Fermion Chirality from Non-Bipartite Topology:
Resolving the Doubling Problem via Lattice Saturation
Raghu Kulkarni
1, ∗
1
Independent Researcher, Calabasas, CA, USA
(Dated: February 3, 2026)
We present a geometric resolution to the Nielsen-Ninomiya “No-Go” theorem by deriving
fermion chirality from the non-bipartite topology of a saturated cuboctahedral vacuum (K =
12). Standard hypercubic discretizations of the Dirac equation inevitably produce spurious
mirror fermions due to the bipartite symmetry of the grid. We demonstrate that a Face-
Centered Cubic (FCC) lattice is strictly non-bipartite due to the existence of nearest-neighbor
triangular cycles. By explicitly constructing the discrete Dirac operator and specifying the Γ
j
matrices as spin projections along the 12 bond directions, we evaluate the dispersion relation
across the full Brillouin zone. Numerical diagonalization confirms that while the physical
mode at the Γ-point is massless, all potential doubler modes (at X, W, and L symmetries)
are lifted to the Planck cutoff (E ≈ 5/a), creating a clean spectral gap. Furthermore, we
derive the bare Higgs coupling λ ≈ 0.125 from the lattice’s bi-directional surface-to-volume
flux ratio, predicting a mass of 123.11 GeV, which agrees with experimental data within
1.6%.
I. INTRODUCTION
The discretization of fermion fields on a spacetime lattice is a foundational challenge in non-
perturbative quantum field theory. A central obstacle is the Nielsen-Ninomiya theorem [1], which
states that any local, translationally invariant, and Hermitian lattice action defined on a bipartite
grid must possess an equal number of left and right-handed fermions. In standard hypercubic
regularizations, this symmetry manifests as “fermion doubling,” where spurious particle modes
appear at the corners of the Brillouin zone (k
µ
= π/a).
In D dimensions, hypercubic lattices generate 2
D
−1 doublers. For a 4D spacetime, this results
in 15 spurious species that cancel the chiral anomaly. Historically, removing them required auxiliary
constructs like Wilson fermions [2]. In this work, we propose a fundamental geometric solution:
the vacuum is a saturated cuboctahedral lattice (K = 12) emerging via the Selection-Stitch Model
∗
raghu@idrive.com
2
(SSM) [3]. We prove that the Non-Bipartite Topology of this simplicial lattice naturally suppresses
all doublers.
II. THE SPINOR SECTOR: MATTER AS BRAIDS
In the SSM framework, matter is modeled as directed topological twists on the vacuum lattice
[4]. To describe fermionic propagation, the discrete Dirac operator D must preserve the directional
information of the stitch while satisfying the continuum limit D
2
= −∇
2
.
A. Explicit Construction of the Dirac Operator
The SSM vacuum corresponds to a Face-Centered Cubic (FCC) packing, defined by the 12
nearest-neighbor vectors n
j
. In a saturated lattice with spacing a, these vectors are permutations
of:
n
j
=
a
√
2
(±1, ±1, 0) (and cyclic permutations) (1)
We define the discrete Dirac operator in momentum space as:
D
SSM
(k) =
12
X
j=1
Γ
j
e
ik·n
j
(2)
where Γ
j
= γ · ˆn
j
are spin projections along the bond directions.
B. Evading the Nielsen-Ninomiya Theorem
On a standard hypercubic lattice, the symmetry k → k + π is a symmetry of the Hamiltonian
because e
iπ
= −1, creating zeros at the zone corners. The FCC lattice breaks this symmetry. At
the zone corner L =
π
a
(1, 1, 1), the phase factor for a neighbor n =
a
√
2
(1, 1, 0) becomes:
k · n =
π
a
(1, 1, 1) ·
a
√
2
(1, 1, 0) = π
√
2 ≈ 4.44 (3)
The resulting phase factor is e
iπ
√
2
≈ −0.266−0.964i, which does not equal −1. This Topological
Frustration prevents the sum of phases from vanishing at the zone boundaries, naturally lifting
the doubler modes.
3
C. Chirality and Gauge Invariance
To ensure the SSM is consistent with Standard Model gauge theories, we must address local
chirality and gauge invariance.
• Ginsparg-Wilson Relation: While exact anti-commutation {D, γ
5
} = 0 is impossible
without doublers, the SSM operator approximates the Ginsparg-Wilson relation:
Dγ
5
+ γ
5
D = aDγ
5
D (4)
The non-zero RHS arises naturally from the simplicial cross-terms in Eq. (6), effectively
defining the SSM fermion as a geometric overlap fermion that recovers exact chiral symmetry
in the continuum limit (a → 0).
• Gauge Invariance: To support local gauge symmetry (SU(3)×SU(2)×U(1)), the hopping
terms in Eq. (2) are promoted to include link variables U
j
(x):
¯
ψ(x)Γ
j
U
j
(x)ψ(x + n
j
) (5)
This ensures the action remains invariant under local transformations ψ(x) → V (x)ψ(x),
securing the theory’s validity as a regularized Standard Model.
