Fermion Chirality from Non-Bipartite Topology:
Geometric Doubler Lifting on the FCC and D4 Lattices
via Holographic U(1)/Z
2
Phase Projection
Raghu Kulkarni
1
1
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA, raghu@idrive.com
May 2026
Abstract
What if the lattice itself protects the chiral fermion? Take the bond-direction Dirac operator on the
face-centered cubic lattice,
D
FCC
(k) =
∑
12
j=1
(γ · ˆn
j
)e
ik·ˆn
j
, summed over all twelve nearest-neighbor
unit bond directions. It anticommutes with
γ
5
exactly at finite lattice spacing. No Wilson term,
no Ginsparg-Wilson modification, no overlap construction. Chiral symmetry comes for free, as an
algebraic identity of the bond geometry.
The non-
Γ
zeros required by Nielsen-Ninomiya sit at irrational FCC fractional coordinates
f
i
=
±1/(2
√
2)
. That is the main analytical result—the Irrational Doubler Theorem—and its consequence
is sharp: no finite lattice of integer side length
L
ever samples them. We prove it via the V-form
factorization, confirm it with a
32
3
Brillouin-zone scan, and identify two doubler classes that arise in
the proof. Type-1: isolated
Z
2
zeros, all bond phases in
{+1,−1}
. Type-2: flat-band U(1) zeros with
generically complex phases. Both sit at irrational coordinates. Holographic projection from the SSM
stabilizer-code vacuum additionally excludes Type-1 from the physical spectrum on the grounds that
real bond phases transport no U(1) charge.
The construction extends without modification to the D4 root lattice in four dimensions. Twenty-
four nearest-neighbor bonds. Four anti-Hermitian gamma matrices. Structure tensor
S
µν
= 6δ
µν
(full SO(4) Euclidean isotropy). Fermi velocity
c
F
= 6
. The V-form gains one cosine factor:
V
µ
(k) = 2
√
2 sin(k
µ
/
√
2)
∑
ν=µ
cos(k
ν
/
√
2)
. Chiral symmetry survives. So does the irrational-
doubler protection. Numerical scans at
L = 8,12,16
(up to 65,536 grid points in 4D) find
Γ
as the
unique zero on every integer-
L
grid. The framework now lives in physical spacetime dimension, not
just three spatial dimensions.
1 Introduction
The Nielsen-Ninomiya theorem [
1
] obstructs single-species lattice fermions. Any local, Hermitian,
translationally invariant Dirac operator on the continuous torus
T
d
must carry equal numbers of left-
and right-handed zeros. On the hypercubic lattice those doublers sit at rational zone-boundary momenta
k
µ
= π/a
, directly accessible to any even-sized simulation grid. Standard remedies all impose a cost:
Wilson fermions [
2
] lift doublers but break chiral symmetry; staggered fermions [
3
] reduce but do not
eliminate doubling; domain-wall [
4
] and overlap [
6
] fermions restore a modified chiral symmetry [
5
] at
substantial computational cost.
We take a different route. Instead of regularizing the doublers away, this paper studies a bond-direction
Dirac operator on the non-bipartite FCC lattice and shows that its doublers sit at irrational fractional
Brillouin-zone coordinates. Theorem 1, the Irrational Doubler Theorem, makes the statement precise:
every non-
Γ
zero of
D
FCC
has at least one component of the form
±1/(2
√
2)
in the FCC fractional basis.
Since
√
2
is irrational, no rational fraction
p/L
ever equals
1/(2
√
2)
, so no integer-
L
grid samples a
doubler. The argument is elementary; the protection it provides is exact.
1
The physical content arrives through the SSM stabilizer-code framework. The FCC vacuum, treated
as a quantum error-correcting code [
10
], supports physical electrons as propagating defects that carry
U(1) electric charge. Propagation requires complex bond phases for Berry-phase accumulation. Type-1
doublers, with all bond phases real, transport nothing. They are static codespace configurations of the
QEC vacuum—present in the spectrum, absent from the physical sector. Nielsen-Ninomiya is respected
on the continuous torus; the physical sector is single-fermion by projection.
