Geometric Wilson Masses for Doubled
Fermions on the FCC and D
4
Lattices
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA raghu@idrive.com
June 2026
Abstract
The bond-direction Dirac operator on the face-centered cubic (FCC) lattice, D
FCC
(k) =
P
12
j=1
(γ·
ˆ
n
j
) e
ik·
ˆ
n
j
summed over the twelve nearest-neighbor unit bonds, is maximally symmetric: its
structure tensor is S
µν
= 4δ
µν
, the long-wavelength limit is an isotropic Dirac cone, and it
anticommutes with γ
5
at finite lattice spacing. It is also doubled. We give the doubler census in
the operator’s own Brillouin zone and show that every doubler sits at a rational momentum that
ordinary even-sided lattices sample exactly, including a one-dimensional set of nodal lines. We
then construct a mass term that removes them. Second-order hopping around the non-bipartite
triangular faces of the cuboctahedral coordination shell generates, through the 120
◦
closure an-
gle and the cancellation of opposite face orientations, a purely scalar correction ∝ ⊮. Summed
over the shell it is the isotropic Wilson term W (k) = r
P
j
1−cos(k·
ˆ
n
j
)
. Because the operator
is anti-Hermitian, this scalar adds in quadrature, |eig|
2
= W
2
+ |V |
2
, and lifts every doubler,
every nodal line included, to a gap that is uniform in [12r, 16r] and independent of lattice size,
while leaving the cone at Γ untouched. The cost is explicit chiral-symmetry breaking, as the
Nielsen–Ninomiya theorem requires. The same scalar-mass construction transfers to the D
4
root lattice in four dimensions (24 bonds, S
µν
= 6δ
µν
, 96 triangular 2-faces). The triangular
faces that generate the mass here are the same faces proposed as color-confinement channels
in the companion FCC-vacuum papers, which ties the regulator to the lattice geometry of that
program. All numerical claims are reproducible from short scripts.
1 Introduction
Under its standard assumptions of locality, translation invariance, Hermiticity, and chiral symmetry,
the Nielsen–Ninomiya theorem [1] forbids a single fermion species: any such Dirac operator on the
torus carries equal numbers of left- and right-handed zeros. On the hypercubic lattice the unwanted
zeros (the doublers) sit at the zone-boundary momenta k
µ
= π/a, which every even-sized grid hits.
The standard cures each pay a price. Wilson fermions [2] give the doublers a momentum-dependent
mass that breaks chiral symmetry; staggered fermions [3] reduce but do not remove the doubling;
domain-wall [4] and overlap [6] fermions restore a modified (Ginsparg–Wilson) chiral symmetry [5]
at a real computational cost. Minimally doubled constructions on non-bipartite geometries [7, 8]
keep two species and exact chiral symmetry while sacrificing some lattice symmetry.
This paper studies the bond-direction Dirac operator on the FCC lattice. The operator is
appealing on symmetry grounds: FCC saturates the three-dimensional kissing number K = 12 [9],
so the twelve bond directions form the most isotropic first shell available in three dimensions, and
the resulting operator has an exactly isotropic continuum cone. It is, however, doubled, and the
1
doublers are not hidden: they sit at rational Brillouin-zone coordinates, and one of the doubler sets
is a one-dimensional family of nodal lines that any even-sided grid samples densely. We make this
census precise in Section 3.
We then take the Wilson route and ask whether the lattice geometry itself supplies the mass
term. It does. Section 4 shows that second-order hopping restricted to the triangular faces of the
cuboctahedral shell generates a purely scalar correction, with the spin-orbit (tensor) part canceling
between the two orientations of each triangular face and the genuine second-neighbor paths silenced
by orthogonality. Summed over the shell the correction is the isotropic Wilson term. Section 5
shows that it lifts every doubler to a size-independent gap and leaves a single massless cone, at
the expected cost of chiral symmetry. Section 6 carries the construction to the D
4
root lattice,
the natural four-dimensional analog. Section 7 records that the mass-generating triangular faces
are the same objects that play the role of confinement channels in the companion lattice-vacuum
papers [11, 12].
