D
4
as Physical Spacetime:
Exact SO(4) Lattice QCD with Quarks as Tetrahedral Defects
Raghu Kulkarni
∗
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
May 2026
Abstract
We identify the D
4
root lattice as physical four-dimensional spacetime in the SSMTheory
framework, where the FCC lattice is three-dimensional physical space and not a discretization
of an underlying continuum. The choice is not free. D
4
is the densest 4D packing, has the
maximum kissing number K = 24 in R
4
, and contains the FCC lattice exactly at every constant-
time slice. No other 4D lattice does. The lattice spacing a is a fixed physical scale of order the
Planck length, not a parameter taken to zero; the continuum action of standard lattice QCD
is what long-wavelength observers see at E ≪ 1/a. From this single substrate we read off,
without adjustable inputs: the 24 nearest neighbors of D
4
partition as 12 + 12 (spatial FCC
plus temporal); the structure tensor is exactly
ˆ
S
µν
= 6δ
µν
; the naive Dirac kinetic field factors
as f
µ
(k) = (4/a) sin(k
µ
)
P
ν=µ
cos(k
ν
), with 144 zero modes in four topological classes whose
chiralities sum to 0 −48 + 64 −16 = 0 (Nielsen–Ninomiya); the Wilson gauge action has 12 link
variables, 32 triangular plaquettes, and a single coupling β per unit cell; the plaquette stiffness
tensor is exactly 48(δ
µρ
δ
νσ
− δ
µσ
δ
νρ
) per origin site, giving SO(4)-symmetric Yang–Mills with
no anisotropy correction at any sub-Planckian scale; Wilson fermion doublers acquire masses
{48, 54, 56, 64}/a, leaving one massless mode at Γ. Quarks are trapped tetrahedral defects in
spatial FCC slices [12]. Appendix A reconstructs the full su(3) Lie algebra acting on the defect’s
color Hilbert space from defect geometry alone — three colors from the skew-pair partition of K
4
,
the Weyl group S
3
from the quotient S
4
/V
4
under anchor selection, the rank-2 maximal torus
from three Bell-pair U(1) phases modulo the diagonal U(1) that gauges to electromagnetism —
with the eight generators closing on su(3) in the fundamental representation 3 via the A
2
root
system. The gauge group of the strong sector is not chosen here; it is read off the geometry.
1 Introduction
1.1 D
4
as physical spacetime in the SSM framework
This paper sits inside the SSMTheory program, whose central postulate is unusual: the Face-
Centered Cubic (FCC) lattice is three-dimensional physical space itself. Not a discretization of
an underlying continuum, not a computational scaffold — the substrate of reality at the most
fundamental level. Particles and gauge structure are defects, bonds, and stabilizer dynamics on
that substrate.
The program’s published foundation rests on three works. The matter paper [12] derives the
FCC lattice itself from a kissing-number K = 12 simulation of a K = 4 → K = 12 phase tran-
sition, identifies quarks as trapped tetrahedral defects with quantized fractional charges, three
∗
raghu@idrive.com
1
color charges, and linear confinement, and computes m
p
/m
e
= 1836 from the structural disruption
count. That is the foundational paper. The mass-energy-information equivalence paper [13] treats
the same lattice as a [[192, 130, 3]] CSS code and identifies particle masses with fault-tolerant verifi-
cation costs C
x
= E
s
×C
s
; filtering 25 candidate defect geometries through four axioms (Minimum
Topological Dimension, Sector Completeness, Boundary Closure, Kinematic Shedding) leaves ex-
actly five stable states with costs {1, 207, 273, 1836, 1839}, matching the empirical mass ratios of
the electron, muon, pion, proton, and neutron to within 0.12%. The same paper establishes emer-
gent Lorentz invariance on FCC from the structure-tensor isotropy S
FCC
µν
= 4δ
µν
and the bond-set
inversion symmetry. The CSS code paper [11] gives the explicit [[192, 130, 3]] stabilizer code on the
FCC bond graph, with weight-12 stabilizers matching the FCC kissing structure. Together, these
three works treat the spatial FCC lattice as fundamental, not as an approximation to anything
else.
A consistent four-dimensional formulation must identify physical spacetime as a 4D lattice whose
spatial slices are FCC. The hypercubic lattice Z
4
fails this requirement immediately — its slices
are Z
3
, not FCC — so it is not a candidate. The 4D lattice this paper identifies is the D
4
root
lattice: D
4
contains FCC at every constant-time slice (Theorem 1), is the densest 4D packing [2],
and has kissing number K = 24, the maximum in R
4
.
What this means operationally. The lattice spacing a is a fixed physical scale — of order
the Planck length — not a parameter to be sent to zero. The lattice points of D
4
are the discrete
events of spacetime; the edges are its elementary causal connections; the plaquettes are its smallest
closed loops. The continuum action that emerges in the large-L limit (L ≫ a) of Sections 7–8
is the effective field theory that long-wavelength observers see at sub-Planckian energies, not the
fundamental dynamics. The analogy is hydrodynamics: the Navier–Stokes equations describe the
long-wavelength behavior of molecular fluids without claiming to be the fundamental theory of
those molecules.
Standard-physics comparison. A reader coming from standard lattice QCD can read the same
mathematical content under the standard interpretation: D
4
as one of many possible discretizations
of continuum 4D Euclidean spacetime, with the a → 0 limit recovering ordinary SU(3) Yang–Mills.
The mathematics of Sections 3–8 and Appendix A is identical under either reading; the physical
meaning of a differs. We adopt the SSM reading as primary because the program’s other elements
— matter as FCC defects, color from tetrahedral defect combinatorics — require it.
1.2 Why the FCC lattice is distinguished in 3D
Three properties make FCC special in 3D. It is the densest sphere packing (Hales’ proof of the
Kepler conjecture [1]); it has the maximum kissing number K = 12 for any 3D Bravais lattice; and
it carries the isotropic structure tensor
S
FCC
µν
=
X
n∈N(FCC)
ˆn
µ
ˆn
ν
= 4δ
µν
, (1)
the largest possible value for any 3D lattice with unit-length kissing vectors. Maximal packing,
maximal kissing, isotropic structure tensor: those three together pick FCC out as a natural physical
substrate. The matter paper [12] supplies the dynamical origin — a kissing-K = 12 simulation in 3D
shows that FCC is what dense-packing dynamics with maximally symmetric coordination actually
selects. The dense bond structure of the FCC kissing graph is what the CSS code construction [11]
uses to define the stabilizer vacuum that the matter paper builds on.
2
1.3 D
4
as the unique 4D extension
For the SSM program to extend consistently to four dimensions, the spacetime lattice must satisfy
three conditions:
• contain the 3D FCC lattice as a constant-time slice (so spatial geometry is preserved at every
time),
• be the densest 4D packing (so the 4D substrate is maximally rigid), and
• carry an isotropic structure tensor (so 4D Euclidean rotational symmetry survives at the
lattice level).
D
4
satisfies all three. It is the densest 4D lattice packing [2], with kissing number K = 24
saturating the maximum in R
4
. Its coordination polytope is the 24-cell, the unique self-dual regular
4-polytope, with full Weyl group F
4
of order 1152 [3]. We show in Section 3 that FCC is exactly the
x
4
= 0 slice of D
4
, and in Proposition 1 that the structure tensor with unit-length normalization is
ˆ
S
µν
= 6δ
µν
— a factor of 3/2 above the FCC value. D
4
is therefore the unique 4D substrate that
extends physical FCC to 4D in a structurally consistent way.
The hypercubic lattice Z
4
fails the first condition: any constant-x
4
slice of Z
4
is Z
3
, not FCC.
It also has only K = 8 nearest neighbors and produces cubic anisotropy at order a
2
[5], breaking
SO(4) to the cubic subgroup at sub-Planckian scales. Under SSM, Z
4
is not a candidate at all;
under standard lattice QCD it remains the conventional choice but with the structural drawbacks
Section 10 catalogs.
Non-hypercubic lattice gauge theory is a small literature. Celmaster studied gauge fields on the
body-centered hypercubic lattice [7]; Kaplan’s domain-wall construction [8] uses a 5D embedding
of Z
4
rather than a non-hypercubic 4D substrate. The present paper differs from both: D
4
is not
introduced here as a discretization but as the physical 4D substrate of the SSM framework.
1.4 Scope of this paper
Treating D
4
as the physical 4D substrate, we construct the lattice gauge theory it supports and
verify that its long-wavelength behavior reproduces standard SU(3) Yang–Mills with no anisotropy
correction. The lattice spacing a is fixed (of order the Planck length); the continuum action that
arises in the large-L limit is the effective theory low-energy observers see, not the fundamental
theory. The main results:
Main results.
1. Slicing structure (Section 3). The 3D FCC lattice sits inside D
4
as the slice x
4
= 0. The 24
nearest neighbors of D
4
partition as 12 + 12: twelve spatial neighbors (the FCC kissing set)
and twelve cross-slice temporal neighbors (six forward, six backward).
2. Factorization theorem (Section 4). The naive Dirac kinetic field on D
4
factors exactly as
f
µ
(k) = (4/a) sin(k
µ
)
P
ν=µ
cos(k
ν
). The 3D specialization to FCC recovers a corresponding
two-cosine factorization for the 3D naive Dirac operator.
3. Zero-mode classification and Nielsen–Ninomiya (Section 5). The zero set decomposes into
four topological classes with chiralities summing to 0 − 48 + 64 − 16 = 0.
4. Wilson gauge action (Section 6). Per unit cell of D
4
: 1 site, 12 unoriented edges (link variables
U
e
), and 32 triangular plaquettes. The 24-cell has only triangular faces, so the Wilson action
has a single coupling.
3
5. Plaquette stiffness and effective SO(4) invariance (Section 7). The plaquette stiffness tensor
is exactly
P
△
b
△
µν
b
△
ρσ
= 48(δ
µρ
δ
νσ
− δ
µσ
δ
νρ
) per origin site, giving full 4D Euclidean SO(4)
isotropy at every order in a. The effective long-wavelength action is therefore exactly SO(4)-
invariant with no anisotropy correction at any sub-Planckian scale.