Section 7 extends every structural result to D4 in four dimensions. The mechanism transfers cleanly.
Exact chiral symmetry, V-form factorization, Type-1/Type-2 phase classification, irrational doubler
positioning, single physical zero at
Γ
. Two features distinguish 4D from 3D: the structure tensor is now
fully SO(4)-isotropic, and the Fermi velocity rises from 4 to 6. The 4D extension brings the construction
into the dimension where lattice gauge theory actually lives.
Scope. The 3D results are for the spatial FCC lattice; the 4D results are for the Euclidean D4 lattice.
Connecting D4 to Lorentzian signature would require a separate axiom that selects one direction as time,
which lies outside the scope of the bond-direction operator presented here. We do not claim a spectral
gap in the strict thermodynamic limit
L → ∞
: U(1) mode energies can be made small by approaching the
Type-2 flat-band surfaces, although never exactly zero on any lattice with finite integer
L
. The mechanism
that protects the physical spectrum is the holographic projection, not a literal absence of zeros in the
Brillouin zone.
2 The FCC Bond-Direction Dirac Operator
2.1 Construction
The FCC lattice with cubic cell parameter a has twelve nearest-neighbor unit bond directions,
ˆn
j
∈
1
√
2
{(±1,±1,0), (±1,0, ±1), (0,±1,±1)}. (1)
The bond-direction Dirac operator in momentum space is
D
FCC
(k) =
12
∑
j=1
(γ · ˆn
j
)e
ik·ˆn
j
, (2)
with
γ
µ
the anti-Hermitian spatial Dirac matrices (
γ
†
µ
= −γ
µ
) and
a = 1
throughout. The bond vectors
satisfy
S
µν
=
∑
j
ˆn
µ
j
ˆn
ν
j
= 4δ
µν
, exact spatial isotropy with Fermi velocity
c
F
= 4
. FCC saturates the 3D
kissing number
K = 12
[
9
], the maximum number of non-overlapping unit spheres that can simultaneously
touch a central sphere. That is what makes
D
FCC
the unique maximally symmetric bond-direction operator
in three dimensions.
Continuum limit. Expand
e
ik·ˆn
j
≈1+ik· ˆn
j
for small
|k|
:
D
FCC
→iγ
µ
k
ν
∑
j
ˆn
µ
j
ˆn
ν
j
= 4iγ ·k
. The standard
massless Dirac equation is recovered, with Fermi velocity 4. The non-bipartite FCC geometry does not
break Lorentz symmetry at long wavelengths.
Relation to the standard FCC tight-binding operator. The standard FCC operator uses physical bond-
position vectors
(a/2)(±1,±1,0)
, of length
a/
√
2
, in the exponent:
D
std
(k) =
∑
j
(γ · ˆn
j
)e
ik·(a/2)(±1,±1,0)
.
D
std
has zeros at the L and X high-symmetry points of the FCC BZ. The bond phase
e
i(π/a)·(a/2)·2
= e
iπ
=
−1
at L makes the vector sum vanish, consistent with Nielsen-Ninomiya. Our
D
FCC
uses unit-length
directions in the exponent and is related to
D
std
by a momentum rescaling
k → k
√
2
, which shifts the
doublers from the rational L, X positions to the irrational positions characterized in Section 4.
2
2.2 Hermiticity and Exact Chiral Symmetry
Proposition 1 (Hermiticity). D
†
FCC
(k) = D
FCC
(k) for all k.
Proof.
The twelve bond vectors form six antipodal pairs,
ˆn
j
′
= −ˆn
j
. Take the adjoint:
D
†
=
∑
j
(−γ ·
ˆn
j
)e
−ik·ˆn
j
. Relabel
j → j
′
and the sign cancels, giving
D
†
= D
. Verified numerically:
∥D −D
†
∥ <
10
−14
.
Proposition 2 (Exact chiral symmetry). {γ
5
,D
FCC
(k)} = 0 for all k and all a.
Proof. {γ
5
,γ
µ
} = 0
for all spatial
µ
, so
{γ
5
,γ · ˆn
j
} = ˆn
µ
j
{γ
5
,γ
µ
} = 0
for every
j
. Hence
{γ
5
,D
FCC
} =
∑
j
0 ·e
ik·ˆn
j
= 0.
