su(3) from a Trapped Tetrahedral Defect in the FCC Lattice

su(3) from a Trapped Tetrahedral Defect in the FCC Lattice
Raghu Kulkarni
∗
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
May 2026
Abstract
The matter paper of the Selection-Stitch Model [1] traces the three QCD color charges to
the three skew-edge pairs of the tetrahedron that bounds a trapped extra node in the FCC
vacuum, but stops short of the su(3) Lie algebra. We close that gap. The skew pairs span a
3-dimensional color space; the quotient S
4
/V
4
∼
=
S
3
generated by anchor selection on the defect
acts as the Weyl group of su(3); and the U(1) phase carried by each Bell-pair valence bond,
modulo the diagonal U(1) that gauges to electromagnetism, supplies a rank-2 maximal torus.
Two diagonal generators built from the bond phases together with six off-diagonal generators
built from the oriented color transitions close on su(3) with the Cartan-Weyl structure constants
of the A
2
root system, verified by explicit computation of all 28 commutators. The fundamental
representation 3, the rank, the Weyl group, the maximal torus, and the full root system emerge
from the defect geometry without adjustable inputs. The QCD coupling g
s
, its running, Λ
QCD
,
and other dynamical features lie outside the static algebraic content of the construction.
1 Introduction
The matter paper of the SSM derives the count of QCD colors from a single combinatorial fact: the
six edges of the tetrahedron K
4
partition uniquely into three pairs of vertex-disjoint (skew) edges,
with no fourth such partition. The matter paper assigns one color to each skew pair and identifies
the count, C(4, 2)/2 = 3, with the dimension of the fundamental representation of QCD’s gauge
group. It is also explicit that this is a derivation of the representation, not of the Lie algebra:
“We do not claim to derive the SU(3) Lie algebra from the lattice geometry; we claim
that the representation of color charges on a baryon-like defect is uniquely fixed at three
by the combinatorial identity C(4, 2)/2 = 3.” [1, Section 5.2]
The present paper supplies the missing step. The defect carries enough structure to reconstruct
the eight-dimensional algebra: three pieces, each grounded in a different feature of the geometry,
combine to give exactly su(3). The first piece is combinatorial (the skew-pair partition of K
4
). The
second piece is symmetry-theoretic (the quotient S
4
/V
4
from anchor selection). The third piece is
continuous: a U(1) phase per valence bond, with the diagonal phase gauged to electromagnetism.
The rank, the maximal torus, the Weyl group, the root system, and the structure constants all fall
out together.
The construction is static and algebraic. The dynamical content of QCD — coupling strength,
running, asymptotic freedom, Λ
QCD
— belongs to the bond energetics of the lattice, which we do
not address.
∗
raghu@idrive.com
1
2 The Defect
We work with the trapped tetrahedral defect of the matter paper [1, Section 3.2]. An extra node
sits at the centroid of a tetrahedral void of the FCC lattice, bonded to four vertices A, B, C, D that
form a regular tetrahedron of edge length L. The four bonds split into one anchor and three valence
bonds when one of the four directions ˆr
A
, ˆr
B
, ˆr
C
, ˆr
D
is selected as the bulk-coupling channel; this
is the matter paper’s anchor-selection step that breaks S
4
→ S
3
on the bond directions.
3 Three colors as skew pairs
3.1 The skew-pair partition
The complete graph K
4
on {A, B, C, D} has
4
2
= 6 edges. A pair of edges is skew if the edges share
no vertex; for K
4
this means the four endpoints are all distinct. Any skew pair specifies a partition
of {A, B, C, D} into two disjoint unordered pairs (the endpoints of each edge). Conversely, any
partition of four elements into two unordered pairs determines a skew pair of edges. The number
of such partitions is
1
2
4
2
!
= 3, (1)
where the factor of 1/2 accounts for the symmetry between the two pairs. The three skew pairs of
K
4
are
P
1
= {AB, CD}, P
2
= {AC, BD}, P
3
= {AD, BC}. (2)
These three pairs partition the six edges of K
4
into three groups of two, and there is no fourth
such partition —
4
2
/2 = 3 is exact, not an approximation. Figure 1 displays them. Following the
matter paper [1], we identify the three skew pairs with the three QCD color charges.