6. Wilson fermion spectrum (Section 8). The Wilson term lifts every doubler to a positive mass;
the masses are {48, 54, 56, 64}/a for the various doubler classes. The all-π Case-1 zero is
identified with Γ through the reciprocal lattice 2πD
∗
4
, leaving exactly one massless physical
fermion.
7. Quarks as tetrahedral defects (Section 9). The Wilson action constructed here is the natural
lattice-wide propagation of the local SU(3) color action of a trapped tetrahedral defect, as
developed in [12]; Appendix A reconstructs the full su(3) Lie algebra from the defect’s geom-
etry alone — three colors from the skew-pair partition of K
4
, the Weyl group S
3
from anchor
selection, the rank-2 maximal torus from Bell-pair phases modulo the diagonal U(1) of elec-
tromagnetism — with the eight generators closing on su(3) in the fundamental representation
3.
Section 10 discusses what is rigorously established here and what remains for further work, including
the exact value of the dimensionless prefactor α in g
2
eff
= α/β, the completeness of doubler removal
after BZ-equivalence reduction, and the hadronic spectrum.
Part I
Geometric Foundation
2 The D
4
Root Lattice
D
4
is the index-2 even sublattice of Z
4
:
D
4
=
n
(x
1
, x
2
, x
3
, x
4
) ∈ Z
4
: x
1
+ x
2
+ x
3
+ x
4
≡ 0 (mod 2)
o
. (2)
Equivalently, D
4
is the integer span of the simple roots
α
1
= e
1
− e
2
, α
2
= e
2
− e
3
, α
3
= e
3
− e
4
, α
4
= e
3
+ e
4
, (3)
of the simply-laced Lie algebra so(8), with e
i
the standard basis vectors of R
4
. A fundamental cell
has volume 2, half the density of Z
4
.
2.1 Nearest neighbors and the kissing set
The minimum norm of D
4
is
√
2, realized by
N(D
4
) = {±e
i
± e
j
: 1 ≤ i < j ≤ 4}, (4)
which contains
4
2
· 2
2
= 24 vectors. That is the kissing number of D
4
, and it equals the kissing
number of R
4
[2]. Each kissing vector has |n|
2
= 2.
4
2.2 Structure tensor and 4D isotropy
Proposition 1 (Isotropy of D
4
). The structure tensor of D
4
,
S
µν
≡
X
n∈N(D
4
)
n
µ
n
ν
, (5)
satisfies S
µν
= 12 δ
µν
. With unit-length normalization ˆn = n/|n| = n/
√
2, this becomes
ˆ
S
µν
= 6 δ
µν
.
Proof. Diagonality and isotropy follow by direct enumeration.
Off-diagonal vanishing. For µ = ν, the only n ∈ N(D
4
) contributing to S
µν
are the four vectors
±e
µ
± e
ν
:
X
±±
(±1)(±1) = (+1)(+1) + (+1)(−1) + (−1)(+1) + (−1)(−1) = 0. (6)
Diagonal equality. For fixed µ, the contributing n are ±e
µ
± e
ν
for ν = µ: three choices of ν
times four sign combinations gives 12 vectors, each contributing n
2
µ
= 1. So S
µµ
= 12, and by index
symmetry the four diagonal entries are all equal.
For unit-length vectors ˆn = n/
√
2, ˆn
µ
ˆn
ν
= n
µ
n
ν
/2, giving
ˆ
S
µν
= 6δ
µν
.
This is the 4D analog of FCC’s S
FCC
µν
= 4δ
µν
. It guarantees that the leading-order discrete
kinetic action on D
4
is Euclidean SO(4)-invariant with no anisotropy correction. The corresponding
result on FCC — emergent 3D rotational invariance from the same structure-tensor argument plus
bond-set inversion symmetry — is the content of [13]; the present construction is its 4D extension,
with the additional content of an isotropic temporal direction in the D
4
bond set.
2.3 The 24-cell and the Weyl group F
4
The 24 nearest-neighbor vectors of D
4
are the vertex set of a 24-cell [3], the unique self-dual regular
4-polytope. It has V = 24 vertices, E = 96 edges, F = 96 triangular faces, and C = 24 octahedral
cells, satisfying the Euler relation
V − E + F − C = 24 − 96 + 96 − 24 = 0. (7)
Every face is an equilateral triangle; there are no square faces. This single-plaquette-type structure
is what makes the gauge action in Section 6 take a single-coupling form.
The 24-cell’s symmetry group is the Weyl group F
4
of order 1152. It contains the FCC point
group O
h
(order 48) as a proper subgroup, with index 1152/48 = 24 — matching the number of
vertices.
3 Slicing Structure: D
4
= FCC × Discrete Time
D
4
contains the 3D FCC lattice as an affine 3D slice.
Theorem 1 (Slicing). Let D
4
∩ {x
4
= 0} denote the points of D
4
with x
4
= 0. Then
D
4
∩ {x
4
= 0} = FCC, (8)
where FCC is the 3D face-centered cubic lattice {(x
1
, x
2
, x
3
) ∈ Z
3
: x
1
+ x
2
+ x
3
≡ 0 (mod 2)}.
Proof. (x
1
, x
2
, x
3
, 0) ∈ D
4
iff x
1
+x
2
+x
3
+0 ≡ 0 (mod 2), which is the FCC condition on (x
1
, x
2
, x
3
).
The same argument works for any slice x
4
= c with c ∈ Z: even-c slices give FCC, odd-c slices give
the shifted FCC sublattice with x
1
+ x
2
+ x
3
odd.
5
3.1 Decomposition of the 24 neighbors
Slicing induces a clean decomposition of the kissing set N(D
4
) by the value of the fourth component.
Proposition 2 (12 + 12 decomposition). The 24 nearest neighbors of D
4
partition by n
4
as
N(D
4
) = N
spatial
⊔ N
+
temporal
⊔ N
−
temporal
, (9)
where
N
spatial
= {±e
i
± e
j
: 1 ≤ i < j ≤ 3}, |N
spatial
| = 12, (10)
N
±
temporal
= {±e
i
+ (±1)e
4
: 1 ≤ i ≤ 3}, |N
±
temporal
| = 6. (11)
Proof. Every n ∈ N(D
4
) is of the form ±e
i
± e
j
for some i < j. If j < 4, then n
4
= 0 and
n ∈ N
spatial
; count is
3
2
· 4 = 12. If j = 4, then n
4
= ±1 and n ∈ N
±
temporal
; count is 3 · 2 = 6 for
each sign.
The spatial subset N
spatial
is the FCC kissing set, exactly. The two temporal subsets connect
adjacent time slices through six forward and six backward bond vectors per site. Figure 1 shows
the decomposition.
1.0
0.5
0.0
0.5
1.0
x
1
1.0
0.5
0.0
0.5
1.0
x
2
1.0
0.5
0.0
0.5
1.0
x
3
(a) Spatial: 12 FCC neighbors at
x
4
= 0
1.0
0.5
0.0
0.5
1.0
x
1
1.0
0.5
0.0
0.5
1.0
x
2
1.0
0.5
0.0
0.5
1.0
time
x
4
x
4
= +1
x
4
= 1
x
4
= 0
(b) Temporal:
6 + 6
cross-slice neighbors
Figure 1: Decomposition of the 24 nearest neighbors of D
4
by time component n
4
. (a) The 12 spatial
neighbors with n
4
= 0 form the FCC kissing set in the x
4
= 0 slice. (b) The 12 cross-slice neighbors,
six with n
4
= +1 (green) and six with n
4
= −1 (blue), connect the origin to neighboring time slices.
The temporal bonds are diagonal rather than straight, in contrast to standard hypercubic lattice
gauge theory.
3.2 Symmetry breaking F
4
→ O
h
× Z
2
Picking the fourth direction as “time” breaks the 24-cell’s F
4
symmetry down to the subgroup
preserving the slicing. That subgroup is O
h
×Z
2
, with O
h
acting on spatial R
3
and Z
2
flipping the
6
time direction. The index
|F
4
|
|O
h
× Z
2
|
=
1152
96
= 12 (12)
matches the number of inequivalent time-direction choices within the 24-cell (the 24 vertices pair
up into 12 antipodal axes).
The breaking F
4
→ O
h
×Z
2
is the standard analog of breaking 4D Euclidean rotational symme-
try to spatial-rotation × time-reversal in any lattice formulation with a preferred time. The full-F
4
case is the Wick-rotated Euclidean version; the O
h
× Z
2
case is the Hamiltonian formulation.
Part II
Naive Dirac Spectrum
4 The Naive Dirac Operator on D
4
4.1 Construction
The naive Dirac operator on D
4
takes the standard lattice-fermion form. For a spinor field ψ(x)
on D
4
, the naive lattice action is
S =
X
x
¯
ψ(x) D(x) ψ(x), D = iγ
µ
f
µ
(∂), (13)
with kinetic field
f
µ
(k) =
1
a
X
n∈N(D
4
)
n
µ
sin(k · n) (14)
in momentum space. Here a is the lattice spacing and γ
µ
are 4D Euclidean Dirac gamma matrices.
Zero modes of D occur at momenta k where f (k) = 0 as a vector in R
4
.
4.2 Factorization theorem
Theorem 2 (D
4
factorization). The kinetic field of Eq. (14) factors as
f
µ
(k) =
4
a
sin(k
µ
)
X
ν=µ
cos(k
ν
) =
4
a
sin(k
µ
) [T (k) − cos(k
µ
)] , (15)
where T (k) ≡
P
4
ν=1
cos(k
ν
).
Proof. The 24 neighbors split into six sheets S
ij
= {±e
i
± e
j
} for 1 ≤ i < j ≤ 4, each holding four
vectors. Only sheets containing index µ contribute to f
µ
; for fixed µ, that is the three sheets S
µν
with ν = µ (treating S
ij
and S
ji
as the same sheet for indexing).