This distinguishes
D
FCC
from Wilson fermions, which add an identity-valued
O(a)
term that com-
mutes with
γ
5
and so breaks chiral symmetry. Here the entire operator anticommutes with
γ
5
at every
lattice spacing.
3 Zero Structure on the Continuous Torus
3.1 The V-form
Antipodal symmetry rewrites the operator as
D
FCC
(k) = iγ
µ
V
µ
(k)
, with
V
µ
(k) =
∑
j
ˆn
µ
j
sin(k · ˆn
j
)
. Direct
expansion factors this further:
V
x
(k) = 2
√
2 sin(k
x
/
√
2)
cos(k
y
/
√
2) + cos(k
z
/
√
2)
, (3)
and cyclically for V
y
and V
z
. D
FCC
= 0 if and only if V
µ
= 0 for all µ.
3.2 The Γ-Point
At
k = 0
every phase equals
+1
and the antipodal cancellation gives
D
FCC
(0) = γ ·
∑
j
ˆn
j
= 0
. Long-
wavelength expansion around the same point yields D
FCC
≈ 4iγ ·k, the standard massless Dirac cone.
3.3 Doubler Zeros and Their FCC Fractional Coordinates
Other zeros required by Nielsen-Ninomiya appear on the continuous torus T
3
in two patterns.
Type-1 (isolated).
k
d
=
π
√
2
(±1,±1,±1)
. At these points
k
d
· ˆn
j
∈ {0, ±π}
for every bond, so
e
ik
d
·ˆn
j
∈
{+1,−1}
. In FCC fractional coordinates
k = f
1
b
1
+ f
2
b
2
+ f
3
b
3
with
b
i
the reciprocal basis, the fractional
components come out to f
i
= ±1/(2
√
2) ≈ ±0.3536. Irrational, by inspection.
Type-2 (flat surfaces). Whole planes such as
k
y
= −π
√
2
,
k
z
= 0
, varying
k
x
. Equation
(3)
forces
V
x
= V
y
= V
z
= 0
on the entire plane:
sin(k
y
/
√
2) = sin(−π) = 0
kills
V
y
;
cos(k
y
/
√
2) + cos(k
z
/
√
2) =
cos(−π)+ cos(0) = 0 kills V
x
and V
z
. The relevant component has |f
i
| = 1/(2
√
2), again irrational.
Both types lie at irrational FCC fractional coordinates and are never sampled by any finite periodic
FCC lattice of integer size
L
. We verified this for
L = 1
to
500
. Nielsen-Ninomiya is satisfied globally:
Γ
carries chiral index
+1
, the doublers carry compensating indices, and all topological charges sum to zero
on T
3
.
3
4 The Irrational Doubler Theorem
Definition 1 (Phase character).
k
is a
Z
2
mode if
e
ik·ˆn
j
∈{+1,−1}
for all twelve bonds. It is a U(1) mode
if at least one bond phase
e
ik·ˆn
j
/∈ R
. Equivalently:
Z
2
modes have
sin(k · ˆn
j
) = 0
for all
j
; U(1) modes
have sin(k · ˆn
j
) = 0 for some j.
Theorem 1 (Irrational Doubler Theorem, FCC). Every non-
Γ
zero of
D
FCC
has at least one irrational
FCC fractional coordinate
f
i
= ±1/(2
√
2)
. No non-
Γ
zero of
D
FCC
is sampled by any finite FCC lattice
of integer size L.
Proof.
V-form structure.
D
FCC
= 0
iff
V
µ
= 0
for
µ = x, y, z
simultaneously. From
(3)
,
V
x
= 0
requires
either
sin(k
x
/
√
2) = 0
or
cos(k
y
/
√
2) + cos(k
z
/
√
2) = 0
(and cyclically for
V
y
,V
z
). In either case, at least
one component satisfies k
i
/
√
2 = n
i
π for a nonzero integer n
i
, giving k
i
= n
i
π
√
2.
