The Tetrahedral Generation Hypothesis
Deriving the 3+1 Flavor Structure and Chiral Asymmetry from the FCC
Lattice
Raghu Kulkarni
∗
Independent Researcher, Calabasas, CA
CEO, IDrive Inc.
February 15, 2026
Abstract
The origin of the three fermion generations and the chiral nature of the weak interaction are
central puzzles of the Standard Model. In this research note, we propose a geometric solution
based on the spectral analysis of the Selection-Stitch Model (SSM) vacuum (K = 12
FCC Lattice).
We demonstrate that the naive Dirac operator on the FCC lattice possesses exactly 4
isolated zero mo des in the interior of the Brillouin Zone, located at the L-points (k =
±π/2, . . . ). These four modes form a reducible permutation representation of the tetrahedral
group S
4
, which naturally decomposes into a singlet (1) and a triplet (3). Furthermore, we
calculate the chirality of these modes and find that the L-point triplet is inherently Left-
Handed (χ = −1), while the fundamental vacuum mode at Γ is Right-Handed (χ = +1).
We show that a momentum-dependent coupling to the vacuum condensate splits the L-
point quartet into a heavy singlet and a light triplet. This mechanism provides a rigorous
geometric origin for the "3 active + 1 sterile" pattern observed in nature and explains why
the three generations participate in the weak interaction as left-handed doublets.
1 Introduction
The Standard Model is characterized by two unexplained integers: the number of generations
(N
g
= 3) and the maximal parity violation of the weak force (Left-Handed fermions). In standard
Grand Unified Theories (GUTs), these are input parameters.
In Lattice Field Theory, the emergence of multiple fermion species is known as the Doubling
Problem. On a simple hypercubic lattice, discretization artifacts create 2
D
= 16 species.
Standard approaches use Wilson terms to remove them entirely.
The Selection-Stitch Model (SSM) [1] takes a different approach. We model the vacuum
as a physical Face-Centered Cubic (FCC) lattice. In this note, we show that the geometry of the
FCC Brillouin Zone naturally filters the doublers into a specific pattern. It preserves exactly four
isolated modes (the L-points) which possess a unique chiral character distinct from the vacuum
state. Under the influence of the vacuum condensate, these modes split into a 3+1 structure,
suggesting that the three generations are geometric necessities of a tetrahedral vacuum.
2 The Spectral Geometry of the FCC Lattice
The First Brillouin Zone (FBZ) of the Face-Centered Cubic (FCC) lattice is a Truncated
Octahedron. This complex polyhedron leads to a fermion spectrum distinct from that of the
simple cubic lattice.
∗
Correspondence: raghu@idrive.com
1
The naive discrete Dirac operator on the lattice is given by:
D(k) =
12
X
j=1
iγ
µ
n
µ
j
sin(k · n
j
) (1)
where n
j
are the 12 nearest-neighbor vectors.
Numerical diagonalization reveals a rich spectral structure consisting of isolated points and
continuous nodal lines.
2.1 The Fundamental Mode (Γ-Point)
At the center of the zone, k = (0, 0, 0), there exists a single massless mode. In the SSM, this
corresponds to the physical vacuum state. It is protected by the condition k ·Φ = 0, ensuring it
remains truly massless relative to the lattice scale.
2.2 The Doublers (L-Points)
In the interior of the zone, away from the boundaries, there are exactly 4 isolated zero modes.
These are located at the centers of the hexagonal faces of the Truncated Octahedron :
L
1
=
π
2
(1, 1, 1)
L
2
=
π
2
(1, −1, −1)
L
3
=
π
2
(−1, 1, −1)
L
4
=
π
2
(−1, −1, 1) (2)
These points correspond to the four distinct body-diagonals of the cubic unit cell.
2.3 Chiral Asymmetry and Nielsen-Ninomiya
A critical feature of the FCC lattice is its natural chiral asymmetry. We compute the chirality
χ of the fermion modes by evaluating the sign of the determinant of the Jacobian matrix J
µν
:
J
µν
=
∂f
µ
∂k
ν
=
12
X
j=1
n
µ
j
n
ν
j
cos(k · n
j
) (3)
where f
µ
(k) is the kinetic vector function.
• At Γ (0,0,0): The Jacobian is proportional to the identity matrix. det(J) > 0. This
corresponds to a single Right-Handed mode (χ = +1).
• At L (±π/2, . . . ): The Jacobian matrix is indefinite (eigenvalues +4, +4, −8), leading to
a negative determinant: det(J) = −128 < 0. Since there are 4 isolated L-points, they
contribute a total chirality of −4. This corresponds to four Left-Handed modes.
The Nielsen-Ninomiya theorem requires the total chirality to vanish (
P
χ = 0).
χ
total
= (+1)
Γ
+ (−4)
L
+ (+3)
boundary
= 0 (4)
The compensating +3 chirality resides on the continuous nodal lines connecting the X and W
points on the zone boundary .
Physical Implication: This geometric structure naturally separates the fermion spectrum
into two chiral sectors. We identify the Left-Handed L-point Triplet with the SU(2)
L
dou-
blets of the three generations, fundamentally opposite in chirality to the vacuum state Γ.
2
3 The Tetrahedral Decomposition (4 → 1 ⊕ 3)
The four L-points form the vertices of a regular tetrahedron inscribed in momentum space. The
symmetry group acting on these momenta is the permutation group of four objects, S
4
.