What this paper does not claim is any evasion of Nielsen–Ninomiya. The doublers are real,
they are sampled, and they are removed by a chiral-symmetry-breaking term. The contribution is
the geometric origin and full isotropy of that term, and the demonstration that the resulting gap
survives the continuum limit.
2 The bond-direction Dirac operator
2.1 Construction and conventions
The FCC lattice with cubic cell parameter a has twelve nearest-neighbor unit bond directions,
ˆ
n
j
∈
1
√
2
(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)
, j = 1, . . . , 12. (1)
We use Euclidean Dirac matrices: three Hermitian γ
µ
with {γ
µ
, γ
ν
} = 2δ
µν
⊮, and a Hermitian
chirality matrix γ
5
with {γ
5
, γ
µ
} = 0. The bond-direction operator in momentum space is
D
FCC
(k) =
12
X
j=1
(γ ·
ˆ
n
j
) e
ik·
ˆ
n
j
, a = 1. (2)
We work throughout in the rescaled variable q
µ
≡ k
µ
/
√
2, in which the bond phases are cos and
sin of integer combinations q
µ
± q
ν
and the operator is 2π-periodic in each q
µ
.
2.2 Isotropy, continuum limit, and chiral symmetry
The bond set is centrosymmetric and its second moment is isotropic:
S
µν
≡
12
X
j=1
ˆ
n
µ
j
ˆ
n
ν
j
= 4 δ
µν
. (3)
Expanding (2) for small |k| and using
P
j
ˆ
n
j
= 0 together with (3),
D
FCC
(k) −→ i γ
µ
k
ν
S
µν
= 4 i γ · k, (4)
the isotropic massless Dirac operator with Fermi velocity c
F
= 4.
Proposition 1 (Anti-Hermiticity). D
†
FCC
(k) = −D
FCC
(k) for all k.
2
Proof. With Hermitian γ
µ
, (γ ·
ˆ
n)
†
= γ ·
ˆ
n, so D
†
FCC
=
P
j
(γ ·
ˆ
n
j
)e
−ik·
ˆ
n
j
. Relabeling
ˆ
n
j
→ −
ˆ
n
j
(the
bonds come in antipodal pairs) sends γ ·
ˆ
n
j
→ −γ ·
ˆ
n
j
while restoring the phase, giving −D
FCC
.
Anti-Hermiticity is the structural fact that makes a scalar mass work later: the eigenvalues of D
FCC
are purely imaginary, ±i|V (k)| (each doubly degenerate), where V
µ
is defined below.
Proposition 2 (Exact chiral symmetry of the bare operator). {γ
5
, D
FCC
(k)} = 0 for all k and all
a.
Proof. {γ
5
, γ ·
ˆ
n
j
} =
ˆ
n
µ
j
{γ
5
, γ
µ
} = 0 for every j; the scalar phases factor out.
2.3 The V -form
Antipodal symmetry collapses (2) to D
FCC
= i γ
µ
V
µ
(k) with V
µ
=
P
j
ˆ
n
µ
j
sin(k ·
ˆ
n
j
). Direct
expansion factorizes each component:
V
x
= 2
√
2 sin q
x
cos q
y
+ cos q
z
, (5)
and cyclically. Since {γ
µ
} are linearly independent, D
FCC
(k) = 0 if and only if V
x
= V
y
= V
z
= 0.