A
B
C
D
Color 1
A
B
C
D
Color 2
A
B
C
D
Color 3
A
B
C
D
All three skew pairs
Figure 1: The three skew-edge pairs of the bounding tetrahedron K
4
. Each pair consists of two
edges that share no vertex; together the three pairs partition the six edges, and there is no fourth
such partition. The first three panels show one pair at a time (red, green, blue); the fourth panel
overlays all three. The matter paper [1] identifies each pair with one QCD color charge.
3.2 The color Hilbert space
We package the three colors into a Hilbert space
H
C
≡ C
3
= span
C
(|1⟩, |2⟩, |3⟩), |i⟩ ↔ P
i
, (3)
2
on which we will construct the su(3) generators. The basis vectors |1⟩, |2⟩, |3⟩ are the three skew-
pair states of the trapped node; the inner product is the standard Hermitian inner product on C
3
.
The states are by construction orthonormal: ⟨i|j⟩ = δ
ij
.
4 The Weyl group from anchor selection
4.1 The action of S
4
on the skew pairs
The full permutation group of {A, B, C, D} is the symmetric group S
4
of order 4! = 24. Each
element σ ∈ S
4
acts on edges by permuting endpoints, σ({X, Y }) = {σ(X), σ(Y )}, and on skew
pairs by acting on each edge:
σ(P ) = σ({e
1
, e
2
}) = {σ(e
1
), σ(e
2
)}. (4)
Since the property of being a skew pair is preserved under permutations (vertex-disjointness is
permutation-invariant), σ(P
i
) is again a skew pair. We obtain a homomorphism
π : S
4
→ Sym({P
1
, P
2
, P
3
})
∼
=
S
3
, σ 7→
P
i
7→ σ(P
i
)
. (5)
Proposition 1. The kernel of π is the Klein four-group
V
4
= {e, (AB)(CD), (AC)(BD), (AD)(BC)} ⊂ S
4
. (6)
Proof. We compute π(g) for each g ∈ S
4
by direct enumeration, focusing on the four elements of
(6).
• π(e) fixes each pair: e(P
i
) = P
i
. Trivially in the kernel.
• π((AB)(CD)): applying (AB)(CD) to P
1
= {AB, CD} gives {(AB)(CD)({A, B}), (AB)(CD)({C, D})} =
{{B, A}, {D, C}} = {AB, CD} = P
1
. Applying it to P
2
= {AC, BD}gives {{B, D}, {A, C}} =
P
2
. Similarly P
3
7→ P
3
. In the kernel.
• π((AC)(BD)) and π((AD)(BC)): by the same computation, each fixes P
1
, P
2
, P
3
as sets. In
the kernel.
For any other element of S
4
, π(g) = e. For example, π((AB)): applying (AB) to P
2
= {AC, BD}
gives {BC, AD} = P
3
, so (AB) sends P
2
to P
3
, and is not in the kernel. The four elements of V
4
are therefore the complete kernel.
By the first isomorphism theorem,
S
4
/V
4
∼
=
Im(π) = S
3
. (7)
The image is all of S
3
because every transposition of skew pairs (P
i
P
j
) is realized by some element
of S
4
(e.g., the swap (P
1
P
2
) is induced by (AC)).
Anchor selection in the matter paper picks one of the four vertices — without loss of generality
A — as the channel coupling the defect to the bulk. The stabilizer of A in S
4
is the symmetric group
on {B, C, D}, isomorphic to S
3
. Composing the inclusion S
3
→ S
4
with π gives an isomorphism
S
3
∼
−→ S
4
/V
4
, (8)
since no non-trivial permutation of {B, C, D} lies in V
4
. The residual symmetry of the defect after
anchor selection is therefore precisely the quotient S
4
/V
4
∼
=
S
3
.
3
4.2 Identification with the Weyl group of su(3)
The Weyl group of a Lie algebra g is the group generated by reflections through the hyperplanes
orthogonal to the roots; it is the symmetry group of the root system. For su(3) the root system is
A
2
, and reflections through any of the three pairs of opposite roots generate
W (A
2
) = S
3
, (9)
the symmetric group on three letters [3, Section 10.3]. The defect’s residual S
3
symmetry is therefore
identical to the Weyl group of su(3) — the same group acting on the same three objects (the three
colors / weight states), not an analog or a finite approximation.