The fermions located at these points transform as a 4-dimensional reducible representation of
S
4
. Group theory dictates that this representation decomposes into irreducible representations
(irreps) as follows:
4
perm
= 1 ⊕ 3 (5)
where 1 is the totally symmetric singlet representation and 3 is the standard triplet representa-
tion.
3.1 The Basis States
We can explicitly construct the eigenstates:
• The Singlet (ψ
S
):
ψ
S
=
1
2
(ψ
1
+ ψ
2
+ ψ
3
+ ψ
4
) (6)
• The Triplet (ψ
T
):
ψ
T 1
=
1
√
2
(ψ
1
− ψ
2
), ψ
T 2
=
1
√
2
(ψ
3
− ψ
4
), . . . (7)
This decomposition exists mathematically regardless of the physics. However, in a perfectly
isotropic vacuum, all four modes would remain degenerate in mass. To realize the distinct "1
heavy + 3 light" spectrum observed in nature, the vacuum must contain a mechanism to lift this
degeneracy.
4 Physical Origin of the Mass Split
We introduce a geometric coupling dependent on the projection of momentum along the vacuum
field Φ (the "time" direction).
L
mass
= g
¯
ψγ
5
(k · Φ)ψ (8)
4.1 Mechanism: Lattice Strain
This momentum-dependent coupling arises physically from anisotropic hopping amplitudes. A
vacuum expectation value (VEV) Φ creates a coherent strain field in the lattice. This displace-
ment u ∝ Φ modifies the bond lengths directionally. Since the hopping parameter depends
exponentially on bond length, this introduces a k-dependent correction to the effective mass.
The γ
5
factor arises from the chiral nature of the triangular loops in the FCC unit cell (as
derived in [2]).
4.2 Calculated Mass Ratio
We choose the bulk time axis along the primary body diagonal, consistent with the crystallo-
graphic growth direction:
Φ = Φ
0
(1, 1, 1) (9)
We now evaluate the mass term |k · Φ| for the Singlet and Triplet modes:
3
4.2.1 1. The Singlet Mass (m
1
)
The Singlet is dominated by the L
1
mode, which is parallel to the vacuum field:
L
1
=
π
2
(1, 1, 1) (10)
The projection is maximal:
m
1
∝ |L
1
· Φ| =
π
2
(1 · 1 + 1 · 1 + 1 · 1)Φ
0
=
3π
2
Φ
0
(11)
4.2.2 2. The Triplet Mass (m
3
)
The Triplet is composed of modes like L
2
, L
3
, L
4
, which are symmetrically arranged around
the field axis. For example, L
2
=
π
2
(1, −1, −1). The projection is minimal due to geometric
cancellation:
m
3
∝ |L
2
· Φ| =
π
2
(1 · 1 − 1 · 1 − 1 · 1)Φ
0
=
π
2
Φ
0
(12)
4.3 The Bare Ratio
Combining these results yields a fundamental geometric prediction for the bare mass ratio at the
lattice scale:
m
singlet
m
triplet
=
3
1
(13)
5 Discussion
5.1 Identification of States
We identify the Triplet (3) with the three active generations of fermions (e, µ, τ ). They remain
light because their momentum vectors are geometrically misaligned with the vacuum time axis.
Crucially, as shown in Section 2.3, these modes are inherently **Left-Handed**, matching the
chiral nature of the Weak Interaction in the Standard Model.
We identify the Singlet (1) with a heavy, sterile state. In the context of neutrino physics,
this corresponds to the right-handed sterile neutrino. Its heavy mass arises from its perfect
alignment with the vacuum geometry.
5.2 Renormalization and Hierarchy
The derived ratio of 3:1 applies at the lattice cutoff. In the full Wilson-gapped spectrum, all
L-point modes acquire mass ∼ 12/a (Planck scale). However, we postulate that the geometric
Wilson coefficient is partially screened by the VEV coupling, reducing the effective mass to an
intermediate scale. The 3:1 split then determines the relative hierarchy of the surviving modes.
6 Conclusion
The "Generation Problem" is solved by recognizing that the fermions of the Standard Model are
the L-point doublers of an FCC vacuum.
1. The number **3** arises from the triplet representation of the tetrahedral group S
4
.
2. The **Left-Handedness** arises from the negative Jacobian determinant (det J < 0) at
the L-points.
3. The **Mass Hierarchy** arises from the geometric projection onto the vacuum time axis.
This framework transforms the arbitrary parameters of flavor physics into geometric necessities
of a crystal lattice.
4
References
[1] Kulkarni, R. "The Cosserat Vacuum: A Unified Lagrangian for Gauge Fields, Fermions,
and Gravity from Saturated Elasticity." Zenodo (2026).
[2] Kulkarni, R. "Fermion Chirality from Non-Bipartite Topology: Resolving the Doubling
Problem via Lattice Saturation." Zenodo (2026).
[3] Kulkarni, R. "Unified Geometric Lattice Theory (UGLT): Deriving Gauge Couplings, Mass
Spectra, and Gravity from a K = 12 Vacuum." Zenodo (2026).
[4] Kulkarni, R. "Structural Correspondence between the Standard Model and Vacuum Geom-
etry: SU(3) × SU(2) × U(1) from the Cuboctahedron." Zenodo (2026).
5