For sheet S
µν
with µ < ν, the four vectors (n
µ
, n
ν
) ∈ {(±1, ±1)} contribute
X
n∈S
µν
n
µ
sin(k · n) = (+1) sin(k
µ
+ k
ν
) + (+1) sin(k
µ
− k
ν
)
+ (−1) sin(−k
µ
+ k
ν
) + (−1) sin(−k
µ
− k
ν
)
= 2 sin(k
µ
+ k
ν
) + 2 sin(k
µ
− k
ν
)
= 4 sin(k
µ
) cos(k
ν
), (16)
7
using sin A + sin B = 2 sin
A+B
2
cos
A−B
2
. The case µ > ν contributes identically by re-indexing.
Summing over the three values of ν = µ gives
f
µ
(k) =
1
a
· 4 sin(k
µ
)
X
ν=µ
cos(k
ν
), (17)
which is Eq. (15). The second form f
µ
= (4/a) sin(k
µ
)[T (k) −cos(k
µ
)] follows from
P
ν=µ
cos(k
ν
) =
T (k) − cos(k
µ
).
Restricting Eq. (15) to a single time slice (3D FCC) gives a two-cosine factorization f
µ
(k) =
(4/a) sin(k
µ
)[cos(k
ν
) + cos(k
ρ
)], with (µ, ν, ρ) a permutation of (1, 2, 3). The 4D version has three
cosines, one per direction transverse to µ.
5 Zero Modes and Topological Indices
5.1 Complete classification
Theorem 3 (Zero-mode classification). A momentum k in the first Brillouin zone of D
4
satisfies
f(k) = 0 if and only if, for each index µ ∈ {1, 2, 3, 4}, at least one of the following holds:
(S
µ
) sin(k
µ
) = 0, i.e., k
µ
∈ {0, π};
(C
µ
) cos(k
µ
) = T (k), where T (k) =
P
ν
cos(k
ν
).
Enumerating the 2
4
= 16 choices of (S
µ
) vs (C
µ
) for each µ yields four classes of solutions:
1. Case 1 (all S): 16 zeros at k ∈ {0, π}
4
.
2. Case 2 with n
C
= 2 (class L
2
): 48 zeros. The two S indices have k-values forming an even
split (one at 0, one at π, so that T
S
= 0), and the two C indices have k = ±π/2.
3. Case 2 with n
C
= 3 (class L
3
): 64 zeros. The single S index has k
S
∈ {0, π} giving T
S
= ±1,
and the three C indices have k
C
= ±arccos(∓1/2), that is, ±π/3 or ±2π/3.
4. Case 2 with n
C
= 4 (class L
4
): 16 zeros, all of the form k
µ
= ±π/2.
Total: 144 zero momenta.
Proof. From Theorem 2, f
µ
(k) = 0 requires either sin(k
µ
) = 0 or T (k) − cos(k
µ
) = 0, that is,
condition (S
µ
) or (C
µ
). Each µ contributes one such condition independently, yielding 2
4
= 16
cases.
Let n
C
be the number of indices satisfying (C
µ
). The C-equations all read cos(k
µ
) = T for µ in
the C-set, so all C-cosines share a common value T . Self-consistency of T =
P
ν
cos(k
ν
) = T
S
+n
C
T
(where T
S
is the cosine sum over the S indices) gives
T (1 − n
C
) = T
S
, i.e., T =
T
S
1 − n
C
(n
C
= 1). (18)
n
C
= 0 (all S). No constraint beyond k
µ
∈ {0, π}; 16 solutions.
n
C
= 1. The self-consistency reduces to T
S
= 0, but T
S
is a sum of three ±1 values (each
cos(k
µ
) for µ ∈ S, |S| = 3) and is therefore odd, so T
S
= 0 has no solution. No solutions in this
case.
n
C
= 2. Then T
S
is a sum of two ±1 values, so T
S
∈ {−2, 0, +2}. The constraint T = −T
S
and |T | ≤ 1 force T
S
= 0, hence T = 0. The two S values must be one 0 and one π (in either
8
order, contributing the factor of 2 below). The two C values satisfy cos(k
C
) = 0, so k
C
= ±π/2.
Counting:
4
2
= 6 choices of which two indices are C, ×2 orderings of 0 and π on the S indices,
×4 sign combinations on the C indices, totalling 6 · 2 · 4 = 48 zeros.
n
C
= 3. Then T
S
is a single ±1 (one S index), and T = −T
S
/2 = ∓1/2. The three C values
satisfy cos(k
C
) = ∓1/2, so k
C
= ±arccos(∓1/2) ∈ {±π/3, ±2π/3}. Counting: 4 choices of which
index is S, ×2 values of k
S
∈ {0, π}, ×2
3
sign combinations on the three C indices, totalling
4 · 2 · 8 = 64 zeros.
n
C
= 4. Then T
S
= 0 (no S indices) and T = 0. All four cos(k
µ
) = 0, so k
µ
= ±π/2 for every
µ. Counting: 2
4
= 16 zeros.
Total: 16 + 0 + 48 + 64 + 16 = 144 zero momenta.
5.2 Chirality computation
The topological index (chirality) at an isolated zero k
∗
is χ(k
∗
) = sgn det J(k
∗
), where the Jacobian
is
J
µν
(k) =
∂f
µ
∂k
ν
=
1
a
X
n∈N(D
4
)
n
µ
n
ν
cos(k · n). (19)
Chirality at Case-1 zeros. At k with each k
µ
∈ {0, π}, the Jacobian is diagonal with entries
J
µµ
(k) =
1
a
X
n∈N(D
4
)
n
2
µ
cos(k · n). (20)
For µ fixed, the contributing neighbors are those with n
µ
= 0, namely the 12 vectors of the form
±e
µ
± e
ν
for ν = µ. At a Case-1 momentum, cos(k · n) = cos(±k
µ
± k
ν
) = cos(k
µ
) cos(k
ν
) (the
cross terms cancel by the ± symmetry). Summing,
J
µµ
=
4
a
cos(k
µ
)
X
ν=µ
cos(k
ν
) =
4
a
cos(k
µ
)
T (k) − cos(k
µ
)
, (21)
so det J =
Q
µ
J
µµ
= (4/a)
4
Q
µ
cos(k
µ
)[T − cos(k
µ
)]. The sign of det J is then determined by the
number of π-values in k.
Direct computation gives chirality χ(k) = (−1)
n
π
(k)
for the 14 isolated Case-1 zeros (excluding
the two zeros where T − cos(k
µ
) = 0 for some µ, which collapse to the BZ identification discussed
below). The 16 Case-1 zeros split as 8 with χ = +1 (n
π
even) and 8 with χ = −1 (n
π
odd),
summing to zero.
Chirality at L
2
, L
3
, L
4
zeros. Direct evaluation of det J at representative momenta in each
class gives, respectively, det J/a
4
= −256, −588, and −768. All zeros within a given class share
the same chirality sign by the invariance of det J under the class-defining sign-flips. The chirality
assignments are:
χ(L
2
) = −1, χ(L
3
) = +1, χ(L
4
) = −1. (22)
5.3 Nielsen–Ninomiya verification
The Nielsen–Ninomiya theorem [9, 10] requires
X
k
∗
:f(k
∗
)=0
χ(k
∗
) = 0 (23)
9
for any local, Hermitian, translation-invariant lattice Dirac operator. Summing over the four classes,
X
χ = 0
|{z}
Case 1
+ 48 · (−1)
| {z }
L
2
+ 64 · (+1)
| {z }
L
3
+ 16 · (−1)
| {z }
L
4
= 0 − 48 + 64 − 16 = 0. ✓ (24)
Figure 2 displays the contributions.
Case 1
(mixed
0/
)
L
2
(
n
C
= 2
)
L
3
(
n
C
= 3
)
L
4
(
n
C
= 4
)
40
20
0
20
40
60
Chirality contribution
i
+8
8
48
+64
16
= +0
,
X
,
M
,
R
= 48
2 of 4 at
± /2
= +64
3 of 4 at
± /3, ± 2 /3
= 16
4 of 4 at
± /2
Nielsen--Ninomiya on
D
4
:
0 48 + 64 16 = 0
i
= +1
i
= 1
Figure 2: Chirality contributions from the four classes of zero modes on D
4
. Case 1 zeros at
high-symmetry corners (Γ, X, M, R) self-cancel (Σχ = 0). The three classes L
2
, L
3
, L
4
at arccos-
determined momenta contribute −48, +64, −16, respectively. The total 0−48+64 −16 = 0 verifies
the Nielsen–Ninomiya theorem.
5.4 Comparison to FCC and hypercubic lattices
Table 1 sets the D
4
zero-mode spectrum against FCC and hypercubic. D
4
has more class structure
than FCC (four classes versus three), and is qualitatively different from hypercubic, where all
doublers are geometrically equivalent under the cubic symmetry.
Part III
Wilson Lattice Gauge Theory on D
4
6 Wilson Gauge Action
6.1 Per-unit-cell data
Proposition 3 (Cell counts). A fundamental unit cell of D
4
contains: 1 site, 12 unoriented edges
(link variables U
e
∈ SU(3)), and 32 triangular plaquettes.
Proof. The fundamental cell has 4-volume 2 in standard units and contains one lattice point. Each
site has 24 oriented edges to its 24 nearest neighbors; double-counting between adjacent sites gives
10
Property Hypercubic Z
4
FCC (3D) D
4
(4D)
Kissing number K 8 12 24
Densest packing in dim? no yes (3D) yes (4D)
Isotropic S
µν
? yes yes yes
Plaquette types 1 (square) 2 (tri + sq) 1 (tri only)
Doubler classes 1 3 4
Zero count (raw) 16 (corners) 1 + 4 + lines 144 (3 isolated classes + Case 1)
P
χ = 0 verification ✓ ✓ (+1 − 4 + 3) ✓ (0 − 48 + 64 − 16)
Table 1: Comparison of naive Dirac spectra on hypercubic, FCC, and D
4
lattices. The D
4
spectrum
has the most class structure, with four topological classes and the largest number of doublers,
reflecting the lattice’s higher coordination number (K = 24).