5 The maximal torus from bond phases
5.1 The U(1) phase of a Bell pair
A Bell-pair entanglement state on two qubits has the general form
|Φ(φ)⟩ =
1
√
2
|00⟩ + e
iϕ
|11⟩
, (10)
parametrized by a single real phase φ ∈ [0, 2π) that distinguishes inequivalent Bell pairs up to local
unitary equivalence [5]. The matter paper’s identification of FCC bonds with Bell-pair entangle-
ment [1, Section 2.3] therefore assigns one continuous phase φ ∈ U(1) to each bond.
For the trapped tetrahedral defect, after anchor selection the three valence bonds ˆr
B
, ˆr
C
, ˆr
D
each carry an independent phase φ
B
, φ
C
, φ
D
∈ U(1). The full phase space is the 3-torus
T
3
= U (1)
B
× U(1)
C
× U(1)
D
. (11)
5.2 Quotienting the diagonal U(1)
The diagonal subgroup
U(1)
diag
= {(e
iθ
, e
iθ
, e
iθ
) : θ ∈ [0, 2π)} ⊂ T
3
(12)
shifts all three valence-bond phases by the same amount. Its action on the trapped node, viewed as
a localized state in H
C
, is by an overall phase |ψ⟩ 7→ e
iθ
|ψ⟩, since each color basis state |i⟩ picks up
the same factor. This overall phase is unobservable on a closed system, but it couples to charged
probes in the standard way of a U(1) gauge symmetry. The matter paper [1, Section 4] identifies
this diagonal U(1) with electromagnetism, assigning it the role of fixing the trapped node’s electric
charge (the integer winding W = w
B
+ w
C
+ w
D
of the matter paper’s Eq. 14 is the conserved
charge associated with U(1)
diag
).
Quotienting U(1)
diag
out of T
3
leaves
T
2
≡ T
3
/U(1)
diag
=
(φ
B
, φ
C
, φ
D
) : φ
B
+ φ
C
+ φ
D
≡ 0 (mod 2π)
. (13)
This is a two-dimensional torus, since the constraint φ
B
+ φ
C
+ φ
D
= 0 removes one direction from
the original T
3
. The dimension matches the rank of su(3).
Lemma 1. The rank of su(3), defined as the dimension of any maximal abelian subalgebra of
diagonalizable elements, equals 2.
4
Proof. The diagonal traceless 3 × 3 Hermitian matrices form a 2-dimensional abelian subalgebra
of su(3): any two diagonal matrices commute, and the traceless condition imposes one constraint
on the three diagonal entries, leaving 2 free parameters. This subalgebra is maximal: any traceless
Hermitian matrix that commutes with all diagonal matrices must itself be diagonal, since the
eigenspaces of a generic diagonal matrix are 1-dimensional. The dimension of this maximal abelian
subalgebra is the rank.
5.3 Cartan generators
A natural basis of infinitesimal generators of T
2
, expressed as diagonal traceless Hermitian matrices
on H
C
, is
H
1
= diag(1, −1, 0), (14)
H
2
=
1
√
3
diag(1, 1, −2). (15)
Verification of the basis properties.
1. Tracelessness. tr(H
1
) = 1 + (−1) + 0 = 0. tr(H
2
) = (1 + 1 − 2)/
√
3 = 0. Both are traceless;
equivalently, both lie in su(3) rather than u(3), having quotiented the diagonal U(1).
2. Commutativity. Diagonal matrices commute: [H
1
, H
2
] = H
1
H
2
−H
2
H
1
= 0, as can be verified
entry-wise since (H
1
H
2
)
ii
= (H
1
)
ii
(H
2
)
ii
= (H
2
H
1
)
ii
.
3. Linear independence. The vectors (1, −1, 0) and
1
√
3
(1, 1, −2) are linearly independent in R
3
(their cross product is non-zero), so H
1
, H
2
span the 2-dimensional traceless diagonal subalge-
bra.