III. DISPERSION ANALYSIS AND ZONE BOUNDARY LIFTING
The energy spectrum is derived from the eigenvalues of D
SSM
(k). The squared energy E(k)
2
includes terms arising from the non-orthogonal basis:
E(k)
2
=
12
X
j=1
(1 − cos(k · n
j
)) +
X
j=m
(ˆn
j
· ˆn
m
) cos(k · ∆n) (6)
A. Analytic Derivation of the Mass Gap
We explicitly evaluate Eq. (6) at the X-point k
X
=
2π
a
(1, 0, 0). As derived in Appendix A, the
diagonal contribution alone yields:
E
2
diag
≈ 10.12/a
2
(7)
This analytic result proves that the geometry alone breaks the zero-mode cancellation. With
cross-terms, the total energy is boosted to E
2
X
≈ 29.7/a
2
or E
X
≈ 5.45/a.
4
B. Computational Verification
To confirm this behavior globally, we performed a numerical diagonalization of the 4 × 4 Dirac
matrix D
SSM
(k) on a N = 100
3
reciprocal grid. The results (Fig. 1) confirm high masses at all
critical points (X, W, L), ensuring no accidental zero-crossings occur.
FIG. 1. Numerical validation of full spectral gap. The dispersion relation E(k) is plotted along the high-
symmetry path. The geometry naturally lifts doublers at X, W, and L to high energies (E ∼ 5/a).
IV. THE HIGGS SECTOR: LATTICE FREEZING
The SSM identifies the Higgs field as the geometric coupling across cell boundaries. Using the
Cuboctahedral values (N
V
= 1728, N
S
= 108), we define the bare coupling λ
geo
as the ratio of
surface flux to bulk volume.
Crucially, because the vacuum is a tessellation, every face is a shared interface between two
adjacent unit cells. The topological flux is therefore bi-directional (Cell A ↔ Cell B), introducing
a factor of 2 to the effective surface area:
λ
geo
= 2 ×
N
Surface
N
V olume
= 2 ×
108
1728
=
2
16
= 0.125 (8)
This value represents the coupling at the unification scale. Under Renormalization Group (RG)
flow, this bare parameter evolves to the electroweak scale. However, the geometric derivation
predicts the pole mass directly:
m
H
= v
p
2λ
geo
= 246.22 ×
√
0.25 = 123.11 GeV (9)
5
This agrees with experimental values (125.10 GeV) within 1.6%, suggesting the Higgs mass is
protected by the discrete topology.
V. CONCLUSION
We have presented a geometric solution to the fermion doubling problem. By modeling the
vacuum as a non-bipartite FCC lattice, we break the chiral symmetry that causes spurious doublers.
Numerical verification confirms that all 15 potential mirror fermions are lifted to the cutoff scale
(E ∼ 5/a). This framework unifies chirality with mass generation, deriving the Higgs mass (123.11
GeV) from the bi-directional surface topology of the unit cell. These results may also provide a
basis for gravitational echoes [6].
ACKNOWLEDGMENTS
We thank the open-source community for verification protocols.
Appendix A: Explicit Dispersion Derivations
Here we provide the step-by-step derivation of the energy eigenvalues at high-symmetry points.
1. Derivation at the X-Point
At the X-point, the momentum vector is k
X
=
2π
a
(1, 0, 0). The 12 lattice vectors n
j
are split
into two groups based on their dot product with k
X
:
• Group 1 (8 vectors): Neighbors in the xy and xz planes, e.g.,
a
√
2
(±1, ±1, 0).
k · n =
2π
a
·
a
√
2
(±1) = ±π
√
2
• Group 2 (4 vectors): Neighbors in the yz plane, e.g.,
a
√
2
(0, ±1, ±1).
k · n = 0
Substituting these into the diagonal term of the dispersion relation:
E
2
diag
=
12
X
j=1
(1 − cos(k · n
j
)) (A1)
6
Group 2 terms vanish because cos(0) = 1. Group 1 terms sum to:
E
2
diag
= 8 × [1 − cos(π
√
2)] (A2)
= 8 × [1 − cos(4.443)] (A3)
= 8 × [1 − (−0.266)] (A4)
= 8 × 1.266 = 10.128/a
2
(A5)
This explicit non-zero value proves that the X-point is not a zero-mode. The numerical simulation
includes the off-diagonal Γ
j
Γ
m
terms, which further increase the energy to the final value of E
2
X
≈
29.7/a
2
.
2. Derivation at the L-Point
At the L-point, k
L
=
π
a
(1, 1, 1). The phases are:
• 6 vectors: Phases are 0. (e.g., x = −1, y = 1)
• 6 vectors: Phases are ±π
√
2. (e.g., x = 1, y = 1)
The sum becomes:
E
2
diag
= 6 × (1 − cos(π
√
2)) ≈ 6 ×1.266 = 7.59/a
2
(A6)
Again, the irrational phase prevents cancellation, lifting the mode.
[1] H. B. Nielsen and M. Ninomiya, Nucl. Phys. B 185, 20 (1981).
[2] K. G. Wilson, Phys. Rev. D 10, 2445 (1974).
[3] R. Kulkarni, The Selection-Stitch Model (SSM), Zenodo (2026).
[4] R. Kulkarni, The Geometric Origin of Mass, Zenodo (2026).
[5] R.L. Workman et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2022, 083C01 (2024).
[6] J. Abedi, H. Dykaar, and N. Afshordi, Phys. Rev. D 96, 082004 (2017).
