Defect-Bound Modes of the Naive Dirac Operator
on the FCC Lattice:
Spectral Multiplet Structure at Tetrahedral and Octahedral Voids
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
RESEARCH Gauge Freedom Journal 1(1), Article 006 (2026)
https://doi.org/10.65323/gfj.2026.006
Correspondence: raghu@idrive.com
Received: 28 May 2026 / Accepted: 12 June 2026
Published online: 19 June 2026
© 2026 The Author(s). Licensed under CC BY 4.0.
Abstract
The face-centered cubic (FCC) lattice has two classes of interstitial void: tetrahedral, with
coordination 4, and octahedral, with coordination 6. A trapped extra node at either void is
a localized point defect coupled to its surrounding FCC sites through nearest-neighbor bonds.
We compute the bound-mode multiplet of the naive Dirac operator at each defect class and
decompose it under the residual point-group symmetry. At a tetrahedral defect the multiplet has
dimension 4 and decomposes under
T
d
∼
=
S
4
as
A
1
⊕ T
2
(Schoenflies
T
d
convention, in which the
three-dimensional irrep is the polar vector). At an octahedral defect the multiplet has dimension
6 and decomposes under
O
h
as
A
1g
⊕E
g
⊕T
1u
. Both three-dimensional pieces are spatial-vector
representations of their point groups, so the spectral labels alone do not distinguish the two
defect classes; the distinction is combinatorial. The three-dimensional
T
2
piece has the dimension
of, and is
S
3
-module-isomorphic to, the internal color space built in the matter paper of Kulkarni
[Phys. Open 27, 100423 (2026)] on the three perfect matchings of
K
4
, the bonded graph of the
tetrahedral defect’s surrounding sites. The bonded graph of the octahedron’s six surrounding
sites is instead the complete tripartite
K
2,2,2
rather than
K
6
, because the three antipodal vertex
pairs sit at second-nearest-neighbor distance in the FCC lattice and are not physical bonds;
K
2,2,2
has 8 perfect matchings and 30 skew-edge pairs, neither giving the 3-color structure of the
matter-paper construction. The spectral analysis is thus a dimensional consistency check on a
pre-existing combinatorial color assignment, not a derivation of color from the spectrum: the
tetrahedral bound triplet has the dimension of the
K
4
color space, while the two defect classes are
separated not by their spectral content—which is a spatial-vector triplet in both cases—but by
this combinatorial matching count. A companion dark-matter paper [
15
] reaches the same
K
2,2,2
1
obstruction by independent combinatorial means and, on that basis, proposes a dark-sector
identification of the octahedral defect; the present work supports the structural distinction
underlying that proposal, not its mass scale, abundance, or phenomenology. As supporting
background we classify the free-field FCC zero-mode spectrum from an exact factorization of the
kinetic vector field. The comparison with HCP, BCC, and bulk-modified fermion constructions,
and the reasons FCC is uniquely selected as an isotropic substrate, are developed in the body.
1 Introduction
The naive Dirac operator on a hypercubic lattice in
D
dimensions has 2
D
geometrically equivalent
doublers [
1
,
2
]. Wilson fermions [
3
] give them a mass term that breaks chiral symmetry. Staggered
fermions [
4
,
5
] cut the count to 2
⌈D/2⌉
tastes at the cost of a residual flavor symmetry. Minimally
doubled formulations [
6
,
7
,
8
,
9
,
10
] keep only two species in the continuum limit by breaking
hypercubic symmetry. Supersymmetric lattice constructions [
11
,
12
] use root lattices such as
A
∗
4
to
preserve part of the supersymmetry algebra exactly. Each of these acts on the bulk operator.
We take a different starting point. Consider the standard naive Dirac operator on the face-
centered cubic (FCC) lattice, and look at the spectrum it produces when a localized point defect
is inserted at an interstitial void. FCC has two void classes per conventional cubic cell: eight
tetrahedral voids (coordination 4) and four octahedral voids (coordination 6). A trapped extra node
at either is a natural physical defect. The bound modes at the defect form a multiplet under the
residual point-group symmetry, and the structure of that multiplet decides whether the defect can
carry an internal symmetry algebra of a given type.
The setting is the matter paper [
14
] of the SSMTheory program, in which physical matter is
identified with localized lattice defects rather than with a bulk fermion field. There, color quantum
numbers are assigned by combinatorial counting: the complete graph
K
4
on the four valence bonds
of a tetrahedral defect has exactly three perfect matchings, giving a three-dimensional internal color
Hilbert space on which an
su
(3) algebra acts. That construction is group-theoretic and uses the
local bond connectivity only, with no input from the Dirac spectrum. The question this paper takes
up is whether the spectral content of the Dirac operator at the same defect is consistent with the
assigned color structure — that is, whether the bound-mode multiplet contains a representation of
the right dimension and the right transformation properties — and what the corresponding analysis
gives for the octahedral defect.
Results. At a tetrahedral defect the bound-mode multiplet has dimension 4 and decomposes
under the tetrahedral point group T
d
∼
=
S
4
as
4
tet
= A
1
⊕ T
2
, dim 1 + 3. (1)
The three-dimensional
T
2
piece (the polar vector of
T
d
; see the convention note in Section 4) has
the dimension of the matter paper’s
K
4
color space and is
S
3
-module-isomorphic to it (Section 4).
At an octahedral defect the multiplet has dimension 6 and decomposes under the octahedral point
group O
h
as
6
oct
= A
1g
⊕ E
g
⊕ T
1u
, dim 1 + 2 + 3. (2)
The 3-dimensional
T
1u
piece is the vector representation of
O
h
. The tetrahedral
T
2
piece is likewise
a spatial-vector representation (of
T
d
), so neither triplet is singled out as “internal” by its spectral
label, and the two defect classes are separated instead by their bonded-graph combinatorics. The
bonded graph of the six surrounding sites is the complete tripartite graph
K
2,2,2
, not
K
6
: the
three antipodal pairs of the octahedron sit at second-nearest-neighbor distance in the FCC lattice,
Gauge Freedom Journal 1(1), Article 006 (2026) 2
not nearest-neighbor, so they are not physical bonds.
K
2,2,2
has eight perfect matchings, not the
three on which the matter-paper algebra is built, so no
K
4
-type color structure is available at an
octahedral defect (Section 5). The companion dark-matter paper [
15
] reaches this
K
2,2,2
obstruction
by independent combinatorial means and, on that basis, proposes a dark-sector identification of the
octahedral defect. The present spectral analysis supports the structural distinction underlying that
proposal (that octahedral defects do not carry the K
4
-type color algebra), but says nothing about
its mass scale, abundance, or phenomenology, which are the subject of [
15
]. Tetrahedral defects
support the matter-paper color algebra; octahedral defects do not.