24/2 = 12 unoriented edges per cell. For plaquettes: every triangular face of the 24-cell coordination
polytope is incident to 3 vertices, so the 96 triangles at a single origin site contribute 96/3 = 32 to
each cell.
These counts can be checked on a 4
4
periodic D
4
lattice with even-parity sites. The lattice
has 128 sites; direct enumeration gives 1536 edges and 4096 triangles — exactly 12 edges and 32
triangles per site. Figure 3 shows the data structure.
6.2 The single-coupling Wilson action
The Wilson formulation assigns a group element U
e
∈ SU(N) to each oriented edge e = (x, x + n),
with U
−e
= U
†
e
for the reverse orientation. For a triangular plaquette △ = (x, x + n
1
, x + n
2
) with
edges e
1
, e
2
, e
3
traversed counterclockwise, the plaquette holonomy is
U
△
= U
e
1
U
e
2
U
e
3
, U
e
1
= U(x → x + n
1
), etc. (25)
The Wilson action is
S
G
= β
X
△∈P
1 −
1
N
Re Tr U
△
(26)
where the sum is over all triangular plaquettes P in the lattice and β is the single inverse-coupling
parameter. For SU(3): N = 3 and β = 2N/g
2
0
= 6/g
2
0
at tree level, with g
0
the bare lattice
coupling.
Why a single coupling. The hypercubic lattice has only square plaquettes and so also gives a
single coupling. The FCC cuboctahedron has both 8 triangular and 6 square faces per coordination
cell — two couplings β
T
and β
S
, whose ratio must be either fixed by the framework or treated as a
renormalization-group parameter [12]. The 24-cell, D
4
’s coordination polytope, has only triangular
faces, so Eq. (26) is the most general D
4
Wilson gauge action consistent with F
4
symmetry on the
coordination polytope (and with O
h
× Z
2
on a sliced 3+1D lattice).
11
1.0
0.5
0.0
0.5
1.0
x
1
1.0
0.5
0.0
0.5
1.0
x
2
1.0
0.5
0.0
0.5
1.0
x
3
x
x
+
n
1
x
+
n
2
U
1
U
2
U
3
(a) Plaquette holonomy
U
=
U
1
U
2
U
3
Sites Edges
(link
U
e
)
Plaquettes
(holonomy
U
)
0
5
10
15
20
25
30
35
Count per
D
4
unit cell
1
12
32
(b) Wilson action data structure
Figure 3: Wilson action data structure on D
4
. (a) An elementary triangular plaquette (x, x +
n
1
, x + n
2
) with link variables U
1
, U
2
, U
3
on its three oriented edges; the plaquette holonomy is
U
△
= U
1
U
2
U
3
. (b) Counts per D
4
unit cell: one site, 12 edges (link variables), and 32 triangular
plaquettes (holonomies).
7 Plaquette Stiffness and the Continuum Limit
7.1 Plaquette stiffness tensor
For a triangular plaquette △ = (x, x+n
1
, x+n
2
), the antisymmetric bivector encoding its orientation
in the lattice is
b
△
µν
= (n
1
)
µ
(n
2
)
ν
− (n
1
)
ν
(n
2
)
µ
. (27)
The plaquette stiffness tensor is the sum of bivector products over all triangles incident to a fixed
site:
T
µνρσ
≡
X
△∋x
b
△
µν
b
△
ρσ
. (28)
This tensor controls the leading-order continuum limit of the Wilson action — it decides whether
the recovered Yang–Mills action is isotropic, and what coefficient relates β to g
2
.
Theorem 4 (Plaquette stiffness on D
4
). For the 96 triangular plaquettes incident to any site of
D
4
,
T
µνρσ
= c
δ
µρ
δ
νσ
− δ
µσ
δ
νρ
, c = 48, (29)
exactly. The proportionality is component-wise, not on-average-only: T
µνρσ
vanishes on every
tensor structure other than (δ
µρ
δ
νσ
− δ
µσ
δ
νρ
).
Proof. Direct enumeration. The 96 triangles incident to the origin are indexed by ordered pairs
(n
1
, n
2
) of D
4
nearest neighbors with n
2
−n
1
∈ N(D
4
) (the third edge of the triangle). For each pair
we compute b
△
µν
from Eq. (27) and accumulate b ⊗b into T
µνρσ
. Numerically, T
0101
= 48 and every
component of T matches 48(δδ −δδ) to machine precision; the Frobenius difference ∥T −c(δδ −δδ)∥
is zero.
12
Analytically, the result follows from F
4
invariance of the sum over triangles at a vertex. Any
rank-(2, 2) tensor antisymmetric in the first pair and the second pair, invariant under F
4
⊃ SO(4),
has to be proportional to the unique SO(4)-invariant rank-(2, 2) tensor of that symmetry type,
(δ
µρ
δ
νσ
− δ
µσ
δ
νρ
).
Per unit cell (1/3 of triangles at any origin site, since each triangle touches 3 vertices), T
cell
µνρσ
=
16(δδ − δδ). Per unit 4-volume (cell volume = 2), the density is T
density
µνρσ
= 8(δδ − δδ).
7.2 Continuum limit
Expanding U
e
≈ exp(igaA
µ
(n)
µ
) for an edge in direction n and evaluating the plaquette holonomy
to second order in ga,
Re Tr U
△
≈ N −
1
2
g
2
Tr(F
µν
F
ρσ
) A
△
µν
A
△
ρσ
+ O(g
4
, a
6
), (30)
where A
△
µν
=
1
2
b
△
µν
is the signed area bivector. Substituting into the Wilson action Eq. (26) gives
S
G
= β
X
△
1
2N
g
2
Tr(F
µν
F
ρσ
)A
△
µν
A
△
ρσ
+ O(g
4
)
=
βg
2
8N
Tr(F
µν
F
ρσ
) T
total
µνρσ
, (31)
with T
total
µνρσ
the stiffness summed over all plaquettes in the lattice. Using the per-volume density
and the exact isotropy of Eq. (29),
S
G
−−−→
a→0
1
4 g
2
eff
Z
d
4
x Tr(F
µν
F
µν
) (32)
with g
2
eff
= α/β for some numerical α that depends on the conventions used in tracking bivector
normalizations.
What the isotropy guarantees. T
µνρσ
∝ (δ
µρ
δ
νσ
− δ
µσ
δ
νρ
) is the only tensor of its symmetry
type compatible with SO(4). The leading-order continuum action is therefore Eq. (32) without
added anisotropy terms like
P
µ
Tr(F
2
µν
), which would break SO(4) to a cubic subgroup the way
the standard hypercubic action does at O(a
2
). On D
4
, the lattice corrections at O(a
2
) are absent
at the level of the gauge action; the first deviations from continuum Yang–Mills enter at O(a
4
)
through higher-derivative terms invariant under the residual lattice symmetry.
What the isotropy does not fix. The dimensionless prefactor α that relates g
2
eff
to β depends
on how the bivector normalization, area normalization, and plaquette density are tracked through
the expansion. We do not pin α to a specific number here. Numerical perturbative matching at
one loop — comparing D
4
lattice perturbation theory to continuum MS at a chosen scale — is the
standard way to fix this constant [6] and is the natural next step toward quantitative lattice work
on D
4
.
13
Part IV
Wilson Fermions and Quarks as Defects
8 Wilson Fermions on D
4
8.1 Action and naive Dirac operator
The naive Dirac action of Section 4 couples to gauge links as
S
naive
F
=
1
2a
X
x
X
n∈N(D
4
)
¯
ψ(x) γ
µ
ˆn
µ
U
x,n
ψ(x + an) + h.c., (33)
where ˆn = n/|n| is the unit vector and U
x,n
is the link variable connecting x to x + an. The
momentum-space kinetic field is given by Theorem 2, with zero modes (doublers) classified by
Theorem 3.
8.2 Wilson term
The standard Wilson term [4] on D
4
adds the lattice-Laplacian-like operator
W ψ(x) =
r
a
X
n∈N(D
4
)
ψ(x) − U
x,n
ψ(x + an)
+ h.c., (34)
giving the momentum-space Wilson mass
m
W
(k) =
2r
a
X
n∈N(D
4
)
1 − cos(k · n)
. (35)
The Wilson parameter r is dimensionless; we work with the standard choice r = 1 throughout.
8.3 Wilson masses at every zero-mode class
Evaluating Eq. (35) at one representative momentum per class:
Proposition 4 (Wilson spectrum). The Wilson mass values at the zero-mode classes of D
4
(with
r = 1, a = 1) are
Class m
W
(units of 1/a) Identification
Γ = (0, 0, 0, 0) 0 physical fermion
X = (π, 0, 0, 0) and perms 48 Case-1 doubler, n
π
= 1
M = (π, π, 0, 0) and perms 64 Case-1 doubler, n
π
= 2
R = (π, π, π, 0) and perms 48 Case-1 doubler, n
π
= 3
all-π = (π, π, π, π) 0 identified with Γ, see below
L
2
56 Case-2 doubler, n
C
= 2
L
3
54 Case-2 doubler, n
C
= 3
L
4
48 Case-2 doubler, n
C
= 4
All non-Γ doublers have m
W
≥ 48/a, and the physical fermion at Γ remains massless.
The full Wilson spectrum is shown in Figure 4.
14
X R
L
4
L
3
L
2
M
0
10
20
30
40
50
60
70
Wilson mass
m
W
(units of
1/
a
,
r
= 1
)
m
W
= 0
48/
a
48/
a
48/
a
54/
a
56/
a
64/
a
Wilson fermion spectrum on
D
4
: physical fermion at , doublers at
m
W
48/
a
Physical fermion
Case 1 doubler (mixed
0/
)
Case 2 doubler (arccos)
Figure 4: Wilson fermion mass spectrum on D
4
at r = 1. The physical fermion at Γ remains
massless; all seven doubler classes acquire Wilson mass m
W
≥ 48/a and decouple in the continuum
limit. Case-1 doublers (mixed 0/π momenta) and Case-2 doublers (arccos-determined momenta)
span comparable mass ranges, with X = R = L
4
= 48/a as the lightest doubler tier.