Geometric meaning. The infinitesimal action of H
1
on a color state |j⟩ is H
1
|j⟩ = (H
1
)
jj
|j⟩, so
|1⟩ acquires phase +1, |2⟩ phase −1, and |3⟩ phase 0. This is the difference of valence-bond phases
on bonds carrying colors 1 and 2 (with color 3 unaffected). Similarly, H
2
assigns phase 1/
√
3 to
colors 1 and 2 and phase −2/
√
3 to color 3, measuring the imbalance between
1
2
(φ
1
+ φ
2
) and φ
3
.
Why 1/
√
3. The factor 1/
√
3 in H
2
is the unique normalization that makes H
1
, H
2
orthonormal
under the Killing form K(X, Y ) = 2 tr(XY ) on su(3). The trace evaluates as
tr(H
2
1
) = 1 + 1 + 0 = 2, (16)
tr(H
2
2
) =
1
3
(1 + 1 + 4) = 2, (17)
tr(H
1
H
2
) =
1
√
3
(1 · 1 + (−1) · 1 + 0 · (−2)) = 0, (18)
giving K(H
i
, H
j
) = 4δ
ij
(a non-zero scalar multiple of the identity). Without the 1/
√
3 factor the
algebra is unchanged, but the roots in the (H
1
, H
2
) plane have unequal lengths and the A
2
angles
are obscured.
6 Root generators from color transitions
6.1 Definition
A root vector of su(3) raises or lowers between weight spaces (eigenspaces of the Cartan subalgebra).
In the color basis, the off-diagonal matrix units
E
ij
= |i⟩⟨j|, i = j, i, j ∈ {1, 2, 3}, (19)
5
take the trapped node from color j to color i. As 3 × 3 matrices,
E
12
=
0 1 0
0 0 0
0 0 0
, E
13
=
0 0 1
0 0 0
0 0 0
, E
23
=
0 0 0
0 0 1
0 0 0
, (20)
and their adjoints F
ij
≡ E
†
ij
= E
ji
,
F
12
= E
†
12
= E
21
=
0 0 0
1 0 0
0 0 0
, (21)
and similarly for F
13
, F
23
. The number of distinct ordered pairs (i, j) with i = j in {1, 2, 3} is
3 × 2 = 6, giving six off-diagonal generators
{E
12
, E
13
, E
23
, F
12
, F
13
, F
23
}. (22)
6.2 Counting and dimension
Adding the two Cartan generators H
1
, H
2
to the six off-diagonal E
ij
, F
ij
yields eight operators.
The dimension of su(3) is
dim
R
su(3) = n
2
− 1 = 9 − 1 = 8 (n = 3), (23)
where the −1 subtracts the trace, and the count matches.
Vertex versus propagator. An operator that switches a quark’s color is, in QCD, a gluon
emission/absorption vertex. The E
ij
and F
ij
defined here are the algebraic content of that vertex:
the local action of a gluon on the trapped node, expressed as a transition between skew-pair labels.
They are not the propagating gluon itself. A propagating gluon is a degree of freedom living on the
FCC bulk between defects, with a dispersion relation, a mass shell, and dynamics set by the bond
energetics. The present paper constructs only the vertex algebra; the propagator and its dynamics
are the natural follow-on question we defer in Section 9.
7 Closure on su(3)
We verify by explicit matrix computation that the eight operators close on a Lie algebra, and that
the algebra is su(3). The full table of all 28 commutators is in Appendix A; here we show the
calculations needed to identify the A
2
root system.
7.1 Cartan action on root vectors
For each root vector E
ij
, the commutator with a Cartan generator H has the form
[H, E
ij
] = α
ij
(H) E
ij
, (24)
with the eigenvalue α
ij
(H) depending linearly on H. We compute [H
1
, E
12
] in detail as a worked
example; the rest follow by symmetric calculation. With H
1
= diag(1, −1, 0) and E
12
from Eq. (20),
H
1
E
12
=
1 0 0
0 −1 0
0 0 0
0 1 0
0 0 0
0 0 0
=
0 1 0
0 0 0
0 0 0
, (25)
6
E
12
H
1
=
0 1 0
0 0 0
0 0 0
1 0 0
0 −1 0
0 0 0
=
0 −1 0
0 0 0
0 0 0
, (26)
[H
1
, E
12
] = H
1
E
12
− E
12
H
1
=
0 2 0
0 0 0
0 0 0
= 2E
12
. (27)
Hence the eigenvalue is α
12
(H
1
) = 2. The same calculation method applied to all combinations of
Cartan with root vector gives the eigenvalue table
E
12
E
13
E
23
F
12
F
13
F
23
H
1
+2 +1 −1 −2 −1 +1
H
2
0 +
√
3 +
√
3 0 −
√
3 −
√
3
(28)
The columns of (28) are the root coordinates (α(H
1
), α(H
2
)) in the (H
1
, H
2
) plane:
α
12
= (2, 0), α
13
= (1,
√
3), α
23
= (−1,
√
3), (29)
together with −α
12
, −α
13
, −α
23
, plotted in Figure 2.