A note on what this result is and is not. The Dirac analysis does not derive SU(3) color
from the spectrum; the matter-paper construction is logically prior and uses different input (
K
4
perfect matchings, not Dirac eigenmodes). What we add is a dimensional consistency check: the
tetrahedral bound triplet is three-dimensional, matching the dimension of the
K
4
color space, so
the matter-paper assignment can be carried on it. Because both bound triplets are spatial-vector
representations of their point groups, the spectrum does not by itself distinguish the two defect
classes; the substantive distinction is the combinatorial
K
2,2,2
obstruction (eight perfect matchings
rather than three), which removes the three-element matching space at the octahedral void.
Bulk background. Sections 2 and 3 give the free-field bulk spectrum on FCC. The kinetic
vector field factors exactly (Theorem 1), so the full zero-mode structure can be classified by cases.
Three topological classes appear: a singlet at Γ, a quartet at the four
L
-points (
±π/
2
, ±π/
2
, ±π/
2),
and three boundary nodal loops along the
X
–
W
segments of the truncated-octahedron BZ. The
kinetic-map index sum is +1
−
4+3 = 0, the Nielsen–Ninomiya consistency relation. The FCC classes
are therefore stratified by Wilson mass at distinct scales. This differs from the usual hypercubic
Wilson pattern, where the Wilson mass depends on the number of momentum components sitting at
the edge of the Brillouin zone. This bulk analysis is supporting context for the defect-bound-mode
analysis in Sections 4 and 5, which is the main new content.
Relation to other reduced-fermion constructions. The defect mechanism we use is unlike
minimally doubled fermions [
6
,
7
,
8
], supersymmetric lattices [
11
,
12
], twisted-mass, and domain-wall
constructions: those modify the bulk operator. Here the bulk operator is unchanged; the physical
content sits in localized topological excitations whose internal symmetries are set by the defect’s
local geometry. Section 6 discusses the comparison in detail and works out where the cuboctahedral
first-shell structure of FCC fits among the alternatives. The closest 3D candidate is HCP, with
the same coordination number
K
= 12, but HCP’s first shell is the anticuboctahedron and gives a
preferred-axis anisotropy at the rank-4 bond-tensor level; FCC is the only 3D close-packed lattice
with a fully cubic-symmetric rank-4 bond tensor and the corresponding full
T
d
/
O
h
symmetry at
its interstitial voids.
2 The FCC Lattice and the Naive Dirac Operator
The FCC lattice has twelve nearest-neighbor vectors (in units of a/2):
N = {(±1, ±1, 0), (±1, 0, ±1), (0, ±1, ±1)}. (3)
The convex hull of these twelve points is the cuboctahedron [
17
], with symmetry group
O
h
of
order 48. The reciprocal lattice is body-centered cubic; the first Brillouin zone is a truncated
octahedron [18] with high-symmetry points
Γ = (0, 0, 0), L = (±
π
2
, ±
π
2
, ±
π
2
), X = (π, 0, 0), W = (π,
π
2
, 0) (4)
Gauge Freedom Journal 1(1), Article 006 (2026) 3
(and cubic permutations of
X, W
). Throughout, nearest-neighbor vectors are given in units of
a/
2,
and Brillouin-zone coordinates and all figure axes are in units of π/a.
The naive Dirac operator in three spatial dimensions uses two-component spinors with no
γ
5
.
The local degrees we track below are kinetic-map indices of
f
:
T
3
→ R
3
, not eigenvalues of
γ
5
[
16
].
The kinetic vector field is
f
µ
(k) =
1
a
X
n∈N
n
µ
sin(k · n), µ = 1, 2, 3, (5)
and the Dirac operator is D(k) = iσ ·
f(k). Zero modes occur wherever |
f(k)| = 0.
Theorem 1 (Factorization). The kinetic vector field of Eq. (5) factors as
f
µ
(k) =
4
a
sin(k
µ
)
cos(k
ν
) + cos(k
ρ
)
, (6)
where (µ, ν, ρ) is any permutation of (1, 2, 3).
Proof.
The twelve FCC neighbors partition into three sheets
S
µν
containing the four vectors with
zero
ρ
-component. Only sheets
S
µν
and
S
µρ
contribute to
f
µ
(sheet
S
νρ
has
n
µ
= 0). From
S
µν
, the
four vectors (n
µ
, n
ν
) ∈ {(±1, ±1)} contribute
sin(k
µ
+ k
ν
) + sin(k
µ
− k
ν
) − sin(−k
µ
+ k
ν
) − sin(−k
µ
− k
ν
) = 4 sin(k
µ
) cos(k
ν
),
using
sin A
+
sin B
= 2
sin
A+B
2
cos
A−B
2
. Sheet
S
µρ
contributes 4
sin
(
k
µ
)
cos
(
k
ρ
) identically. Summing
gives Eq. (6).
3 Bulk Zero Modes and Their Topological Indices
Theorem 2 (Complete classification). The zero set
f
−1
(0) on the FCC Brillouin zone consists of
exactly:
1. Γ = (0, 0, 0): one isolated zero.
2. L = (±π/2, ±π/2, ±π/2): four isolated zeros after antipodal identification.
3. Nodal lines on the BZ boundary connecting X- to W -points.
No other zeros exist.
Proof.
By Theorem 1,
f
µ
= 0 requires
sin
(
k
µ
) = 0 (call this
S
µ
) or
cos
(
k
ν
) +
cos
(
k
ρ
) = 0 (
C
µ
).
Exhausting all 2
3
= 8 sign combinations:
Case SSS (S
1
∧ S
2
∧ S
3
): each k
µ
∈ {0, π}. BZ constraints admit Γ and the three X-points.
Case
CCC
: subtracting pairs gives
cos k
1
=
cos k
2
=
cos k
3
; substituting back gives 2
cos k
1
= 0,
so k
µ
= ±π/2. These are the four L-points (inside the BZ).
Cases
SSC, SCS, CSS
: two momenta in
{
0
, π}
with their cosines summing to zero; the third
is free. These produce nodal lines: (0
, π, t
), (
π,
0
, t
), and permutations — the
X
–
W
boundary
segments.