3 Doubler structure in the Brillouin zone
3.1 The reciprocal cell
The operator depends on k only through q
µ
= k
µ
/
√
2, and as a function of q it is a finite sum of
phases e
iq·m
j
over the twelve integer bond vectors m
j
∈ {(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)}. It
is therefore invariant under q → q + G precisely when e
iG·m
j
= 1 for every bond, i.e. G ·m
j
∈ 2πZ
for all j. Applying this to (1, 1, 0) and (1, −1, 0) gives 2G
x
, 2G
y
∈ 2πZ, so each G
µ
∈ πZ; applying
it to (1, 1, 0) then forces G
x
+ G
y
∈ 2πZ, and cyclically, so the three integers G
µ
/π share a common
parity. The invariance lattice is thus
Λ
∗
= 2πZ
3
∪
π(1, 1, 1) + 2πZ
3
, (6)
the body-centered-cubic lattice, which is the reciprocal of FCC. The displayed 2π-periodic cube
[0, 2π)
3
contains exactly two points of Λ
∗
, the origin and the body center π(1, 1, 1), so the true
Brillouin zone is half the cube and
D
FCC
q + π(1, 1, 1)
= D
FCC
(q). (7)
In particular Γ = (0, 0, 0) and R = π(1, 1, 1) are the same physical point; both are zeros of D
FCC
and both carry the long-wavelength cone. There is one Dirac cone, not two, and this identification
must be respected when counting doublers.
3.2 Complete doubler census
By the V -form, D
FCC
(q) = 0 iff V
x
= V
y
= V
z
= 0, with V
x
= 2
√
2 sin q
x
(cos q
y
+ cos q
z
) and
cyclically. The zero set is exhausted by a case split on how many of the three sines vanish.
Proposition 3 (Zero set of D
FCC
). In the displayed cube [0, 2π)
3
, the zeros of D
FCC
are exactly:
the points Γ = (0, 0, 0) and R = π(1, 1, 1), which are the single physical cone; eight isolated Type-1
points q = (±
π
2
, ±
π
2
, ±
π
2
); and six nodal lines, on each of which two of the q
µ
are fixed to one 0
and one π while the third runs freely. Modulo the identification (7) the eight Type-1 points reduce
to four and the six nodal lines to three. No other zeros exist.
3
Proof. Each component vanishes iff sin q
µ
= 0 or cos q
ν
+ cos q
ρ
= 0 for the complementary pair.
Three sines zero: q
µ
∈ {0, π}, giving the eight cube-corner points—Γ, R, and the six X/M -type
points, which are the endpoints of the nodal lines below. Exactly two sines zero, say sin q
x
= sin q
y
=
0 with sin q
z
= 0: the z-equation then requires cos q
x
+ cos q
y
= 0, so {q
x
, q
y
} = {0, π} and q
z
is
free—a nodal line. The three choices of free axis and the two orderings of {0, π} give six such lines.
Exactly one sine zero, say sin q
x
= 0: the y- and z-equations force cos q
y
= cos q
z
= −cos q
x
= ∓1,
which sets sin q
y
= sin q
z
= 0, contradicting the assumption; this case is empty. No sine zero:
all three cosine sums must vanish, giving cos q
x
= cos q
y
= cos q
z
= 0, i.e. the Type-1 points. A
numerical search over 4 × 10
5
random momenta finds no zero outside these classes.
The two ordering choices on each axis pair are the ones exchanged by (7). The high-symmetry
points X = π(1, 0, 0) and M = π(1, 1, 0) are nodal-line endpoints.
3.3 Sampling
None of these coordinates is irrational. A nodal line is reached when one q
µ
= π, i.e. m/L ·2π with
m = L/2, so every even-sided grid samples the nodal lines, at Θ(L) points per line. Counting in
the displayed cube with the two cone images Γ and R removed, the on-grid zeros number 6L + 2
for L divisible by four: the six lines carry 6L points, less 6 for the X/M endpoints each shared by
two lines, plus the 8 Type-1 points. The Type-1 points need q
µ
= π/2, hence L divisible by four.
Either way the doublers are present on ordinary grids in the continuum limit; the regulator must
remove them, and a “they live at unsampled momenta” argument is false.