8.4 The all-π identification
The Wilson term vanishes at k = (π, π, π, π) as well as at Γ = (0, 0, 0, 0), which would naively give
two physical massless fermions. But the reciprocal lattice of D
4
is
D
∗
4
= D
4
∪ (D
4
+ ω
4
) = Z
4
∪
Z
4
+ (
1
2
,
1
2
,
1
2
,
1
2
)
, (36)
with fundamental weight ω
4
= (
1
2
,
1
2
,
1
2
,
1
2
) generating the half-integer coset. The reciprocal vector
g = 2πω
4
= π(1, 1, 1, 1) therefore lies in 2πD
∗
4
, and the all-π point is BZ-equivalent to Γ:
(π, π, π, π) − π(1, 1, 1, 1) = (0, 0, 0, 0). (37)
The two zeros of the Wilson mass correspond to a single momentum in the first Brillouin zone,
leaving one physical massless fermion. The same BZ-equivalence identification arises in the 3D
FCC case via the corresponding half-integer reciprocal vector.
8.5 Doubler removal: status
The Wilson term as written lifts every doubler to a positive mass, and the all-π Case-1 zero is
identified with Γ. The result, at the level of the explicit Wilson masses computed in Eq. (35), is
exactly one physical fermion.
A complete analysis of doubler removal also requires the BZ equivalences within each Case-2
class. The 48 L
2
, 64 L
3
, and 16 L
4
raw zeros include pairs related by reciprocal-lattice translations
— notably the π(1, 1, 1, 1) shift, which permutes signs of ±π/2 entries and shifts arccos angles.
Enumerating BZ-inequivalent doubler points within each class, and checking that Nielsen–Ninomiya
15
holds on the reduced set, is a technical calculation that we leave to numerical lattice work. The raw
chirality sum of Eq. (24) establishes the topological consistency of the spectrum; the equivalence-
class accounting is the more delicate finishing step.
9 Quarks as Trapped Tetrahedral Defects
9.1 The two viewpoints
The lattice gauge theory constructed above admits two views of what a quark is. In the standard
bulk-field viewpoint, ψ(x) is a spinor at every D
4
site, carrying color, flavor, and spin indices, gauge-
coupled through the link variables U
e
. That is the conventional lattice QCD setup adapted to D
4
,
and the Wilson fermion analysis of Section 8 is its concrete realization.
In the defect viewpoint of the matter paper [12], a physical quark is a localized lattice defect
— a trapped extra node at the centroid of a tetrahedral void in a spatial FCC slice x
4
= t. The
defect bonds to four FCC sites by the edges of the surrounding tetrahedron, and the matter paper
builds color, charge, and confinement on that localized object. The point of this section is not to
re-derive the matter paper but to check that its defects are compatible with the D
4
Wilson gauge
action of Sections 6–8.
9.2 Compatibility with the Wilson action
The slicing theorem (Theorem 1) makes the compatibility almost immediate: any spatial defect
configuration on the 3D FCC lattice embeds unchanged into the x
4
= t slice of D
4
. A defect at
spatial position x in slice t couples to two kinds of gauge link from the D
4
Wilson action:
• In-slice links: the spatial valence-bond links from the defect site to its FCC neighbors. These
are the 12 spatial neighbors of Proposition 2.
• Cross-slice links: the 12 temporal neighbors connecting (x, t) to (x+ δ, t ±1) for δ ∈ N
spatial
/2
(the diagonal time bonds).
The in-slice links carry the spatial gauge structure — the same SU(3) link variables that act
on the defect’s color state in the matter paper [12]. The cross-slice links propagate that color state
from one time slice to the next.
Two structural conditions are needed for the viewpoints to agree:
1. The color Hilbert space of a single quark transforms under SU(3) as a fundamental (3). This
is the universal property of quarks across every formulation of QCD; it holds by construction
in the matter paper [12].
2. The gauge dynamics on the defect’s valence bonds is the dynamics generated by the Wilson
action Eq. (26) on those same bonds. This is automatic — the valence bonds are bonds of
the underlying D
4
lattice, and the Wilson action assigns SU(3) link variables to every such
bond.
Both conditions hold. The first says that the local color rotations of the defect generate su(3)
in the 3; the second says that those rotations are propagated across the lattice by the D
4
Wilson
field. The two viewpoints describe the same SU(3) gauge theory at different operational levels: the
matter paper’s defect framework says what a quark is as a localized lattice configuration; the D
4
Wilson action says how the gauge field acts on it.
16
Appendix A closes the second half of the picture. It rebuilds su(3) on the defect’s color Hilbert
space from defect geometry alone — three colors from the skew-pair partition of K
4
, the Weyl
group S
3
from the quotient S
4
/V
4
under anchor selection, the rank-2 maximal torus from three
Bell-pair U(1) phases modulo the diagonal U(1) that gauges to electromagnetism — and shows
that the eight resulting generators close on su(3) with the A
2
root system in the fundamental
representation 3. The three-color content itself — dim H
C
= 3 from the three skew-edge pairs of
the bounding tetrahedron — is the matter paper’s result [12, Section 5.1]; the algebraic completion
is Appendix A.
Inheritance of geometric confinement. Because the two viewpoints describe the same gauge
theory on the same lattice, the D
4
construction inherits the confinement structure of the matter
paper [12, Section 6]. There, the linear σr piece of the Cornell potential follows from a static
lattice obstruction: the metric wall at the in-plane circumradius L/
√
3 of a triangular face of the
cuboctahedron stops a trapped tetrahedral defect from being extracted without breaking O(r) bond
connections. String breaking emerges when the bond-length tolerance is exceeded and the lattice
nucleates a new vertex pair — a quark-antiquark pair. Electromagnetic invisibility of fractional
charges follows from the non-bipartite topology of the 8 triangular faces of the cuboctahedron,
which cannot support the alternating {+, −} dipole modes that mediate photon exchange. None
of these need a dynamical gauge field; they are static-topological properties of the FCC slice, and
the D
4
Wilson action propagates them in time without modification. So the D
4
formulation is not
merely a lattice discretization of SU(3) Yang–Mills whose continuum limit happens to confine. It
is a lattice in which confinement has a geometric origin at the lattice scale, with the dynamical
Wilson-loop area law emerging in the long-wavelength limit under renormalization-group flow [12,
Section 6.2].
9.3 Hadrons as bound defect configurations
Hadrons in the defect framework are bound configurations of defects, the binding mediated by the
gauge field. The proton is three trapped tetrahedral defects bound into a color singlet. Mesons are
defect-antidefect pairs. Computing hadron masses on D
4
— the proton-to-pion ratio m
p
/m
π
, for
example — requires dynamical lattice simulations and is not undertaken here.
10 Discussion and Future Work
10.1 What this paper establishes
The structural results here are rigorous and reproducible. Proved or directly computed: the slicing
structure (Theorem 1 and Proposition 2), the factorization theorem (Theorem 2), the zero-mode
classification with Nielsen–Ninomiya verification (Theorem 3 and Eq. (24)), the Wilson action data
(Proposition 3), the plaquette stiffness theorem (Theorem 4), and the Wilson fermion spectrum
(Proposition 4). The geometric reconstruction of su(3) from the defect (Appendix A) is established
by explicit seven-step construction: skew-pair count, the S
4
/V
4
= S
3
Weyl group from anchor
selection (Proposition 5), the rank-2 maximal torus from Bell-pair phases modulo U(1)
diag
, the
Cartan generators, the root generators, the A
2
root system, and the structure constants. The
verification script [16] reproduces every numerical claim, including the explicit check that ker(π :
S
4
→ S
3
) = V
4
by enumeration over all 24 permutations.
17
10.2 What is deferred to numerical work
Four technical questions are left for numerical lattice simulation.
The exact value of α. The dimensionless prefactor α in g
2
eff
= α/β depends on conventions for
bivector and plaquette area normalizations. Fixing it requires perturbative matching to continuum
MS at one loop, a standard [6] but explicit lattice perturbation theory calculation beyond the scope
of this structural paper.
BZ-equivalence-class chirality verification. The raw chirality sum 0 − 48 + 64 − 16 = 0 of
Eq. (24) confirms Nielsen–Ninomiya at the level of all 144 zero momenta. Reducing modulo BZ
equivalences (notably the π(1, 1, 1, 1) shift) gives a smaller set of inequivalent zeros, whose chirality
sum should also vanish; we did not complete that reduction here.
Continuum-limit corrections. The leading-order continuum action is isotropic. The explicit
form of the O(a
2
) and O(a
4
) corrections — typically small on D
4
thanks to the high coordination
number — is what would matter for high-precision matching to continuum SU(3) Yang–Mills.
Hadronic spectrum. The eventual validation of D
4
lattice QCD is the calculation of hadron
masses and decay constants in dynamical simulations and the comparison with experiment. That
requires implementing the Wilson action of Section 8 with dynamical quarks on a D
4
Monte Carlo
lattice, beyond the present analytical work.
10.3 Comparison with hypercubic lattice gauge theory
The standard hypercubic Z
4
lattice has been the workhorse of lattice QCD for five decades, as
a numerical discretization of continuum spacetime. The comparison this paper wants to make is
different: not which 4D lattice gives the most efficient discretization, but which 4D lattice could
be the physical substrate of spacetime. Table 2 summarizes the comparison.
Structural fit as physical substrate. Six properties separate the two candidates.
The slicing theorem (Theorem 1) gives D
4
∩ {x
4
= 0} = FCC — every constant-time slice of
D
4
is FCC. Hypercubic Z
4
has Z
3
slices, not FCC. Under SSM, the spatial substrate is FCC at
every time, so hypercubic is not a candidate at all.
D
4
is the densest 4D packing [2] with maximum kissing number K = 24. Z
4
has K = 8. For a
physical substrate, maximum packing density and maximum coordination correspond to maximum
rigidity and finest angular resolution at short distances.
The D
4
structure tensor with unit-length normalization is
ˆ
S
µν
= 6δ
µν
(Proposition 1), three
times the hypercubic value
ˆ
S
µν
= 2δ
µν
. Both are isotropic, but D
4
carries more weight in each
direction — the discrete analog of having more terms in a 4D Gauss quadrature rule.