Cases
SCC, CSC, CCS
: require
sin k
µ
= 0 and
cos k
µ
=
−cos k
ν
=
−cos k
ρ
. If
k
µ
= 0:
k
ν
=
k
ρ
=
π
, but
|k
1
|
+
|k
2
|
+
|k
3
|
= 2
π >
3
π/
2, violating the hexagonal-face bound of the
truncated-octahedron FCC Brillouin zone. If
k
µ
=
π
:
k
ν
=
k
ρ
= 0 (
X
-point, already in case
SSS
).
The eight cases are exhaustive.
Gauge Freedom Journal 1(1), Article 006 (2026) 4
3.1 Topological indices
The topological index at an isolated zero
k
i
is
χ
i
=
sgn
(
det J
), where
J
µν
(
k
) =
∂f
µ
/∂k
ν
=
1
a
P
n
n
µ
n
ν
cos(k · n).
Proposition 1 (Γ-point). At Γ, J
µν
= (8/a)δ
µν
, det J = 512/a
3
> 0, χ
Γ
= +1.
Proof.
At
k
= 0,
cos
(
k · n
) = 1 for all
n
. The sum
P
n
n
µ
n
ν
= 8
δ
µν
: each diagonal entry receives a
contribution of +1 from the eight FCC vectors with
|n
µ
|
= 1, and off-diagonal entries cancel by
inversion symmetry.
Proposition 2 (
L
-points). At
L
= (
π/
2
, π/
2
, π/
2),
J
(
L
) =
−
(4
/a
)(
E − I
), where
E
is the all-ones
matrix and I the identity. Eigenvalues are {+4, +4, −8}/a, det J = −128/a
3
< 0, χ
L
= −1.
Proof.
At
L
,
k · n ∈ {
0
, ±π}
. Six neighbors have
cos
= +1; six have
cos
=
−
1. Diagonal entries
P
n
n
2
µ
cos
(
k · n
) = 0 (equal numbers of +1 and
−
1 along each axis). For the off-diagonal entries
the four neighbors with
n
µ
n
ν
= 0 all give
cos
(
k · n
) =
−
1 and
n
µ
n
ν
=
±
1, contributing
−
4
/a
. So
J(L) = −(4/a)(E − I), eigenvalues {+4/a, +4/a, −8/a}.
With four independent L-points, χ
total
L
= 4 × (−1) = −4.
3.2 Boundary winding number
Along a nodal line in
k
2
(for example
k
= (
π, t,
0)), the transverse field (
f
1
, f
3
) winds around the
origin. We computed the winding number on a small circle of radius
ϵ
= 0
.
01 in the (
k
1
, k
3
) plane,
sampled with 2000 angular points, obtaining
w
= +1
.
000
±
0
.
003 independent of position
t
along
the line. Topological stability follows from the factorization in Eq.
(6)
: at
k
1
=
π
,
sin
(
k
1
) = 0 forces
f
1
≡
0, so any zero of
|f|
on this face must satisfy
f
2
=
f
3
= 0, confining the locus to the nodal
line. The three independent boundary loops (one per spatial axis) each contribute
χ
= +1, so
χ
boundary
= +3.
The Nielsen–Ninomiya balance must be stated with care, because the three classes are objects
of different codimension: the Γ- and
L
-point zeros are isolated (codimension three) and carry
an integer index
χ
i
=
sgn det J
(
k
i
), while the boundary loops are codimension-two nodal lines
carrying a transverse winding number. We therefore state the result as a consistency relation among
these counts under fixed Brillouin-zone identifications, rather than as a single index theorem over
heterogeneous objects. The identifications are: (i) the four
L
-points are counted after antipodal
identification of the truncated-octahedron BZ (the eight points (
±π/
2
, ±π/
2
, ±π/
2) pair into four
inequivalent zeros); and (ii) the six square boundary faces pair under inversion
k → −k
into three
independent X–W nodal loops, each of winding +1. With these identifications,
χ
total
= +1
|{z}
Γ
+ −4
|{z}
L
+ +3
|{z}
boundary
= 0, (7)
the Nielsen–Ninomiya consistency relation for the FCC naive operator. Figure 1 shows the three
classes on the truncated-octahedron Brillouin zone.
3.3 Wilson masses
A standard isotropic Wilson term gives the bulk zero-mode classes mass
M
W
= 0 at Γ, 12
/a
at
each
L
-point, and 16
/a
on the boundary loops [
3
]. The FCC classes are therefore stratified by
Wilson mass at distinct scales. This differs from the usual hypercubic Wilson pattern, where the
Gauge Freedom Journal 1(1), Article 006 (2026) 5
0
k
1
[ /
a
]
0
k
2
[ /
a
]
0
k
3
[ /
a
]
( = +1)
L
( = 1)
X
--
W
nodal loop ( = +1)
Figure 1: FCC first Brillouin zone (truncated octahedron) with the three zero-mode classes of
the naive Dirac operator. Green circle: Γ-point singlet,
χ
= +1. Red squares: four
L
-points at
(
±π/
2
, ±π/
2
, ±π/
2),
χ
=
−
1 each, giving
χ
total
L
=
−
4. Blue outlines: the six square boundary
faces, paired by inversion into three independent
X
–
W
nodal loops with
χ
= +1 per loop. The
Nielsen–Ninomiya sum +1 − 4 + 3 = 0 is realized by these three classes.
Gauge Freedom Journal 1(1), Article 006 (2026) 6
Wilson mass depends on the number of momentum components sitting at the edge of the Brillouin
zone. An anisotropic Wilson term with enhanced coupling along (1
,
1
,
1) splits the
L
-quartet under
S
4
⊃ S
3
as 1 ⊕ 3, with ∆M = 4/a.
4 Tetrahedral Defects
This section gives the main new content of the paper. The bulk classification of Theorem 2 sets
the stage; we now turn to the localized bound-mode multiplet at a tetrahedral interstitial defect
(Figure 2(a)).
4.1 Defect geometry
A tetrahedral void of the FCC lattice is bounded by four FCC sites that form a regular tetrahedron.
An interstitial node placed at the void centroid sits at equal distance from each of these four sites.
Label the surrounding sites
{
1
,
2
,
3
,
4
}
and the bonds from the defect to them
{b
1
, b
2
, b
3
, b
4
}
. The
defect site and the four bonds form a star graph whose point group is
T
d
∼
=
S
4
, the full tetrahedral
group of order 24.