Irrational fractional coordinates. Inverting
(1)
to convert
k
into FCC fractional coordinates gives
f
j
= ±n
j
/(2
√
2)
for the component(s) with
k
j
= n
j
π
√
2
. Since
√
2
is irrational,
f
j
= n
j
/(2
√
2) = p/L
for any integers p, L with n
j
= 0. Every non-Γ zero lies at an irrational FCC fractional coordinate.
Classification of zero types. Type-1 (isolated): all three sines vanish,
f
1
= f
2
= f
3
= ±1/(2
√
2)
, all
bond phases in
{+1,−1}
(
Z
2
). Type-2 (flat surfaces): one sine and one cosine-sum cancel simultaneously,
with one fractional coordinate at
±1/(2
√
2)
and the others free along the surface. On these surfaces the
bond phases are generically complex (U(1)), but the irrational coordinate persists for all points on the
surface, guaranteeing they are never sampled by any integer-L grid.
Numerical confirmation. A dense
32
3
scan of rational FCC BZ momenta (Section 6) finds no non-
Γ
zero.
Within the Z
2
class, two subclasses are physically distinct:
Remark 1 (Zero classification). • Uniform Z
2
(k = 0, all phases = +1): the physical Dirac cone.
•
Staggered
Z
2
(Type-1 isolated, all phases
∈ {+1,−1}
): at
f
i
= ±1/(2
√
2)
, excluded by holo-
graphic projection and by irrational coordinates.
•
Type-2 flat-band zeros (generically U(1) complex phases): at
f
i
= ±1/(2
√
2)
in at least one
component, never sampled by finite integer-L grids.
5 Holographic Projection and the Physical Spectrum
5.1 Physical Motivation in the SSM Framework
In the SSM stabilizer-code vacuum [
10
], the FCC lattice carries a quantum error-correcting code. Physical
electrons are propagating topological defects with U(1) electric charge. Charge transport requires Berry-
phase accumulation along the bond network, which requires genuinely complex U(1) hopping amplitudes.
A mode with all bond phases in
{+1,−1}
is something different. The hopping amplitudes are
real-valued, no net current flows, no U(1) charge is transported. Such modes correspond to standing-wave
interference patterns whose phase cancellations are exact—static configurations of the QEC vacuum, not
propagating particles. They are present in the spectrum of
D
FCC
; they are absent from the physical sector.
Definition 2 (Holographic projection). The physical (U(1)) sector of D
FCC
consists of:
1. all U(1) modes (e
ik·ˆn
j
/∈ R for at least one bond), and
2. the unique uniform Z
2
mode k = 0 (the Dirac vacuum).
Non-uniform Z
2
(staggered) modes are excluded as non-propagating.
4
5.2 Consequence: Single Physical Dirac Mode
Theorem 2 (Single zero on finite FCC lattices). On any finite FCC lattice with integer size
L
,
D
FCC
has
exactly one zero: the Γ-point. After holographic projection, this is the unique physical massless mode.
Proof.
By Theorem 1, every non-
Γ
zero has
f
i
= ±1/(2
√
2)
for at least one component. Irrationality
of
1/(2
√
2)
forbids
f
i
= p/L
for any integers
p
,
L
. No non-
Γ
zero falls on the
L
-site grid. Holographic
projection additionally excludes staggered
Z
2
modes from the physical spectrum on physical grounds.
5.3 Dual Protection: Irrational BZ Coordinates
Two independent mechanisms exclude the doublers from physical observability. Kinematically: irrational
FCC coordinates are never sampled by any integer-
L
lattice (Theorem 1). Physically: holographic
projection removes Type-1
Z
2
modes, which carry no U(1) charge. Type-2 flat-band zeros are U(1)
modes and survive the holographic projection; for them, only the irrational-coordinate argument applies.
Together the two arguments cover both doubler classes.
5.4 Comparison with Existing Approaches
Wilson fermions [
2
]: add
r
∑
µ
(1 −cos k
µ
a) ·⊮
, breaking chiral symmetry through an identity-valued
mass-like term. D
FCC
has no such term.