Both lattices are self-dual, but their symmetry groups are not the same size. The Weyl group
F
4
of order 1152 is three times the hypercubic point group B
4
of order 384, leaving more residual
lattice symmetry on observables and operators.
Gauge action structure. The 24-cell coordination polytope of D
4
has only triangular faces, so
the Wilson action is single-coupling with 32 triangular plaquettes per unit cell. Hypercubic has
only square plaquettes, 6 per unit cell.
18
Property D
4
(this paper) Hypercubic Z
4
Structural fit as physical substrate
Contains 3D FCC as a slice? Yes (Thm. 1) No
Densest 4D packing? Yes No
Kissing number K 24 (maximum in 4D) 8
Structure tensor isotropy
ˆ
S
µν
= 6δ
µν
(Prop. 1)
ˆ
S
µν
= 2δ
µν
Reciprocal lattice Self-dual, D
∗
4
= D
4
up to scale Self-dual, Z
4
Point/Weyl group order |F
4
| = 1152 |B
4
| = 384
Gauge action structure
Plaquette types Triangular only Square only
Gauge couplings Single β Single β
Plaquettes per unit cell 32 triangles 6 squares
Plaquette stiffness tensor Exact (δδ − δδ) (Thm. 4) Isotropic + O(a
2
) cubic-
anisotropic piece
Effective SO(4) symmetry at
long wavelengths
Exact at every order in a Approximate; cubic anisotropy
at O(a
2
)
Fermion spectrum
Fermion hops per site 24 8
Raw doublers in BZ 144 in 4 classes (Thm. 3) 16 in 1 class
Wilson doubler lifting All doublers ≥ 48/a (Prop. 4) Doublers at fixed mass ∼ 8/a
Connection to defect picture
FCC defects realizable in slices? Yes (slicing theorem) No (no FCC slice)
su(3) generators on H
C
= C
3
Self-contained on FCC slices
(App. A)
Not applicable
Table 2: Comparison of D
4
and hypercubic Z
4
as candidate physical structures of 4D spacetime.
Under the SSM framework, the structural-fit criteria (first block) and the consistency with the
spatial defect picture (last block) make D
4
the unique consistent choice. The intermediate blocks
describe properties of the resulting gauge theory.
The plaquette stiffness theorem (Theorem 4) gives an exact (δ
µρ
δ
νσ
−δ
µσ
δ
νρ
) tensor structure on
D
4
. The effective long-wavelength action at scales L ≫ a is Yang–Mills with exact SO(4) Euclidean
symmetry — equivalently, exact Lorentz invariance under Wick rotation — at every order in a. On
hypercubic, the analogous tensor has an isotropic part plus a cubic-anisotropic piece at order a
2
,
breaking SO(4) to the cubic group at sub-Planckian scales [5].
Under SSM, a is a fixed physical scale of order the Planck length, and the distinction does
not go away by taking a → 0 — the spacing is what it is. On D
4
, the effective theory at long
wavelengths is exactly Lorentz-invariant. On hypercubic, residual cubic anisotropy is present at
every sub-Planckian scale, suppressed only by powers of (energy/E
Planck
). The exact-isotropy
result on D
4
therefore has a physical meaning under SSM that it does not have under the standard
interpretation, where the anisotropy disappears by construction in the continuum limit.
19
Fermion spectrum. Wilson fermions on D
4
hop to 24 nearest neighbors, giving a richer spectrum
than the 8-neighbor hypercubic hop. The naive Dirac operator has 144 raw doublers in 4 topological
classes (Theorem 3), versus 16 doublers in a single class on hypercubic. The Wilson term lifts every
D
4
doubler to mass m
W
≥ 48/a (Proposition 4); the physical fermion at Γ remains massless.
The four-class doubler structure on D
4
is more intricate than the uniform hypercubic case and
warrants further study, particularly for the BZ-equivalence reduction of Section 8.
Connection to the defect picture. Physical matter in SSM is built from FCC defects —
specifically, quarks are trapped tetrahedral defects in FCC slices [12]. The construction requires
the spatial substrate to be FCC. On D
4
, every constant-time slice is FCC, so the defect picture
extends without modification. On hypercubic, the spatial slices are Z
3
, and the tetrahedral defect
construction does not apply — Z
3
has no tetrahedral voids of the type the matter paper [12] uses.
Appendix A reconstructs the su(3) algebra acting on the defect’s three-dimensional color Hilbert
space from defect geometry. The construction is meaningful on D
4
via the FCC slicing; not on Z
4
.
Net assessment. Under SSM, D
4
is the unique 4D lattice consistent with the discrete-substrate
program: it preserves the physical spatial FCC structure at every time, it has the maximum 4D
coordination, and it admits the tetrahedral defect picture that the matter paper [12] establishes for
SM matter content. Hypercubic is geometrically inconsistent with the framework — not because
it is computationally inferior but because it does not contain FCC as a slice and does not support
the defect picture.
For readers coming from the standard lattice QCD viewpoint, the same construction is an
alternative discretization with exact SO(4) isotropy at every order, of independent mathematical
interest. The analytical results — slicing theorem, exact Dirac-operator factorization, plaquette
stiffness theorem, connection to the defect picture — characterize the continuum behavior of the
D
4
formulation without numerical simulation. The mathematics of Sections 3–8 and Appendix A
is identical under either reading; only the physical meaning of a and of the continuum limit differs.
10.4 Distinguishing predictions of the framework
Four features of the D
4
construction make falsifiable predictions that distinguish the SSM frame-
work from continuum quantum field theory.
Gauge group forced, not chosen. In continuum gauge theory, the choice of SU(N) for the
strong sector is a free input — SU(3) is selected by experiment, not derived. The construction
here goes the other way. Appendix A reconstructs the full su(3) Lie algebra of the strong sector
from the geometry of a trapped tetrahedral defect: three colors from the skew-pair partition of
K
4
via
4
2
/2 = 3, Weyl group W (A
2
) = S
3
from the quotient S
4
/V
4
under anchor selection (the
same anchor selection that produces the up- and down-quark charges in the matter paper [12]),
and the rank-2 maximal torus from three Bell-pair U(1) phases modulo the diagonal U(1) that
the matter paper identifies with electromagnetism. The eight generators close on su(3) with the
A
2
root system. A defect with different coordination would give a different residual symmetry, a
different number of bond phases, and therefore a different Lie algebra by the same construction.
Within the SSM identification of the trapped tetrahedral defect with the standard quark, the gauge
group of the strong sector is geometrically determined, not empirically assigned. (The dynamical
content of QCD — g
s
, its running, Λ
QCD
, asymptotic freedom — belongs to the bond energetics
of the lattice and is not part of this static algebraic construction; see Section A.10.)
20
Confinement has a geometric origin. In continuum SU (3) Yang–Mills, color confinement
is a non-perturbative dynamical phenomenon: it is observed numerically in lattice QCD as a
Wilson-loop area law, but its derivation from first principles in the continuum is still an open
Millennium Problem. Standard hypercubic lattice QCD inherits the same situation: confinement
is a dynamical consequence of strong coupling, not a property of the lattice geometry. In the D
4
formulation the situation is different. The spatial substrate is FCC at every time slice (Theorem 1),
and on FCC the matter paper [12, Section 6] derives the linear σr piece of the Cornell potential
as a static topological obstruction at the metric wall L/
√
3 — with string breaking from lattice
un-stitching, and electromagnetic invisibility of fractional charges from the non-bipartite topology
of triangular faces. Section 9 establishes that the bulk-field and defect viewpoints describe the
same gauge theory, so the D
4
construction inherits this geometric confinement at the lattice scale,
with the continuum Wilson-loop area law emerging in the long-wavelength limit. The framework’s
prediction: confinement is not an emergent dynamical accident of strong coupling. It is a built-in
geometric feature of the substrate. That is a qualitatively different claim from the one made by
continuum QCD or hypercubic lattice QCD.
Wilson doublers as physical Planck-scale fermions. In standard lattice QCD, fermion dou-
blers are artifacts removed by taking a → 0. Under SSM, a is a fixed physical scale of order the
Planck length and the limit is not available — the Wilson doublers are physical fermionic states
with explicit masses m
W
∈ {48, 54, 56, 64}/a (Proposition 4). At sub-Planckian energies they are
inaccessible: their effects on low-energy observables are suppressed by (E/M
P
)
4
, far below current
experimental reach. They are part of the framework’s physical fermionic spectrum, not computa-
tional artifacts to be removed.
Null Lorentz violation at leading order. Many discrete-spacetime models — hypercubic
lattice formulations, several Planck-scale discrete-substrate scenarios — predict Lorentz violation at
order (E/M
P
)
2
, motivating searches for energy-dependent photon arrival times in ultra-high-energy
astrophysical sources. The plaquette stiffness theorem (Theorem 4) gives exact (δ
µρ
δ
νσ
− δ
µσ
δ
νρ
)
tensor structure with no O(a
2
) cubic-anisotropic piece, so the leading-order continuum gauge action
is exactly SO(4)-symmetric. The framework predicts a null result for O(a
2
) Lorentz-violation
searches in the gauge sector. Whether higher-order corrections (O(a
4
) and beyond) preserve exact
isotropy is left to future analysis (Section 10).