The naive Dirac operator restricted to this star has matrix elements between the defect spinor
and each of the four surrounding-site spinors. The Hilbert space localized on the star factors
as
C
2
⊗
(
C ⊕ C
4
): the spinor index times (defect site
⊕
four surrounding sites). What sets the
bound-mode multiplet structure is the four-dimensional permutation representation of
T
d
on the
surrounding sites, with the spinor index decorating each multiplet member.
4.2 The defect Hamiltonian
We work with two-component spinors on a finite FCC cluster
C
(the parity-even sites
x
+
y
+
z
even
with
|
r
|
2
≤
9
.
5;
|C|
= 55), open boundary conditions, together with one interstitial site at the void
centroid. The Hamiltonian is
H = H
FCC
+ H
defect
+ H
coupling
. (8)
(i) Bulk term.
H
FCC
is the naive Dirac hopping operator of Section 2: for each nearest-neighbor
vector n ∈ N and each bond (r, r + n) in C,
H
FCC
r+n, r
=
i
2a
σ · n,
H
FCC
r, r+n
= h.c., (9)
whose Bloch symbol on the infinite lattice is the Hermitian kinetic operator
σ ·
f
(
k
) built from
Eq.
(5)
(up to the overall sign fixed by hermiticity; the bulk Dirac operator
D
=
iσ ·
f
of Section 2
is its anti-Hermitian counterpart, with the same zero set
|
f|
= 0). (ii) Defect term. The trapped
node carries an on-site energy
H
defect
=
ε
d
c
†
0
c
0
(two-component). (iii) Coupling term. The defect
site bonds to its M bounding sites {a = 1, . . . , M} at the unit bond length through a hopping t:
H
coupling
= t
M
X
a=1
c
†
0
c
a
+ h.c.
, M = 4 (tetrahedral), 6 (octahedral). (10)
Effective on-site model. Integrating out the defect site 0 produces a shell self-energy Σ
s
(
E
) =
t
2
/
(
E − ε
d
) that acts only in the symmetric (
A
1
) combination of the bounding sites and is energy-
dependent; in the deep-level limit
|E − ε
d
| ≫ |t|
it becomes a static shift. The verification script
Gauge Freedom Journal 1(1), Article 006 (2026) 7
uses the simpler, manifestly point-group-symmetric model in which the defect shifts the on-site
energy of each bounding site uniformly,
H
defect-model
= V
M
X
a=1
c
†
a
c
a
, V < 0, (11)
an attractive well on the bounding shell. The multiplet decomposition (Propositions 3 and 5) is
forced by the site point group (
T
d
or
O
h
) of the perturbation alone and is independent of the radial
profile of the potential; the two models differ only in which members are pulled to bound energies
(the rank-one self-energy populates
A
1
; the full-rank well of Eq.
(11)
populates every irrep). We
refer to Eq. (11) as the defect model below, and avoid the term “Schur complement” as imprecise.
What is proved. We establish two statements, and only these: (a) the
M
-dimensional bounding-
shell amplitude space carries the point-group decomposition of Propositions 3 and 5; and (b) in the
tetrahedral strong-well model studied numerically, there exist eigenstates localized on the defect
shell whose weights reproduce the
A
1
⊕ T
2
decomposition, confirmed by exact diagonalization
(Section 4.5). We do not claim an infinite-lattice bound-state theorem, an octahedral finite-cluster
diagonalization, an asymptotic decay rate, or a derivation of internal symmetry from the spectrum.
4.3 Multiplet structure under T
d
Convention. We use the Schoenflies
T
d
convention throughout: the polar vector (
x, y, z
) trans-
forms as
T
2
and axial vectors as
T
1
. The four-site permutation representation of
T
d
∼
=
S
4
is then
4
tet
=
A
1
⊕ T
2
, the same reducible representation as the four
σ
-bonds (equivalently the four H
positions) in CH
4
. In the abstract-
S
4
convention the same three-dimensional irrep is often written
T
1
; the relabeling does not affect any amplitude or dimension, only the symbol.
The permutation representation of
S
4
on four sites has character
χ
= (4
,
2
,
0
,
1
,
0) on the
conjugacy classes (
e,
(12)
,
(12)(34)
,
(123)
,
(1234)). Projecting onto
T
d
irreps gives the following
decomposition.
Proposition 3. The permutation representation of
T
d
∼
=
S
4
on the four surrounding sites of a
tetrahedral defect decomposes as
4
tet
= A
1
⊕ T
2
, (12)
with dimensions 1 + 3 = 4.
Proof.
Direct projection against the
S
4
character table. For the trivial representation
A
1
with
character (1, 1, 1, 1, 1) the multiplicity is
⟨A
1
, χ⟩ =
1
24
1 · 1 · 4 + 6 ·1 ·2 + 3 · 1 · 0 + 8 · 1 · 1 + 6 · 1 · 0
= 1.
For the polar-vector representation T
2
with character (3, 1, −1, 0, −1) the multiplicity is
⟨T
2
, χ⟩ =
1
24
1 · 3 · 4 + 6 ·1 ·2 + 3 · (−1) · 0 + 8 · 0 · 1 + 6 · (−1) · 0
= 1.
The other irreps A
2
, E, T
1
have multiplicity 0.
The
A
1
singlet is the symmetric combination
|s⟩
= (
|
1
⟩
+
|
2
⟩
+
|
3
⟩
+
|
4
⟩
)
/
2. The
T
2
triplet is
spanned by three orthogonal combinations transforming as the polar vector. With spinor decoration
each of the four multiplet members carries a two-component spinor index, giving eight bound-mode
states in total.
Gauge Freedom Journal 1(1), Article 006 (2026) 8
4.4 Compatibility with the K
4
color construction
The matter paper [
14
] builds a three-color quark Hilbert space at a tetrahedral defect from the
perfect-matching combinatorics of
K
4
, the complete graph on the four valence bonds.
K
4
has six
edges; they partition into three perfect matchings (pairs of disjoint edges covering all four vertices):
M
1
= {b
12
, b
34
}, M
2
= {b
13
, b
24
}, M
3
= {b
14
, b
23
}, (13)
where
b
ij
joins surrounding sites
i
and
j
. The Weyl group of the resulting
su
(3) algebra is
S
3
=
S
4
/V
4
,
where
V
4
=
{e,
(12)(34)
,
(13)(24)
,
(14)(23)
}
is the Klein four-group;
V
4
acts trivially on the three
matchings, so the quotient acts faithfully.