Overlap fermions [
6
]: realize the Ginsparg-Wilson relation
{γ
5
,D} = Dγ
5
D
at high computational
cost. D
FCC
achieves the stronger {γ
5
,D} = 0 through geometry alone.
Creutz / Kimura-Misumi [
7
,
8
]: exploit non-bipartite geometry in 2D and 4D. The present 3D
construction additionally supplies the Z
2
/U(1) phase mechanism.
6 Numerical Results
6.1 32
3
BZ Scan with Phase Classification
We scanned
N = 32
in each FCC BZ direction—
32
3
= 32,768 k
-points, fractional coordinates
f
i
∈
[−0.5,0.5)—and classified each by phase character:
•
U(1) modes: 32,767 sampled points, all with
E > 0
. Minimum
E
across the full scan is
0.0013/a
,
arising from grid proximity to the Type-2 flat-band surfaces at irrational BZ positions (not exact
zeros). Minimum E on the standard high-symmetry path is E
K
= 0.01372/a at the K-point.
• Z
2
modes: 1 sampled point—the Γ-point, all phases = +1, E = 0.
•
Non-
Γ
doubler zeros: all at irrational BZ coordinates (
f
i
= ±1/(2
√
2)
in at least one component),
absent from the integer-L grid.
This confirms Theorem 1 numerically: no non-Γ zero is sampled by the rational grid.
6.2 Zone-Boundary Spectrum
Table 1 gives E(k) = min|eig(D
FCC
)| at the standard FCC BZ high-symmetry points.
The K-point value
0.01372/a
is a genuine local minimum of the dispersion. Displacing by
ε = 0.05/a
along each of the six principal reciprocal directions raises the gap to
≈ 0.114/a
, which rules out a saddle.
Because K has fully rational FCC fractional coordinates (
3
8
,
3
8
,
3
4
), this gap is independent of lattice size:
any L divisible by 8 samples K exactly, with the same gap 0.01372/a.
The global minimum grid gap behaves differently. By Dirichlet’s approximation theorem, rational
fractions
p/L
approximate
1/(2
√
2)
to within
O(1/L)
, so grid points can approach the Type-2 flat-band
zero surfaces arbitrarily closely as
L → ∞
. The minimum grid gap therefore approaches zero, although no
5
Table 1: Minimum singular value at FCC BZ high-symmetry points. All zone-boundary modes are U(1)
and strictly gapped. The K-point value 0.01372/a is the minimum on the standard high-symmetry path.
Point Fractional coordinates ( f
1
, f
2
, f
3
) E Type
Γ (0,0,0) 0 Z
2
uniform (physical)
X (0,
1
2
,
1
2
) 5.45/a U(1), gapped
W (
1
4
,
1
2
,
3
4
) 1.97/a U(1), gapped
L (
1
2
,
1
2
,
1
2
) 4.72/a U(1), gapped
K (
3
8
,
3
8
,
3
4
) 0.01372/a U(1), path minimum
individual non-
Γ
grid point ever reaches it exactly at finite
L
. Numerically:
E
min
(L = 48) ≈ 3 ×10
−7
/a
,
E
min
(L = 32) ≈ 1.3 ×10
−3
/a
, with the value oscillating irregularly as
L
varies—governed by the quality
of rational approximation to 1/(2
√
2) at each L.
6.3 Gauge Field Robustness
A U(1) background field
A
µ
inserts link phases
e
iA·ˆn
j
into the operator. For all tested strengths
|A|≤2.0/a
,
the spectral gap at zone-boundary points persists, and no new zero-crossings appear. The
Γ
-point shifts in
momentum, as expected from gauge response, but the physical spectrum structure is preserved.
7 Extension to the D4 Root Lattice
What changes when the lattice gains a fourth dimension? The natural 4D analog of FCC is D4, the densest
4D lattice with kissing number
K = 24
[
11
]. This section shows that every structural property of
D
FCC
transfers to D4 without modification, and that two new features emerge: full SO(4) Euclidean isotropy
and Fermi velocity c
F
= 6.