4 A geometric Wilson term from second-order hopping
4.1 Selection of paths by triangular closure
A second-order hop
ˆ
n
a
followed by
ˆ
n
b
carries the matrix (γ ·
ˆ
n
b
)(γ ·
ˆ
n
a
) and the phase e
ik·(
ˆ
n
a
+
ˆ
n
b
)
.
A Wilson term must renormalize the nearest-neighbor dispersion, i.e. carry phases cos(k ·
ˆ
n
j
); this
requires the net displacement to be itself a nearest-neighbor bond,
ˆ
n
a
+
ˆ
n
b
=
ˆ
n
c
,
ˆ
n
a
,
ˆ
n
b
,
ˆ
n
c
all unit bonds. (8)
On FCC this is exactly the closure of a triangular face of the cuboctahedral shell; for instance
1
√
2
(1, 1, 0) +
1
√
2
(−1, 0, 1) =
1
√
2
(0, 1, 1) (Figure 1). Closure forces the bond angle:
ˆ
n
a
·
ˆ
n
b
= −
1
2
, the
120
◦
angle of the cuboctahedron’s triangular faces.
4.2 Cancellation of the tensor part
For Hermitian γ,
(γ · u)(γ · v) = (u · v) ⊮ +
X
µ<ν
(u
µ
v
ν
− u
ν
v
µ
) γ
µ
γ
ν
. (9)
The two orientations of a triangular face contribute (γ ·
ˆ
n
b
)(γ ·
ˆ
n
a
) and (γ ·
ˆ
n
a
)(γ ·
ˆ
n
b
). Their scalar
parts are equal; their tensor parts are built on
ˆ
n
b
∧
ˆ
n
a
and
ˆ
n
a
∧
ˆ
n
b
, equal and opposite. Hence
(γ ·
ˆ
n
b
)(γ ·
ˆ
n
a
) + (γ ·
ˆ
n
a
)(γ ·
ˆ
n
b
) = 2(
ˆ
n
a
·
ˆ
n
b
) ⊮ = −⊮. (10)
The set of ordered pairs reaching a given target
ˆ
n
c
is closed under a ↔ b, because
ˆ
n
a
+
ˆ
n
b
is sym-
metric, so the antisymmetric piece cancels pairwise everywhere. Clockwise and counterclockwise
4
(a) FCC shell: cuboctahedron,
K
= 12
ˆ
n
a
ˆ
n
b
ˆ
n
c
ˆ
n
a
+
ˆ
n
b
=
ˆ
n
c
,
ˆ
n
a
·
ˆ
n
b
=
−
1
/
2
(b) Triangular closure
→
scalar mass
Figure 1: (a) The first FCC coordination shell is a cuboctahedron with K = 12 vertices, eight triangular
faces (shaded) and six square faces. (b) A triangular face realizes the closure
ˆ
n
a
+
ˆ
n
b
=
ˆ
n
c
with
ˆ
n
a
·
ˆ
n
b
=
−
1
2
. Summed over both traversal orientations, the second-order hop around the face yields a purely scalar
contribution.
traversal of each triangular face carry opposite spin-orbit torque, so only the scalar (mass) content
survives. Both orientations are present only because the face is an odd cycle, which is the role
played by the non-bipartite triangular faces.
4.3 Second neighbors are silenced; third neighbors renormalize
Two unit bonds satisfy
ˆ
n
a
·
ˆ
n
b
∈ {−1, −
1
2
, 0,
1
2
, 1}, sorting the second-order paths by net displace-
ment. The genuine second-neighbor displacement, of cube-axis type ∝ (2, 0, 0), is reachable only
by orthogonal bond pairs (for example
1
√
2
(1, 1, 0) and
1
√
2
(1, −1, 0)), so
ˆ
n
a
·
ˆ
n
b
= 0 and the scalar
part (10) vanishes. Second-neighbor processes contribute nothing, neither a scalar nor, after the
orientation sum, a tensor. The leading anisotropic scalar one might have feared is absent for a
structural reason: a cube-axis second neighbor on FCC requires a right-angle pair, and a right-
angle pair carries no scalar. Third-neighbor paths (∝ (2, 1, 1)) do carry a nonzero scalar, but the
(2, 1, 1) star obeys
P
t
µ
t
ν
∝ δ
µν
by cubic symmetry, so they add an isotropic constant (absorbed
into the on-site term) plus a subleading isotropic O(k
2
) correction.