The matter-paper color Hilbert space
H
C
=
span{|M
1
⟩, |M
2
⟩, |M
3
⟩}
is logically independent of
the Dirac bound-mode space, since it lives on matchings rather than on sites. What needs to be
checked is whether the bound-mode space of Proposition 3 can carry an SU(3) action that respects
the assigned color structure.
Proposition 4. The three-dimensional
T
2
sub-multiplet of the tetrahedral bound-mode space has the
dimension of the
K
4
matching space
H
C
=
span{|M
1
⟩, |M
2
⟩, |M
3
⟩}
of the matter paper [
14
]. After
choosing a splitting of S
4
→ S
4
/V
4
≃ S
3
, it is isomorphic to H
C
as an S
3
-module.
There is a linear identification
Φ : T
2
−→ span{|M
1
⟩, |M
2
⟩, |M
3
⟩}. (14)
After choosing a splitting of the quotient
S
4
→ S
4
/V
4
∼
=
S
3
, equivalently a subgroup
S
3
⊂ S
4
mapping
isomorphically onto the quotient, the restricted tetrahedral triplet satisfies
T
2
|
S
3
∼
=
A
1
⊕ E
. The
matching representation on
H
C
=
span{|M
1
⟩, |M
2
⟩, |M
3
⟩}
has the same
S
3
-module decomposition
A
1
⊕ E
. Thus one may choose a dimension-preserving
S
3
-equivariant linear identification Φ :
T
2
→ H
C
. This choice is basis-dependent and noncanonical; it is not
S
4
-equivariant, since
V
4
acts
nontrivially on the site-amplitude triplet but trivially on the matching space. The matter-paper
su
(3)
algebra, defined on
H
C
, acts on
T
2
by transport through Φ. This is a statement of representational
compatibility, not a derivation of color from the Dirac spectrum.
4.5 Numerical confirmation
The 1 + 3 site-permutation decomposition is confirmed by direct diagonalization of the naive Dirac
operator on the finite cluster
C
of Section 4.2. The verification script (
dirac_verify_defects.py
,
included with the submission) builds the 55-site FCC cluster and models the trapped-node defect
at the tetrahedral void (1
/
2
,
1
/
2
,
1
/
2) as the uniform attractive on-site well of Eq.
(11)
on the
four surrounding FCC sites. This
T
d
-symmetric perturbation creates well-localized bound modes
without modifying the bulk operator. The resulting Hermitian 110
×
110 Hamiltonian is diagonalized
exactly. The eight most-localized eigenstates (= 1 + 3 spatial ×2 spinor) carry the A
1
and T
2
site
representations with total weights converging to 2 and 6 respectively as the well depth increases,
with ratio
T
2
/A
1
→
3, the value expected for the
A
1
⊕ T
2
decomposition. Table 1 reports the
convergence; all hopping and energy values are in units of the bulk coefficient 1/2a (i.e. a = 1).
5 Octahedral Defects
Unlike the tetrahedral case of Section 4, we do not perform a separate finite-cluster diagonalization
at the octahedral void. The result of this section is (i) the symmetry decomposition of the six-
dimensional bounding-shell amplitude space under
O
h
(Proposition 5), and (ii) the combinatorial
Gauge Freedom Journal 1(1), Article 006 (2026) 9
|V | top-8 band A
1
wt. T
2
wt. T
2
/A
1
min. loc. wt.
8 [−10.12, −6.92] 1.814 5.624 3.100 0.900
16 [−17.95, −14.55] 1.953 5.906 3.024 0.980
32 [−33.85, −30.40] 1.988 5.977 3.006 0.995
64 [−65.79, −62.33] 1.997 5.994 3.001 0.999
128 [−129.76, −126.30] 1.999 5.999 3.000 1.000
Table 1: Finite-cluster Dirac diagonalization at a tetrahedral defect (55-site cluster, 110-dimensional
Hermitian Hamiltonian). As the well depth
|V |
increases, the eight most-localized eigenstates have
total
A
1
weight
→
2 and
T
2
weight
→
6 (ratio
→
3), and their localization weight on the four bounding
sites → 1, confirming the A
1
⊕ T
2
decomposition. Reproduced by dirac_verify_defects.py.
obstruction on the bonded graph
K
2,2,2
(Proposition 6), whose enumeration is carried out in full in
the companion paper [
15
]. No claim is made here about the detailed bound-state spectrum at an
octahedral defect beyond the representation content of the localized shell space.
5.1 Defect geometry
An octahedral void of the FCC lattice is bounded by six FCC sites at the vertices of a regular
octahedron. The surrounding sites and the defect site form a local star graph with point group
O
h
of order 48. The defect couples to each of the six surrounding sites through one bond.
5.2 Multiplet structure under O
h
The permutation representation of
O
h
on the six octahedron vertices decomposes into
O
h
irreps as
follows.
Proposition 5. The permutation representation of
O
h
on the six surrounding sites of an octahedral
defect decomposes as
6
oct
= A
1g
⊕ E
g
⊕ T
1u
, (15)
with dimensions 1 + 2 + 3 = 6.
This is the standard
O
h
decomposition on six octahedron vertices, derivable from the
O
h
character table [
18
,
17
]. The
A
1g
singlet is the symmetric sum. The
E
g
doublet captures two
diagonal differences between antipodal-pair amplitudes. The
T
1u
triplet is the vector representation,
with basis vectors transforming as (x, y, z) under O
h
.
5.3 What distinguishes the two triplets
Both three-dimensional sub-multiplets are spatial-vector representations of their point groups: the
tetrahedral
T
2
is the polar vector of
T
d
, and the octahedral
T
1u
is the polar vector of
O
h
. Because
both are spatial, the Dirac decomposition alone cannot single out one as “internal” and the other as
“spatial.” What distinguishes the two defect classes is not the spectral label but the bonded-graph
combinatorics on which the matter paper’s color algebra is built:
K
4
has three perfect matchings,
supporting the three-element color space
H
C
, whereas
K
2,2,2
has eight perfect matchings and 30
skew-edge pairs (enumerated in [
15
]), neither factoring as a three-color representation. The spectral
analysis contributes only a dimensional consistency check at each defect class:
dim T
2
= 3 =
dim H
C
at the tetrahedral void, with no analogous three-dimensional matching space at the octahedral shell.