7.1 Construction
The D4 nearest-neighbor unit bond vectors are the 24 unit-length permutations with exactly two nonzero
entries ±1:
ˆn
j
∈
1
√
2
(±1,±1,0, 0) and coordinate permutations
, |ˆn
j
| = 1, j = 1,...,24. (4)
Six coordinate pairs
(µ,ν)
with
µ < ν
, four sign combinations each, total
4
2
·4 = 24
. These are the 24
vertices of the 24-cell—the unique regular self-dual 4-polytope, with no Euclidean analog in any other
dimension.
The bond-direction Dirac operator in 4D is
D
D4
(k) =
24
∑
j=1
(γ · ˆn
j
)e
ik·ˆn
j
, (5)
with four anti-Hermitian gamma matrices
γ
0
,γ
1
,γ
2
,γ
3
satisfying
{γ
µ
,γ
ν
} = −2δ
µν
⊮
and a Hermitian
chirality matrix
γ
5
with
{γ
5
,γ
µ
} = 0
for all
µ ∈ {0,1,2,3}
. The three spatial gammas
γ
1
,γ
2
,γ
3
are
inherited unchanged from the FCC construction. The fourth gamma is
γ
0
=
0 i⊮
2
i⊮
2
0
, (6)
anti-Hermitian, with γ
2
0
= −⊮
4
and {γ
0
,γ
i
} = 0 for spatial i.
6
Structure tensor and 4D isotropy. Direct enumeration over the 24 unit vectors gives
S
µν
=
24
∑
j=1
ˆn
µ
j
ˆn
ν
j
= 6 δ
µν
. (7)
Trace:
trS = 24 ·|ˆn|
2
= 24
. Isotropy across all four indices follows from the
S
4
symmetry of the 24
vectors under coordinate permutations. No direction is privileged. The Lorentzian interpretation, with
one direction selected as time, requires a separate axiom external to the bond-direction operator and lies
outside the scope of the present analysis.
Continuum limit.
D
D4
→ iγ
µ
k
ν
S
µν
= 6iγ ·k
for small
|k|
. Fermi velocity
c
F
= 6
, compared with
4
for
FCC.
7.2 Hermiticity and Exact Chiral Symmetry
Proposition 3. D
†
D4
(k) = D
D4
(k) and {γ
5
,D
D4
(k)} = 0 for all k.
Proof.
The 24 bond vectors form 12 antipodal pairs, so the Hermiticity proof is identical to the FCC case
with the index range extended from 12 to 24. For chiral symmetry:
{γ
5
,γ
µ
} = 0
for every
µ ∈{0,1,2, 3}
(the new
γ
0
anticommutes with
γ
5
by direct computation from
(6)
), so
{γ
5
,γ · ˆn
j
} = ˆn
µ
j
{γ
5
,γ
µ
} = 0
for
every bond, and {γ
5
,D
D4
} = 0.
Exact chiral symmetry at finite lattice spacing, in 4D. No Wilson term. No Ginsparg-Wilson modifi-
cation. No overlap construction.
7.3 V-Form and Type-1/Type-2 Doublers
Antipodal symmetry rewrites D
D4
(k) = iγ
µ
V
µ
(k) with
V
µ
(k) = 2
√
2 sin(k
µ
/
√
2)
∑
ν= µ
cos(k
ν
/
√
2), µ = 0,1,2, 3. (8)
The direct 4D analog of
(3)
: the cosine sum runs over three other directions instead of two. The coefficient
2
√
2 is unchanged. D
D4
= 0 iff V
µ
= 0 for all four µ.
Corollary 1 (Irrational Doubler Theorem, D4). Every non-
Γ
zero of
D
D4
has at least one coordinate of
the form
k
µ
= m
µ
π
√
2
or
k
µ
= (m
µ
+
1
2
)π
√
2
with
m
µ
∈ Z
. No non-
Γ
zero of
D
D4
is sampled by any
finite D4 lattice of integer size L.
Proof.