4.4 The geometric Wilson term
Collecting the closure paths and the antipodal phases, the second-order operator is proportional to
the identity and carries the cos structure,
D
2
(k) ∝
12
X
j=1
cos(k ·
ˆ
n
j
) ⊮. (11)
The back-and-forth paths
ˆ
n → −
ˆ
n are also second order and contribute the momentum-independent
(on-site) piece, so the combination 1 −cos arises naturally; the geometry fixes the form of the term
5
(an isotropic scalar with this cosine structure) but not its overall magnitude or sign, which we
absorb into a single Wilson parameter r. Writing the result with the conventional normalization,
W (k) = r
12
X
j=1
1 − cos(k ·
ˆ
n
j
)
⊮ = 4r
h
3 − cos q
x
cos q
y
− cos q
y
cos q
z
− cos q
z
cos q
x
i
⊮ (12)
with q
µ
= k
µ
/
√
2 and r > 0 chosen so the doublers are lifted upward. The closed form on the
right follows from cos(q
µ
+ q
ν
) + cos(q
µ
−q
ν
) = 2 cos q
µ
cos q
ν
. Fixing r from the underlying lattice
action, rather than treating it as a free parameter, is left open (Section 5).
Remark 1 (Why a scalar gaps this operator). For an anti-Hermitian D
FCC
with eigenvalues
±i|V |, adding the real scalar W ⊮ produces eigenvalues W ± i|V |, so |eig|
2
= W
2
+ |V |
2
. This
vanishes only when W = 0 and |V | = 0 together; the scalar adds in quadrature and gaps. (Had one
chosen anti-Hermitian γ, D
FCC
would be Hermitian with real eigenvalues ±|V |, and a scalar would
merely shift them, producing a spurious surface of zeros at W = |V |. The Euclidean Hermitian-γ
convention is what turns the geometric scalar into a genuine Wilson mass.)
5 Spectrum with the Wilson term
The full operator D
FCC
(k) + W (k)⊮ has |eig| =
p
W
2
+ |V |
2
. Its behavior at the high-symmetry
points is collected in Table 1, computed directly from the 4 × 4 matrices.
point q/π bare |V | gap at r
Γ (0, 0, 0) 0 0 (physical cone)
R ≡ Γ (1, 1, 1) 0 0 (image of Γ)
T (Type 1) (
1
2
,
1
2
,
1
2
) 0 12r
X (nodal line) (1, 0, 0) 0 16r
M (nodal line) (1, 1, 0) 0 16r
Table 1: Bare and gapped spectra at high-symmetry momenta (q
µ
= k
µ
/
√
2). Every doubler is lifted; Γ
and its image R remain gapless.
Three properties follow.
The physical cone survives. Near Γ, W ≈ 4r|q|
2
while |V | ≈ 4
√
2 |q|, so the scalar enters only
at O(k
2
) and the linear cone is preserved at leading order, with the usual O(a) Wilson shift. At
R ≡ Γ both W and |V | vanish, consistent with (7): one massless mode.
Every doubler is lifted, uniformly. At a Type-1 point W = 12r; on every nodal line W = 16r
(constant along the line, since q
x
= π, q
z
= 0 forces the bracket in (12) to 4 for all q
y
). The doubler
set is therefore raised into the band [12r, 16r]. Figure 2 shows the dispersion along a high-symmetry
path: the bare operator touches zero along the nodal line through X and M and at T , and the
Wilson operator gaps all of it while keeping the cone at Γ and R.