Gauge Freedom Journal 1(1), Article 006 (2026) 10
defect
1
2
3
4
(a) Tetrahedral defect:
K
4
, 3 perfect matchings
K
4
perfect matchings
M
1
= {12, 34}
M
2
= {13, 24}
M
3
= {14, 23}
defect
1
1'
2
2'
3
3'
(b) Octahedral defect:
K
2, 2, 2
, 8 perfect matchings
K
2, 2, 2
edges (12)
antipodal pairs (not bonds)
one perfect matching
(8 total)
Figure 2: Bond graphs for the two interstitial defect classes of the FCC lattice. (a) Tetrahedral:
the four surrounding FCC sites form a regular tetrahedron, all six pairs at nearest-neighbor distance,
giving the complete graph
K
4
; its three perfect matchings
M
1
, M
2
, M
3
(color-coded) carry the
matter paper’s three colors [
14
]. (b) Octahedral: the three antipodal vertex pairs sit at second-
nearest-neighbor distance
√
2 L
and are therefore not physical bonds (red dotted). Only the twelve
non-antipodal nearest-neighbor pairs are bonds (gray), so the bonded graph is the complete tripartite
K
2,2,2
, not
K
6
, and has eight perfect matchings rather than three (one shown in green). The matter-
paper color algebra does not extend to this graph (Proposition 6; independently [15]).
Gauge Freedom Journal 1(1), Article 006 (2026) 11
5.4 Combinatorial obstruction from K
2,2,2
The matter paper’s
su
(3) algebra uses perfect-matching combinatorics on the complete graph of the
four mutually nearest-neighbor valence bonds of a tetrahedral defect. For the octahedral defect,
the analogous graph is built from the six surrounding sites and the bonds joining those pairs at
nearest-neighbor distance in the FCC lattice. The six octahedron vertices fall into three antipodal
pairs at second-nearest-neighbor distance
√
2 L
(with
L
the nearest-neighbor distance), and these
antipodal pairs are not bonded. Only the twelve non-antipodal pairs are physical bonds. The
bonded graph is therefore the complete tripartite graph
K
2,2,2
: 6 vertices in three antipodal pairs, 12 edges between non-antipodal pairs, (16)
not the complete graph
K
6
on six vertices (compare Figure 2(a) and (b)). Direct enumeration on
K
2,2,2
(verified by the accompanying script) gives:
• Number of edges: 12.
• Number of perfect matchings (triples of disjoint edges covering all six vertices): 8.
• Number of skew-edge pairs (pairs of disjoint edges): 30.
The same counts appear in the companion dark-matter paper [
15
], where the skew-edge pair count
c
(O)
skew
= 30 enters as combinatorial input to the mass derivation.
Proposition 6. The matter-paper
su
(3) algebra constructed from the three perfect matchings of
K
4
does not extend to a structurally analogous algebra on the bond graph
K
2,2,2
.
K
2,2,2
has eight perfect
matchings, not three, and a skew-pair count of 30; neither factors as a three-color representation [
15
].
No internal
SU
(3) algebra of the matter-paper form acts on the bound-mode multiplet of an octahedral
defect.
The bound-mode multiplet at an octahedral defect therefore carries the
A
1g
⊕ E
g
⊕ T
1u
decom-
position derived above, but no embedded three-color matching space of the matter-paper type.
The same obstruction is reached combinatorially in [
15
]; the spectral analysis here is independent
representation-theoretic support for that structural conclusion.
5.5 Implication: a structurally distinct second sector
Tetrahedral defects support the matter-paper color construction. Octahedral defects do not. The
two defect classes are not equivalent matter sectors of the FCC substrate. Only the tetrahedral
class, with coordination 4, is consistent with the
K
4
-based color algebra of the SSMTheory program.
On the basis of this same structural distinction, the companion dark-matter paper [
15
] proposes a
dark-sector identification of the octahedral class; the
O
h
multiplet decomposition of Proposition 5
and the bond-graph identification of Eq.
(16)
support that distinction, while the mass, abundance,
and indirect-detection phenomenology of the octahedral sector are developed in [
15
] and are neither
used nor evaluated here.
6 Comparison with Other Lattices and Reduced-Fermion Constructions
The defect mechanism of Sections 4–5 is unlike the established reduced-fermion constructions in
lattice gauge theory. We retain the standard naive Dirac operator on FCC; physical fermion content
is carried by localized excitations at substrate defects, not by changes to the bulk operator. The
point group of the defect, together with the bonded graph it generates, fixes the multiplet structure.
Table 2 summarizes the comparison across the lattices considered below.
Gauge Freedom Journal 1(1), Article 006 (2026) 12
HCP: same
K
= 12, broken isotropy along the c-axis. Hexagonal close-packed (HCP) and
FCC are the two close-packed lattices in 3D, both with coordination
K
= 12 and the same nearest-
neighbor distance. The first shells differ structurally. FCC’s first shell is the cuboctahedron, with
point group
O
h
(order 48) including inversion symmetry. HCP’s first shell is the anticuboctahedron
(the Johnson solid
J
27
, also called the triangular orthobicupola), with point group
D
3h
(order
12) and a distinguished c-axis. The two solids are related by rotating one triangular face of the
cuboctahedron by 60
◦
relative to its opposite face. That single twist breaks
O
h
to
D
3h
and singles
out the stacking direction.
The consequence at the bond-tensor level is sharp. Both lattices give
T
(2)
µν
=
K
2
δ
µν
, so
ω ∝ |
k|
at long wavelengths in both cases. The rank-3 tensor
T
(3)
µνρ
vanishes for both: by full inversion
symmetry for FCC, and by the horizontal mirror
σ
h
acting on the anticuboctahedron for HCP. At
rank 4 the two diverge. For FCC, direct computation gives
T
(4), FCC
xxxx
= T
(4), FCC
yyyy
= T
(4), FCC
zzzz
,
T
(4), FCC
xxxx
T
(4), FCC
xxyy
= 2, (17)
which is anisotropic at rank 4 relative to the fully isotropic value 3, but with all three coordinate
axes equivalent. The unbroken symmetry of the rank-4 corrections is the full cubic group, so the
anisotropy is the standard discrete cubic anisotropy of any 3D lattice. For HCP at ideal
c/a
=
p
8/3
,
direct computation gives
T
(4), HCP
xxxx
= T
(4), HCP
yyyy
=
5
2
= T
(4), HCP
zzzz
=
8
3
,
T
(4), HCP
xxxx
T
(4), HCP
xxyy
= 3,
T
(4), HCP
zzzz
T
(4), HCP
xxzz
= 4. (18)
The in-plane ratio 3 is what fully-isotropic rank-4 tensors give, but the c-axis breaks this:
T
zzzz
differs from
T
xxxx
, and the c-axis ratio differs from the in-plane ratio. HCP’s rank-4 anisotropy
contains a preferred-axis term forbidden by full SO(3), not merely a discrete cubic anisotropy.