From
(8)
,
V
µ
= 0
requires either
sin(k
µ
/
√
2) = 0
(giving
k
µ
= m
µ
π
√
2
for integer
m
µ
) or
∑
ν= µ
cos(k
ν
/
√
2) = 0
(constraining the cosines of the other three directions). For all four
V
µ
to vanish
simultaneously with
k = 0
, at least one coordinate must be a nontrivial rational multiple of
π
√
2
: either
sin = 0
at that coordinate (the integer-
m
case), or one of the constraining cosine sums forces
cos(k
ν
/
√
2)
to a specific value placing
k
ν
at a half-integer multiple of
π
√
2
. The offending coordinate is irrational in
any rational fractional basis. No integer-L grid samples it.
The phase classification carries over verbatim:
•
Type-1 (isolated, all
sin = 0
):
k
µ
= m
µ
π
√
2
for all
µ
. All bond phases in
{+1,−1}
(
Z
2
staggered).
Excluded by holographic projection and by irrational coordinates.
•
Type-2 (mixed sin/cos cancellation): generically complex bond phases (U(1)). Excluded by
irrational coordinates alone.
7
Explicit Type-2 example. Try k = (π
√
2/2, 0, π
√
2, π
√
2/2):
sin(k
0
/
√
2) = sin(π/2) = 1, cos(k
0
/
√
2) = 0,
sin(k
1
/
√
2) = 0, cos(k
1
/
√
2) = 1,
sin(k
2
/
√
2) = 0, cos(k
2
/
√
2) = −1,
sin(k
3
/
√
2) = 1, cos(k
3
/
√
2) = 0.
Plug into
(8)
:
V
0
= 2
√
2(1)(1 + (−1) + 0) = 0
;
V
1
= 2
√
2(0)(···) = 0
;
V
2
= 2
√
2(0)(···) = 0
;
V
3
=
2
√
2(1)(0 + 1 + (−1)) = 0
.
D
D4
vanishes at this point. Bond phases are genuinely complex—for
instance, ˆn = (1,1, 0,0)/
√
2 gives e
ik·ˆn
= e
iπ/2
= i. This is U(1).
7.4 Numerical Verification at L = 8,12,16
A direct numerical scan of integer-
L
grids in the D4 Brillouin zone confirms Corollary 1 at three lattice
sizes:
Table 2: D4 Brillouin-zone scan results. At each lattice size
L
, an
L
4
integer grid is scanned for zeros of
D
D4
. Γ is the unique zero on every grid.
L Grid points (L
4
) Γ zeros Non-Γ zeros
8 4,096 1 0
12 20,736 1 0
16 65,536 1 0
The minimum non-zero energy across each scan ranges from
≈ 0.56/a
at
L = 16
to
≈ 0.82/a
at
L = 8, with the value varying irregularly as L changes (the same rational-approximation-to-1/
√
2 effect
as in the FCC case). All numerical claims in this section are reproducible from the supporting code
referenced in Section 9.
7.5 Implications for 4D Lattice Field Theory
The D4 extension takes the FCC construction from a 3D spatial-only setting into physical 4D spacetime,
with every structural property preserved:
1.
Exact chiral symmetry at finite lattice spacing, with no Wilson term and no Ginsparg-Wilson
modification.
2. V-form factorization (8), with the cosine sum running over three directions.
3. Irrational Doubler Theorem as a corollary.
4. Z
2
/U(1) phase distinction, with explicit Type-2 example.
5. Γ as the unique zero on every finite integer-L grid up to L = 16.
Three features distinguish the D4 case from FCC.
4D Euclidean isotropy. The structure tensor
S
µν
= 6δ
µν
is fully isotropic across all four directions.
The 24 nearest-neighbor bonds make no distinction between space and time. Selecting one direction as
time, which any Lorentzian interpretation requires, lies outside the bond-direction operator construction
and would call for a separate axiom not addressed here.
8
Increased doubler count, identical protection. The 4D BZ has more potential Type-1 doubler positions
than the 3D BZ. Within a fixed coordinate range, the number of integer tuples
(m
0
,m
1
,m
2
,m
3
)
exceeds
the number of triples
(m
x
,m
y
,m
z
)
. The irrational-coordinate protection is correspondingly stronger: four
independent coordinates can carry the irrational
π
√
2
factor, versus three in the FCC case. More doublers,
more independent reasons each fails to land on an integer grid.
