arises from the
hexagonal face symmetry of the unit cell. Under this constraint, unitarity (|U
e1
|
2
+ |U
e2
|
2
+
|U
e3
|
2
= 1) forces the solar component |U
e2
|
2
to absorb the probability ”stolen” by the reactor
angle.
4
|U
e2
|
2
= 1 − |U
e1
|
2
− |U
e3
|
2
= 1 −
2
3
− sin
2
θ
13
=
1
3
− sin
2
θ
13
(8)
Converting this to the physical mixing angle sin
2
θ
12
:
sin
2
θ
12
=
|U
e2
|
2
cos
2
θ
13
=
1 − 3 sin
2
θ
13
3(1 − sin
2
θ
13
)
(9)
Using the experimental input sin
2
θ
13
≈ 0.022:
sin
2
θ
12
≈
1 − 0.066
3(0.978)
=
0.934
2.934
≈ 0.318 (10)
Result: The corrected value 0.318 is in agreement with the global fit (0.307 ±0.013) within
1σ. This demonstrates that the geometric prediction of 1/3, when constrained by unitarity and
the measured reactor angle, naturally yields the observed solar mixing parameter.
V. CONCLUSION
We have derived the structure of the PMNS Matrix from the first principles of lattice geom-
etry:
1. Atmospheric: θ
23
= 45
◦
(Maximal Mixing from Face Symmetry).
2. Solar: sin
2
θ
12
≈ 0.318 (Body-to-Face Projection with Unitarity Correction).
3. Reactor: θ
13
is taken as an experimental input associated with lattice packing defects.
This derivation explains the ”Flavor Puzzle” as a consequence of topological dimensionality:
Quarks are anchored to 2D surface defects (small mixing), while Neutrinos propagate through
3D bulk defects (large mixing).
[1] R. Kulkarni, ”Unified Geometric Lattice Theory (UGLT): Deriving Gauge Couplings, Mass Spectra,
and Gravity from a K=12 Vacuum,” Zenodo, doi:10.5281/zenodo.18520623 (2026).
[2] R. Kulkarni, ”Micropolar Neutrinos: Deriving Mass Suppression and Oscillation from the Cosserat
Lagrangian of the Vacuum,” Zenodo, doi:10.5281/zenodo.18503146 (2026).
[3] R. Kulkarni, ”Structural Correspondence between the Standard Model and Vacuum Geometry:
SU(3)xSU(2)xU(1) from the Cuboctahedron,” Zenodo, doi:10.5281/zenodo.18503168 (2026).