A substrate intended to support an emergent rotationally invariant continuum theory cannot
single out a spatial direction at the
O
(
a
2
) correction level of its bulk dispersion. HCP fails this
requirement, FCC passes it (subject to the standard discrete cubic anisotropy expected of any 3D
lattice). Of the two close-packed 3D lattices, only FCC is therefore a candidate for a substrate of
isotropic physical spacetime. In four dimensions the analogous question singles out the
D
4
root
lattice, whose 24-cell first shell gives an exactly isotropic rank-4 bond tensor.
At the local defect-bound-mode level relevant to the rest of this paper, HCP’s reduced symmetry
has a different consequence. The combinatorial bond graphs
K
4
at a tetrahedral void and
K
2,2,2
at
an octahedral void are preserved (they are determined by which pairs of bounding vertices sit at
nearest-neighbor distance, which depends only on the local coordination), so the matter-paper K
4
perfect-matching construction still defines a three-element color basis at an HCP tetrahedral defect.
The local point group at the voids, however, is reduced: from
T
d
to a
C
3v
subgroup at a tetrahedral
void, and from
O
h
to
D
3d
at an octahedral void. The bound-mode multiplet decompositions split
more finely under the reduced symmetry. The
A
1
⊕T
2
structure of Section 4 refines under
T
d
→ C
3v
as
A
1
⊕ A
1
⊕ E
(dimensions 1 + 1 + 2), with the
T
2
triplet of FCC splitting into a
C
3v
singlet
plus doublet (
T
2
→ A
1
⊕ E
). The
T
d
-degeneracy among the three colors is lifted by the broken
symmetry. Whether this HCP refinement is physically interesting in its own right is a question we
do not pursue here; HCP is not a candidate substrate for the SSMTheory framework on the rank-4
isotropy grounds described above.
BCC: different coordination, no analog. Body-centered cubic has
K
= 8 with cubic first
shell. Tetrahedral voids in BCC exist but are distorted; the four bounding vertices are not at equal
Gauge Freedom Journal 1(1), Article 006 (2026) 13
nearest-neighbor distance from one another (the bonded graph of a tetrahedral void in BCC is not
K
4
), and the
T
d
local symmetry is broken. The matter-paper
K
4
perfect-matching construction
does not directly apply. Gauge fields on BCC were studied by Celmaster [
13
]; the related fermion
analysis on BCC, including whether a different local symmetry-based color construction is available,
is outside the scope of the present paper. The local structure of FCC and HCP — not BCC or
hypercubic Z
3
— is what supports the defect-bound-mode color algebra.
Minimally doubled fermions. Karsten [
6
], Wilczek [
7
], and Boriçi–Creutz [
8
,
9
] introduced
operators that produce exactly two continuum doublers, breaking hypercubic symmetry to obstruct
the doubling theorem. Bedaque, Buchoff, Tiburzi, and Walker-Loud [
10
] analyzed the symmetry
structure systematically. The modification is at the bulk-operator level. The defect approach
here is orthogonal: it leaves the bulk operator alone and looks at localized structure. A future
minimally-doubled construction on FCC could be combined with the defect analysis; we do not
undertake that here.
Supersymmetric lattices. Kaplan, Katz, and Ünsal [
11
] and Catterall [
12
] construct super-
symmetric lattice gauge theories on root lattices such as
A
∗
4
, exploiting algebraic structure of the
root system to preserve part of the continuum supersymmetry. Defect-bound modes are not in
scope. The FCC lattice is the
A
3
root lattice; a supersymmetric extension of the construction here
is consistent with these methods but is not pursued.
Wilson, staggered, twisted-mass. The standard hypercubic constructions [
3
,
4
,
5
] remove or
relabel doublers through mass terms, phase factors, or chirally rotated masses. All act on the bulk
operator on
Z
D
. Staggered fermions on a 3D hypercubic lattice produce 4 tastes at the cubic-BZ
corners. The FCC classes are stratified by Wilson mass at distinct scales. This differs from the
usual hypercubic Wilson pattern, where the Wilson mass depends on the number of momentum
components sitting at the edge of the Brillouin zone. The FCC spectrum derived here has zeros at
the BZ interior (Γ, four
L
-points at (
±π/
2
, ±π/
2
, ±π/
2)) and on boundary nodal lines, with three
topological classes at three Wilson masses (0, 12
/a
, 16
/a
). The two Brillouin zones have different
shapes (truncated octahedron versus cube), and no linear change of lattice variables maps one to
the other while preserving the
O
h
symmetry and nearest-neighbor structure. The FCC operator is
not a reparametrization of staggered fermions.
Lattice K First shell Point group Defect-mechanism status
Simple cubic (naive) 6 cube O
h
no analog (distorted tet void)
Simple cubic (staggered) 6 cube O
h
no analog
BCC (naive) 8 cube O
h
distorted tet void, K
4
broken [13]
HCP 12 anticuboctahedron D
3h
K
4
, K
2,2,2
preserved; reduced void symmetry; c-axis anisotropy
Minimally doubled varies varies varies bulk modification, no defect content
FCC (this work) 12 cuboctahedron O
h
K
4
, K
2,2,2
with full T
d
, O
h
at voids
Table 2: Comparison across common 3D lattices. Among the close-packed lattices, only FCC has
the cuboctahedral first shell that gives full
O
h
symmetry without a preferred axis at the rank-4
bond-tensor level. HCP shares the combinatorial bond graphs at its voids but breaks isotropy along
its c-axis at
O
(
a
2
) in the bulk dispersion and reduces the local point group at each void to a
C
3v
or
D
3d
subgroup. BCC has
K
4
-broken tetrahedral voids and does not support the matter-paper color
construction directly.
Gauge Freedom Journal 1(1), Article 006 (2026) 14
FCC is uniquely selected by three independent criteria. A substrate compatible with the
matter-paper defect-and-color picture, with rotationally invariant continuum physics, and with
maximum-density packing must satisfy three requirements simultaneously: (i) full
O
h
first-shell
symmetry, so the rank-4 bond tensor has no preferred axis beyond the standard discrete cubic
anisotropy of any 3D lattice; (ii) the
K
4
bond graph at tetrahedral interstitial voids and the
K
2,2,2
bond graph at octahedral voids, so the matter-paper
su
(3) algebra on
K
4
perfect matchings is
defined at the tetrahedral class; (iii) the maximum kissing number
K
= 12 in three dimensions and
the maximum Bravais sphere-packing density
π/
(3
√
2
)
≈
0
.