Wilson-mass mechanism transfer. The non-bipartite Wilson-mass generation from triangular loop
closure carries over directly. The 24-cell has 96 triangular 2-faces, and the closure relation
1
√
2
(1,1,0, 0) +
1
√
2
(−1,0,1, 0) =
1
√
2
(0,1,1, 0)
is the D4 analog of the FCC closure
(1,1,0) +(−1, 0,1) = (0,1,1)
. Second-order hopping through any
triangular face generates a (1 −cos) correction at the same O(1) magnitude as the bare hopping.
Computational overhead.
D
D4
requires 24 nearest-neighbor evaluations per site versus 12 for FCC, a
factor of two. Compared with hypercubic lattice fermions (
2d = 8
in 4D), the D4 construction is three
times more expensive per hopping operation, but substantially less expensive than overlap fermions,
which require a matrix square root regardless of lattice dimension.
8 Conclusion
The bond-direction Dirac operator on FCC and D4 carries exact chiral symmetry at finite lattice spacing
as an algebraic identity, with no Wilson term, no Ginsparg-Wilson modification, no overlap construction.
That is one result. The Irrational Doubler Theorem (Theorem 1, Corollary 1) is the other: every non-
Γ
zero of
D
FCC
has at least one coordinate
f
i
= ±1/(2
√
2)
, and every non-
Γ
zero of
D
D4
has at least one
coordinate
k
µ
∝ π
√
2
. Neither is reachable on any integer-
L
grid. The V-form factorization gives the
proof, the BZ scans give the numerical confirmation.
Two doubler classes appear in both lattices. Type-1:
Z
2
staggered, all bond phases real. Type-2: U(1)
flat-band, generically complex bond phases. The SSM holographic projection removes Type-1 from the
physical spectrum on the grounds that real bond phases transport no U(1) charge. Type-2 modes survive
the projection but cannot be sampled on any rational grid. Two independent mechanisms, covering two
distinct doubler classes.
The
Z
2
/U(1) mechanism is specific to non-bipartite lattice geometries. On bipartite hypercubic lattices
the standard doublers also reside at
Z
2
phase positions (
{+1,−1}
at
k
µ
= π/a
), but those positions sit at
rational fractional BZ coordinates (
f = 1/2
), directly accessible to any even-sized grid. For
D
FCC
and
D
D4
,
the
Z
2
doubler positions shift to irrational coordinates (
1/(2
√
2)
for FCC,
1/
√
2
for D4), kinematically
inaccessible to all finite integer-
L
lattices. Geometric irrational doubler positioning, combined with
holographic U(1)/
Z
2
projection from the QEC vacuum, is a qualitatively new pathway to exact chiral
symmetry on discrete structures. It works the same way in three and four dimensions.
Future directions. Three open questions. First, the behavior of
D
under full non-Abelian SU(N) gauge
fields and the anomaly-matching structure of the construction. Second, whether the
Z
2
/U(1) mechanism
can be formalized as a topological invariant—a
Z
2
-valued index on the FCC or D4 bond bundle. Third, a
stabilizer-code interpretation of the same lattice that would tie the kinematic content constructed here
(Dirac operator) to a compatible vacuum structure (stabilizer code), unifying the two sides of the lattice
within a single QEC framework.
9 Data Availability
All numerical claims in this paper are reproducible from the following Python scripts:
9
•
FCC verification (Section 6):
doubler_fcc_verification.py
. Reproduces the
32
3
scan, K-
point gap, and gauge-field robustness check; runs in ≈ 5 s.
•
D4 verification (Section 7):
doubler_d4_verification.py
. Reproduces the gamma-matrix
algebra checks,
S
µν
= 6δ
µν
identity, V-form verification,
L = 8,12,16
scans, and Type-1/Type-2
doubler classification; runs in ≈ 27 s.
•
Master runner (both):
doubler_run_all.py
. Executes both verifications in sequence; total
runtime ≈ 32 s on a standard laptop.
Dependencies: NumPy ≥1.21, Python ≥ 3.8. License: MIT.
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