The gap survives the continuum limit. Because the lift is set by W at fixed doubler momenta,
it does not depend on the lattice size L. We verified this by scanning even-sided grids: the bare
operator has exact zeros on the grid at the doubler momenta (gap 0, with the count of on-grid
6
Γ
X M
Γ
T R
≡
Γ
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
|
eig
D
(
k
)
|
(
a
−
1
)
12
r
16
r
Dirac cone at
Γ
survives; every doubler is lifted
bare
|
V
(
k
)
|
(doublers gapless) with Wilson term,
r
= 1
Figure 2: Smallest |eig| along Γ–X–M–Γ–T –R. The bare operator (dashed) is gapless along the nodal line
through X and M and at T . The Wilson term (r = 1, solid) lifts the doublers to 16r (nodal lines) and 12r
(Type 1) while leaving the linear cone at Γ and at its image R.
nodal-line zeros growing linearly with L), while the value of the gapped operator at every one of
those momenta lies in [12r, 16r], with minimum 12r, for all L tested (Figure 3). The minimum
doubler gap is thus 12r, independent of L. This is the property the bare nodal-line zeros denied:
as a → 0 the doublers stay pinned at the cutoff scale rather than returning as light modes.
5 10 15 20 25 30 35
lattice size
L
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
doubler gap
(
a
−
1
)
,
r
= 1
12
r
16
r
FCC: lift is uniform in
[12
r,
16
r
]
, independent of
L
bare doublers (on grid):
0
Wilson-lifted doublers
Figure 3: FCC doubler gap versus lattice size at r = 1. Bare doublers sit on every even grid at gap 0
(triangles). The Wilson-lifted doublers cluster in [12r, 16r], independent of L.
The cost: chiral symmetry. Since W ⊮ commutes with γ
5
, {γ
5
, D
FCC
+W ⊮} = 2W γ
5
= 0: the
Wilson term breaks chiral symmetry at finite a. A single massless cone with all doublers gapped is
the configuration Nielsen–Ninomiya forbids for a chirally symmetric operator, so a chiral-breaking
term is mandatory. The geometric content of this paper is the origin and isotropy of that term,
not an escape from the theorem.
7
Scope. The derivation establishes that the second-order contribution is a pure isotropic scalar,
with the second-neighbor paths silenced. We do not prove that the operator remains free of an
induced tensor (spin-orbit) mass to all orders in the hopping expansion; that is a plausible conse-
quence of the same orientation cancellation but is left as a conjecture rather than a result.
6 Extension to the D
4
root lattice
The four-dimensional analog of FCC is the D
4
root lattice, the densest lattice in four dimensions,
with kissing number K = 24 [10]. Its nearest-neighbor unit bonds are the 24 vertices of the 24-cell,
ˆ
n
j
∈
1
√
2
(±1, ±1, 0, 0) and coordinate permutations
, j = 1, . . . , 24. (13)
With four Hermitian γ
µ
(µ = 0, 1, 2, 3) and the operator D
D
4
(k) =
P
24
j=1
(γ ·
ˆ
n
j
)e
ik·
ˆ
n
j
, every
structural property carries over. The structure tensor is fully isotropic across all four directions,
S
µν
=
24
X
j=1
ˆ
n
µ
j
ˆ
n
ν
j
= 6 δ
µν
, (14)
giving D
D
4
→ 6 i γ ·k in the continuum, Fermi velocity c
F
= 6. The operator is anti-Hermitian and
anticommutes with γ
5
by the same antipodal and Clifford arguments. The V -form and Wilson term
have the same shape as in three dimensions, now summed over four directions (with q
µ
= k
µ
/
√
2):
V
µ
(q) = 2
√
2 sin q
µ
X
ν=µ
cos q
ν
, W
D
4
(q) = 4r
h
6 −
X
i<j
cos q
i
cos q
j
i
⊮. (15)
The 24-cell has 96 triangular 2-faces, with closure
1
√
2
(1, 1, 0, 0) +
1
√
2
(−1, 0, 1, 0) =
1
√
2
(0, 1, 1, 0) and
again
ˆ
n
a
·
ˆ
n
b
= −
1
2
, so the orientation cancellation of Section 4 produces the scalar term (15). The
cube-axis second neighbors of D
4
are likewise reached only by orthogonal 24-cell pairs (
ˆ
n
a
·
ˆ
n
b
= 0),
so they too are silenced. The operator is invariant under q → q + π(1, 1, 1, 1), which identifies Γ
with the corner (π, π, π, π) as in three dimensions, leaving a single physical cone. Representative
gaps are collected in Table 2; scans of L
4
grids at L = 8, 12, 16 (up to 65,536 points) confirm one
gapless mode after the Wilson term, with minimum doubler gap 24r independent of L. We do not
give a D
4
analog of Proposition 3; the four-dimensional zero set is richer, and the claims here are
the structural transfer of the Wilson construction together with the numerical grid checks, not a
full analytic classification.