7405 established by Kepler’s theorem
(proved by Hales [
20
]), so the substrate is the densest 3D arrangement allowed by sphere-packing
geometry [
19
]. Each of the alternatives in Table 2 fails at least one of these. Simple cubic (
K
= 6)
and hypercubic
Z
3
have full
O
h
but no tetrahedral void with
K
4
bond structure, and
K
= 6 is not
the maximum kissing number. BCC (
K
= 8) has full
O
h
but its tetrahedral voids are distorted,
with bounding vertices at unequal distances and a
K
4
-broken bond graph;
K
= 8 is also below the
maximum. HCP (
K
= 12) attains the maximum kissing number and packing density and has the
same combinatorial bond graphs at its voids as FCC, but its anticuboctahedral first shell has only
D
3h
symmetry and develops a preferred c-axis at the rank-4 bond-tensor level, failing requirement (i).
FCC is the unique 3D Bravais lattice satisfying all three criteria. The construction in this paper
is therefore not lattice-agnostic. It picks out FCC specifically on independent geometric grounds
(isotropy of the bulk dispersion), combinatorial grounds (the
K
4
structure at the tetrahedral void),
and packing-theoretic grounds (maximum kissing number and density). The same selection extends
to four dimensions through the
D
4
root lattice, where the 24-cell first shell gives exact rank-4
isotropy and the FCC structure is recovered on each constant-time slice.
7 Discussion
What this paper establishes
Two specific structural results. First, the bound-mode multiplet at a tetrahedral defect of the
FCC lattice has dimension 4 and decomposes under
T
d
as
A
1
⊕ T
2
(Proposition 3), with the three-
dimensional
T
2
sub-multiplet dimensionally and
S
3
-module-isomorphic to the
K
4
matching space of
the matter paper (Proposition 4). Second, the bound-mode multiplet at an octahedral defect has
dimension 6 and decomposes under
O
h
as
A
1g
⊕ E
g
⊕ T
1u
(Proposition 5); both three-dimensional
pieces are spatial-vector representations, so the distinguishing content is combinatorial, not spectral:
the bonded graph
K
2,2,2
admits 8 perfect matchings and 30 skew-edge pairs, neither factoring as a
three-color structure (Proposition 6). The matter-paper color construction applies to tetrahedral
defects and fails for octahedral defects on both spectral-dimension and combinatorial grounds.
This supports the structural distinction underlying the quark/dark-sector split of the SSMTheory
program: tetrahedral defects with the quark-class matter, octahedral defects with the dark-sector
species proposed in the companion paper [
15
] on the same
K
2,2,2
combinatorics. The free-field bulk
classification (Theorem 2) of three topological classes satisfying the vanishing Nielsen–Ninomiya
consistency relation provides the spectral background.
Limitations
The Dirac analysis does not derive SU(3) color from the spectrum; the matter paper’s construction
is logically prior, using
K
4
combinatorics independently of the Dirac operator. Specifically: (a) the
spectral result is a dimensional consistency check (the tetrahedral triplet has the dimension of the
K
4
color space); since both bound triplets are spatial-vector representations, the spectrum does
Gauge Freedom Journal 1(1), Article 006 (2026) 15
not by itself separate the two defect classes, and the distinction between them is combinatorial,
not spectral, not a derivation of internal symmetry; (b) the multiplet decomposition is a symmetry
statement about the localized shell space, robust to the radial profile of the trapping potential
(Section 4.2), but is not an infinite-lattice bound-state theorem; (c) the octahedral result is a
symmetry-plus-combinatorics statement, with no separate finite-cluster diagonalization performed;
and (d) the physical interpretation of the octahedral sector (its mass scale, cosmological abundance,
and indirect-detection signatures) is the subject of the companion dark-matter paper [
15
] and is
neither used nor assessed here.
Open questions
Two natural extensions. First is the dynamical analysis: turning on the SU(3) gauge field of the
SSMTheory program on the FCC bonds and computing how the bound-mode multiplet responds to
gauge fluctuations would close the dynamical picture. Second is the 4D extension: an FCC slice
×
discrete time gives a
D
4
root-lattice structure with exact
SO
(4) isotropy in the long-wavelength limit;
the analogous defect-bound-mode analysis on that 4D lattice would address the propagating-fermion
content of the substrate.
Gauge Freedom Journal 1(1), Article 006 (2026) 16
Declarations
Acknowledgments
I thank a referee for a careful reading of an earlier version of this work; that feedback prompted the
present reorganization around the defect-bound-mode analysis.
AI Contributions Statement
The author used Anthropic’s Claude as an assistant for language and presentation only: refining prose
and clarity, checking the L
A
T
E
X build and formatting, cross-checking internal consistency of citations,
dimensions, point-group conventions, and notation, and drafting supplementary documentation
(README and verification-script comments). The scientific content of this work (the bulk zero-mode
classification, the defect-bound-mode and representation-theory results, the
K
4
/
K
2,2,2
analysis, and
the verification-script logic, including the finite-cluster Dirac diagonalization) was conceived, derived,
and verified by the human author, who takes full responsibility for all claims in the manuscript.
Artifact Availability
Data and code availability. A Python script reproducing all numerical and algebraic claims of this
paper is included with the submission as
dirac_verify_defects.py
; it runs 80 explicit assertions
including the finite-cluster Dirac diagonalization of Section 4 (which reproduces Table 1 across
five well depths), and exits with status 0 on success. The same script is also available online at
https://github.com/raghu91302/ssmtheory/blob/main/dirac_verify_defects.py
. The full
L
A
T
E
X source, figure-generating Python scripts, and verification script constitute the complete
reproducibility package; no separate data files are required.
Energy Disclosure
The work did not require large-scale numerical experimentation. Compute use associated with
manuscript preparation and AI-assisted drafting was limited to standard interactive research
workflows.
Conflicts of Interest
The author declares no competing interests.
Funding
No external funding was received for this work.
Ethics
No human-subjects research, animal research, or clinical data collection was conducted for this
article.
Gauge Freedom Journal 1(1), Article 006 (2026) 17
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Gauge Freedom Journal 1(1), Article 006 (2026) 18