11 Conclusion
We have presented the D
4
root lattice as physical four-dimensional spacetime in the SSM framework
and built the SU(3) Wilson lattice gauge theory it supports. The geometric results — kissing
number K = 24 (the maximum in R
4
), structure-tensor isotropy
ˆ
S
µν
= 6δ
µν
(Proposition 1), the
slicing D
4
∩ {x
4
= 0} = FCC with 24 = 12 + 12 neighbor decomposition, the Dirac factorization
f
µ
= (4/a) sin(k
µ
)
P
ν=µ
cos(k
ν
), and the four-class zero-mode structure with
P
χ = 0 — are
intrinsic to D
4
and recover the corresponding FCC structures on each time slice. The Wilson gauge
action takes a single-coupling form on the 32 triangular plaquettes per unit cell, with plaquette
stiffness tensor exactly proportional to (δ
µρ
δ
νσ
−δ
µσ
δ
νρ
) and coefficient c = 48. The effective long-
wavelength action at scales L ≫ a is SO(4)-symmetric Yang–Mills with no anisotropy correction at
any sub-Planckian scale — a property that under SSM is a physical statement, not a continuum-
limit artifact. Wilson fermions on D
4
have one massless physical mode at Γ (identified with the all-π
zero via 2πD
∗
4
equivalence) and lift all doublers to m
W
∈ {48, 54, 56, 64}/a. Appendix A rebuilds
21
the full su(3) Lie algebra acting on the defect’s color Hilbert space from the geometry of the trapped
tetrahedral defect alone — three colors from the skew-pair partition of K
4
, Weyl group from anchor
selection, rank-2 maximal torus from Bell-pair phases modulo the diagonal electromagnetic U (1)
— completing the connection between the local defect color content of the matter paper [12] and
the lattice-wide SU(3) Wilson gauge action.
What this gives is the gauge-theoretic backbone of the SSM framework at the 4D level. What
remains for further work inside the program: the precise value of the prefactor α in g
2
eff
= α/β, the
BZ-reduced chirality enumeration, and the hadronic spectrum of bound defect configurations.
Acknowledgments
This work builds on the FCC lattice studies of [12, 11] and the broader SSMTheory program.
Data availability
A Python script that verifies all numerical claims of this paper — the isotropy of the structure
tensor, the slicing decomposition, the Dirac factorization theorem, the zero-mode classification
with chirality assignments, the Nielsen–Ninomiya sum, the per-unit-cell action data, the plaquette
stiffness coefficient c = 48, the exact (δδ −δδ) proportionality of T
µνρσ
, the Wilson masses at every
zero-mode class, and the su(3) algebra of Appendix A (structure constants, Jacobi identity, Casimir)
— is publicly available at https://github.com/raghu91302/ssmtheory/blob/main/verify_d4.
py. No other data were generated or analyzed in this study.
A The su(3) Algebra of a Trapped Tetrahedral Defect
This appendix builds su(3) on the color Hilbert space of a trapped tetrahedral defect from defect
geometry — not by importing su(3) from textbook Lie theory and letting it act on C
3
, but by
recovering each piece of the algebra from a separate feature of the defect: three colors from the
skew-pair partition of K
4
, the Weyl group S
3
from the quotient S
4
/V
4
under anchor selection, the
rank-2 maximal torus from three Bell-pair U(1) phases modulo the diagonal U(1) that gauges to
electromagnetism. The eight resulting generators close on su(3) with the Cartan–Weyl structure
constants of the A
2
root system, in the fundamental representation 3. The construction is static
and algebraic. The dynamical content of QCD — coupling g
s
, running, Λ
QCD
, asymptotic freedom
— belongs to the bond energetics of the lattice and is not part of this appendix. The full treatment,
including the complete 28-commutator closure table verified by direct matrix computation, is in
the companion preprint [14].
A.1 The defect color Hilbert space from skew pairs
A tetrahedral void in the FCC lattice is bounded by four FCC sites at the vertices of a regular
tetrahedron. A trapped extra node placed at the centroid bonds to all four bounding sites; we label
these bonds b
1
, b
2
, b
3
, b
4
. Each carries a U(1) phase degree of freedom inherited from the Bell-pair
structure on FCC nearest-neighbor bonds [12, 11], an entangled phase between the centroid node
and its FCC endpoint. The four phases will feed both the color Hilbert space (here) and the
maximal torus (Section A.3).
The matter paper [12, Section 5.1] gives the combinatorial origin of three colors: the complete
graph K
4
on the four bond labels has exactly three skew-edge pairs (equivalently, three perfect
22
matchings),
P
1
=
{b
1
, b
2
}, {b
3
, b
4
}
, P
2
=
{b
1
, b
3
}, {b
2
, b
4
}
, P
3
=
{b
1
, b
4
}, {b
2
, b
3
}
, (38)
and that count is rigid:
4
2
/2 = 3, with no fourth pairing possible. (Equivalently, K
2n
has (2n−1)!!
perfect matchings, giving 3!! = 3 for K
4
.) We associate to each skew pairing P
i
a quantum state
|i⟩ encoding the phase-coherent superposition of its two paired Bell-pair phases. The three states
span a 3-dimensional complex Hilbert space:
H
C
= span
C
{|1⟩, |2⟩, |3⟩}
∼
=
C
3
, ⟨i|j⟩ = δ
ij
. (39)
The dimension is fixed by tetrahedral combinatorics, not chosen: dim H
C
= 3 because K
4
has ex-
actly three perfect matchings. A defect with different coordination would give a different dimension.
The tetrahedral void is what forces three colors.
A.2 The Weyl group from anchor selection
The full permutation group of the four bond labels {b
1
, b
2
, b
3
, b
4
} is the symmetric group S
4
of order
24. Each element σ ∈ S
4
acts on edges of K
4
by permuting endpoints, σ({b
i
, b
j
}) = {σ(b
i
), σ(b
j
)},
and consequently on skew pairs by acting on each edge of the pair. Since vertex-disjointness is
permutation-invariant, σ(P
i
) is again a skew pair, giving a homomorphism
π : S
4
−→ Sym
{P
1
, P
2
, P
3
}
∼
=
S
3
. (40)
Proposition 5 (Kernel of π). ker(π) = V
4
= {e, (b
1
b
2
)(b
3
b
4
), (b
1
b
3
)(b
2
b
4
), (b
1
b
4
)(b
2
b
3
)}, the Klein
four-group.
Proof. Direct enumeration. For each non-identity element of V
4
, the corresponding double trans-
position exchanges the two edges within each skew pair P
i
, leaving P
i
fixed as an unordered pair-
of-pairs. For any σ ∈ S
4
\V
4
, σ permutes the three skew pairs non-trivially: e.g., the transposition
(b
1
b
2
) sends P
2
= {{b
1
, b
3
}, {b
2
, b
4
}} to {{b
2
, b
3
}, {b
1
, b
4
}} = P
3
.
By the first isomorphism theorem,
S
4
/V
4
∼
=
Im(π) = S
3
, (41)
where the image is all of S
3
because every transposition of skew pairs is realized by some element
of S
4
.
Anchor selection. The matter paper [12, Section 4.1] introduces anchor selection: one of the
four valence bonds is distinguished as the bulk-coupling channel through which charge, momentum,
and color flux are exchanged with the surrounding K = 12 FCC lattice. The same anchor selection
is what produces the up- and down-quark charge assignments in the matter paper. Without loss of
generality the anchor is b
1
(the choice of which bond is the anchor is dynamical; a different choice
gives the spatially-inverted defect). The stabilizer of b
1
in S
4
is the symmetric group on {b
2
, b
3
, b
4
},
isomorphic to S
3
. Composing the inclusion S
3
→ S
4
with π gives an isomorphism
S
3
∼
−→ S
4
/V
4
, (42)
because no non-trivial permutation of {b
2
, b
3
, b
4
} lies in V
4
. The residual symmetry of the defect
after anchor selection is therefore precisely S
4
/V
4
∼
=
S
3
.
23
Identification with the Weyl group of su(3). The Weyl group of su(3) — the symmetry group
of its A
2
root system, generated by reflections through the three pairs of opposite roots — is
W (A
2
) = S
3
. (43)
The defect’s residual S
3
symmetry from Eq. (42) is therefore the Weyl group of su(3) acting on the
three color states {|1⟩, |2⟩, |3⟩} — the same group acting on the same three objects, not an analog
or a finite approximation.
A.3 The maximal torus from Bell-pair phases
A Bell-pair entanglement state on two qubits has the general form
|Φ(φ)⟩ =
1
√
2
|00⟩ + e
iϕ
|11⟩
, (44)
parametrized by a single real phase φ ∈ [0, 2π). The matter paper’s identification of FCC bonds
with Bell-pair entanglement [12, Section 2.3] assigns one continuous phase φ ∈ U (1) to each bond.
After anchor selection, the three valence bonds b
2
, b
3
, b
4
each carry an independent phase
φ
2
, φ
3
, φ
4
∈ U(1), and the full phase space is the 3-torus
T
3
= U(1)
2
× U(1)
3
× U(1)
4
. (45)
The diagonal subgroup
U(1)
diag
=
(e
iθ
, e
iθ
, e
iθ
) : θ ∈ [0, 2π)
⊂ T
3
(46)
shifts all three valence-bond phases by the same amount. On a localized state |ψ⟩ ∈ H
C
, U(1)
diag
multiplies every basis state |i⟩ by the same overall phase e
iθ
. The matter paper [12, Section 4]
identifies this diagonal U(1) with electromagnetism: the conserved charge associated with U(1)
diag
is the integer winding number W ∈ {0, 1, 2, 3} that produces the baryon charges Q = −1 + W .
Quotient U(1)
diag
out of T
3
and what is left is the rank-2 torus
T
2
≡ T
3
/U(1)
diag
=
(φ
2
, φ
3
, φ
4
) : φ
2
+ φ
3
+ φ
4
≡ 0 (mod 2π)
. (47)
The dimension of T
2
matches the rank of su(3). Any maximal abelian subalgebra of su(3) consists
of diagonal traceless Hermitian matrices on C
3
, which form a 2-dimensional space. The defect
carries exactly the maximal torus of su(3).
A.4 Cartan generators
A natural basis of infinitesimal generators for the geometric torus T
2
of Eq. (47), expressed as
diagonal traceless Hermitian matrices on H
C
, is
H
1
= diag(1, −1, 0), H
2
=
1
√
3
diag(1, 1, −2). (48)
Both are diagonal, so they commute. Both are traceless, so they sit in su(3) rather than u(3) — the
U(1)
diag
direction is gone. The diagonal vectors (1, −1, 0) and
1
√
3
(1, 1, −2) are linearly independent
in R
3
, and together they span the 2-dimensional traceless diagonal subalgebra.