point q/π bare |V | gap at r
Γ ≡ (π, π, π, π) (0, 0, 0, 0) 0 0 (physical cone)
isolated (
1
2
,
1
2
,
1
2
,
1
2
) 0 24r
nodal (1, 0, 0, 0) 0 24r
nodal (1, 1, 0, 0) 0 32r
Table 2: Representative bare and gapped spectra for D
4
(q
µ
= k
µ
/
√
2). Doubler gaps lie in [24r, 32r], with
minimum 24r.
Selecting one direction as time, which any Lorentzian reading requires, lies outside the bond-
direction construction and is not addressed here.
8
7 Relation to the SSM lattice vacuum
The cuboctahedral shell carries eight non-bipartite triangular faces and six bipartite square faces.
In the companion lattice-vacuum papers, the triangular faces are the channels that forbid single-
color-mode propagation and thereby implement color confinement, while the square faces host the
screening sector [11, 12]. The present construction assigns a second role to the same eight faces:
their closure relation (8), summed over both orientations, generates the fermion mass term. The
same faces proposed as confinement channels in that program are the faces that remove the doublers
here. This is a structural coincidence; we do not import any of the mass-ratio identifications of
those papers, and the results of the present work stand on the lattice-fermion analysis alone.
8 Conclusion
The FCC bond-direction Dirac operator is the most isotropic nearest-neighbor Dirac operator in
three dimensions, with an exact continuum cone and exact chiral symmetry at finite spacing, but
it is doubled: its doublers sit at rational momenta that every even grid samples, including a one-
dimensional set of nodal lines. Second-order hopping around the non-bipartite triangular faces of the
coordination shell generates a purely scalar, fully isotropic Wilson term: the 120
◦
closure angle fixes
the scalar, opposite face orientations cancel the spin-orbit part, and orthogonal second-neighbor
pairs contribute nothing. Because the operator is anti-Hermitian, this scalar gaps in quadrature,
lifting every doubler, nodal lines included, into a size-independent band [12r, 16r] while leaving a
single massless cone. The same geometric Wilson mechanism transfers to the D
4
lattice in four
dimensions. The price is the explicit chiral-symmetry breaking that Nielsen–Ninomiya demands of
any such single-cone regulator.
Two questions remain open. Whether the operator stays free of an induced tensor mass beyond
second order is the immediate technical one. Whether the same triangular-face structure that
supplies confinement channels and the fermion mass also fixes the relative normalization r from
the lattice action, rather than leaving it a free Wilson parameter, is the one that would tie this
regulator most tightly to the underlying geometry.
Data availability
All numerical claims are reproducible from short Python scripts (NumPy only): operator assembly
and the high-symmetry and L-scan tables (d4 doubled fcc wilson verify.py), the second-order
Clifford algebra checks (d4 doubled second order check.py), and the D
4
scans (d4 doubled d4 verify.py).
The scripts run in under a minute on a laptop and are available at https://github.com/raghu91302/
ssmtheory.
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