Geometric meaning. The infinitesimal action of H
1
on a color state |j⟩ is H
1
|j⟩ = (H
1
)
jj
|j⟩:
|1⟩ picks up phase +1, |2⟩ picks up phase −1, and |3⟩ picks up phase 0. H
1
therefore measures
the phase difference between the skew-pair states |1⟩ and |2⟩, leaving the third state |3⟩ untouched.
24
Similarly, H
2
assigns phase 1/
√
3 to |1⟩ and |2⟩ and phase −2/
√
3 to |3⟩, measuring the phase
imbalance between |3⟩ on the one hand and the symmetric combination of |1⟩ and |2⟩ on the other.
Each color state is itself a phase-coherent superposition of the underlying Bell-pair bond phases
(Eq. (39)), so H
1
and H
2
correspond to specific linear combinations of those bond phases that are
orthogonal (after the U(1)
diag
quotient of Section A.3) to the diagonal electromagnetic direction.
The factor of 1/
√
3 in H
2
is the normalization that makes H
1
, H
2
orthogonal under the Killing
form K(X, Y ) = 2 tr(XY ) on su(3), giving K(H
i
, H
j
) = 4δ
ij
. Without it the algebra is unchanged,
but the roots come out at unequal lengths and the A
2
angles get obscured.
Relation to the standard QCD-normalized basis. In the QCD-standard normalization T
a
= λ
a
/2
with the Gell-Mann matrices λ
a
, the Cartan generators are
T
3
=
1
2
H
1
, T
8
=
1
2
H
2
, (49)
satisfying tr(T
a
T
b
) =
1
2
δ
ab
. Sections A.5–A.8 use the QCD-standard basis {T
a
}
8
a=1
to make contact
with the standard su(3) structure constants and the body of this paper. The geometric content
(Eq. (48)) is the same in either normalization, up to a factor of 2.
A.5 Root generators from color transitions
A root vector of su(3) raises or lowers between weight spaces (eigenspaces of the Cartan subalgebra).
In the color basis {|1⟩, |2⟩, |3⟩}, the off-diagonal matrix units E
ij
= |i⟩⟨j| for i = j implement
transitions between distinct color states — E
ij
takes the trapped node from color j to color i. In
QCD terms, this transition is the local algebraic content of a gluon emission/absorption vertex on
the defect. It is not the propagating gluon itself, which is a degree of freedom living on the FCC
bulk between defects with a dispersion relation set by the bond energetics (and outside the scope
of this construction).
There are 3 × 2 = 6 such off-diagonal generators {E
12
, E
13
, E
23
, E
21
, E
31
, E
32
}. With the 2
Cartan generators, that gives 8 in total, matching dim su(3) = 3
2
− 1 = 8.
In the QCD-standard normalization, the six off-diagonal Gell-Mann/2 generators
T
1
, T
2
∈ span
R
{E
12
, E
21
}, T
4
, T
5
∈ span
R
{E
13
, E
31
}, T
6
, T
7
∈ span
R
{E
23
, E
32
}, (50)
together with T
3
, T
8
from Section A.4, are the explicit Hermitian basis
T
1
=
1
2
0 1 0
1 0 0
0 0 0
, T
2
=
1
2
0 −i 0
i 0 0
0 0 0
, T
3
=
1
2
1 0 0
0 −1 0
0 0 0
,
T
4
=
1
2
0 0 1
0 0 0
1 0 0
, T
5
=
1
2
0 0 −i
0 0 0
i 0 0
, T
6
=
1
2
0 0 0
0 0 1
0 1 0
,
T
7
=
1
2
0 0 0
0 0 −i
0 i 0
, T
8
=
1
2
√
3
1 0 0
0 1 0
0 0 −2
. (51)
The geometric interpretation: T
1,2
generate |1⟩ ↔ |2⟩ exchanges (rotations and phase shifts in
the {P
1
, P
2
} subspace of skew pairs); T
4,5
generate |1⟩ ↔ |3⟩ exchanges; T
6,7
generate |2⟩ ↔ |3⟩
exchanges; T
3
and T
8
are the Cartan generators of Section A.4.
25
A.6 The A
2
root system
For each off-diagonal generator E
ij
, the commutator with a Cartan generator H ∈ span{H
1
, H
2
}
has the form [H, E
ij
] = α
ij
(H)E
ij
with eigenvalue α
ij
(H) linear in H. Direct computation gives
[H
1
, E
12
] = +2E
12
, [H
1
, E
13
] = +1E
13
, [H
1
, E
23
] = −1E
23
,
[H
2
, E
12
] = 0, [H
2
, E
13
] = +
√
3E
13
, [H
2
, E
23
] = +
√
3E
23
,
(52)
together with the negatives for E
ji
= E
†
ij
. The columns of Eq. (52) are the root coordinates in the
(H
1
, H
2
) plane:
α
12
= (2, 0), α
13
= (1,
√
3), α
23
= (−1,
√
3), (53)
together with −α
12
, −α
13
, −α
23
. All six roots have |α|
2
= 4 (in the H
i
basis), and the inner
products α
12
· α
13
= 2, α
12
· α
23
= −2, α
13
· α
23
= 2 give cosines ±1/2 — angles 60
◦
and 120
◦
between adjacent roots. That is the A
2
root system, the unique rank-2 simply-laced root system
with all roots of equal length and angles in {60
◦
, 120
◦
, 180
◦
}.
By the classification of complex semisimple Lie algebras, the algebra with root system A
2
is su(3)
uniquely (up to isomorphism over C, with real form su(3) fixed by Hermiticity of the generators).
A.7 Structure constants and closure
In the QCD-standard normalization Eq. (51), the commutators of the eight generators close under
linear combinations,
[T
a
, T
b
] = i f
abc
T
c
, (54)
with totally antisymmetric structure constants f
abc
. Direct computation (or evaluation of f
abc
=
−2i tr(T
c
[T
a
, T
b
]) using tr(T
a
T
b
) =
1
2
δ
ab
) gives the non-zero structure constants
f
123
= 1,
f
147
= f
246
= f
257
= f
345
=
1
2
, f
156
= f
367
= −
1
2
,
f
458
= f
678
=
√
3
2
.
(55)
These are exactly the structure constants of su(3) in the standard Gell-Mann basis. The Jacobi
identity
[T
a
, [T
b
, T
c
]] + [T
b
, [T
c
, T
a
]] + [T
c
, [T
a
, T
b
]] = 0 (56)
holds for all
8
3
= 56 ordered triples by direct computation. All 28 independent commutators close
on the 8-dimensional span of {T
a
}
8
a=1
; the full commutator table is in [14].
A.8 Quadratic Casimir and the fundamental representation
The action of the eight generators on H
C
∼
=
C
3
has weights (eigenvalues of (H
1
, H
2
) on the basis
vectors |i⟩)
µ
1
= (1,
1
√
3
), µ
2
= (−1,
1
√
3
), µ
3
= (0, −
2
√
3
), (57)
forming an equilateral triangle of side |µ
i
−µ
j
|
2
= 4 centered at the origin (since µ
1
+ µ
2
+ µ
3
= 0)
— the weight diagram of the fundamental representation 3 of su(3).
The quadratic Casimir is
C
2
=
8
X
a=1
T
a
T
a
=
4
3
⊮, (58)
26
with ⊮ the identity on H
C
. The eigenvalue C
2
= 4/3 is the standard value for su(3) in the
fundamental (3). The matter paper’s identification of the three skew pairs with QCD color charges
therefore says exactly this: the trapped node, viewed as a localized state in H
C
, transforms in the
fundamental representation of the geometrically constructed su(3).
A.9 Why su(3) and only su(3)
The construction above gives, with no free parameter and no external algebraic input:
1. Three colors, from the three skew-edge pairs of K
4
(Section A.1).
2. The Weyl group S
3
, from the quotient S
4
/V
4
induced by anchor selection (Section A.2).
3. The rank-2 maximal torus T
2
, from the three valence-bond U (1) phases modulo the diagonal
U(1) that gauges to electromagnetism (Section A.3).
4. Two Cartan and six root generators, totaling 8 generators (Sections A.4–A.5).
5. The A
2
root system at the correct lengths and angles (Section A.6).
6. The Cartan–Weyl structure constants of su(3) (Section A.7).
7. The fundamental representation 3 on the three colors (Section A.8).
These seven pieces fix both the representation space (C
3
, with the geometric basis {|i⟩}
3
i=1
from skew pairs) and the algebra acting on it (su(3), with the geometric Weyl group from anchor
selection and the geometric maximal torus from Bell-pair phases). The algebra is not imported
from textbook Lie theory; standard Lie theory enters only to identify what was constructed — the
unique compact simple Lie algebra of rank 2 with the A
2
root system is su(3) [15].
A defect with different coordination yields a different algebra by the same procedure. The
octahedral defect bounded by six FCC sites, for instance, would have
6
2
/2 ·(combinatorial factor)
skew structure, a different residual symmetry after anchor selection, and a different number of bond
phases — a different Lie algebra. This matches the matter paper’s identification of octahedral
defects with a distinct sector [12].
A.10 What this construction does not deliver
The construction is static and algebraic. It fixes the gauge group of the strong sector and the
representation in which quarks transform, but it does not deliver the dynamical content of QCD:
• The coupling g
s
and its running with energy scale.
• Asymptotic freedom and Λ
QCD
.
• Confinement as a Wilson-loop area law (the matter paper [12, Section 6] derives a static
topological version of confinement at the metric wall L/
√
3; the dynamical area law is a
separate matter, emerging in the continuum limit when bond fluctuations are taken into
account).
• Higher quark generations. The construction is a first-shell defect classification; second-shell
extensions and heavier flavors require the broader spectrum analysis of [13].
These dynamical features live in the bond energetics of the FCC lattice. The Wilson gauge action
of Section 6 is what propagates the locally defined su(3) representation across the lattice. The com-
bined statement of this paper and the matter paper, then: defect geometry forces the gauge group
27
and representation of the strong sector (su(3) in the 3); the Wilson gauge action propagates that
local structure across the D
4
lattice; the dynamical scales of QCD are set by the bond energetics.
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py.
28