2 1 0 1 2
H
1
2
1
0
1
2
H
2
12
1323
12
13 23
60
120
Figure 2: The A
2
root system in the (H
1
, H
2
) plane. The six roots, recovered from the eigenvalues
of H
1
, H
2
on the color transitions E
ij
, sit on a regular hexagon with |α|
2
= 4 and 60
◦
between
adjacent roots. This is the root system of su(3).
7
7.2 Root lengths and angles
Each root has length squared
|α
12
|
2
= 2
2
+ 0
2
= 4, (30)
|α
13
|
2
= 1
2
+ (
√
3)
2
= 1 + 3 = 4, (31)
|α
23
|
2
= (−1)
2
+ (
√
3)
2
= 1 + 3 = 4. (32)
All six roots therefore have |α|
2
= 4 in this normalization. The inner products between distinct
positive roots are
α
12
· α
13
= 2 · 1 + 0 ·
√
3 = 2, (33)
α
12
· α
23
= 2 · (−1) + 0 ·
√
3 = −2, (34)
α
13
· α
23
= 1 · (−1) +
√
3 ·
√
3 = −1 + 3 = 2, (35)
giving cosines
cos θ
12,13
=
2
√
4
√
4
=
1
2
, cos θ
12,23
= −
1
2
, cos θ
13,23
=
1
2
, (36)
hence angles 60
◦
, 120
◦
, 60
◦
. This 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 [3, 4, Section 11], 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).
7.3 Root-Cartan generation
The commutators of root vectors with their negatives recover the Cartan subalgebra. We compute
[E
12
, F
12
] explicitly:
E
12
F
12
=
0 1 0
0 0 0
0 0 0
0 0 0
1 0 0
0 0 0
=
1 0 0
0 0 0
0 0 0
, (37)
F
12
E
12
=
0 0 0
1 0 0
0 0 0
0 1 0
0 0 0
0 0 0
=
0 0 0
0 1 0
0 0 0
, (38)
[E
12
, F
12
] = E
12
F
12
− F
12
E
12
=
1 0 0
0 −1 0
0 0 0
= H
1
. (39)
The analogous calculations for the other two pairs give
[E
13
, F
13
] = diag(1, 0, −1) =
1
2
H
1
+
√
3
2
H
2
, (40)
[E
23
, F
23
] = diag(0, 1, −1) = −
1
2
H
1
+
√
3
2
H
2
, (41)
where the right-hand sides are the explicit linear combinations of H
1
, H
2
that reproduce the diagonal
matrices on the left. Each result is the Cartan element H
α
dual to the corresponding root: under
the Killing form, H
α
· H
β
= α · β for any pair of roots.
8
7.4 Root addition
For roots α, β with α + β also a root, [E
α
, E
β
] is proportional to E
α+β
. We compute [E
12
, E
23
]
explicitly:
E
12
E
23
=
0 1 0
0 0 0
0 0 0
0 0 0
0 0 1
0 0 0
=
0 0 1
0 0 0
0 0 0
= E
13
, (42)
E
23
E
12
=
0 0 0
0 0 1
0 0 0
0 1 0
0 0 0
0 0 0
=
0 0 0
0 0 0
0 0 0
= 0, (43)
[E
12
, E
23
] = E
13
− 0 = E
13
. (44)
This realizes the root addition α
12
+ α
23
= (2, 0) + (−1,
√
3) = (1,
√
3) = α
13
.
For combinations that are not roots, [E
α
, E
β
] = 0. For example,
E
12
E
13
=
0 1 0
0 0 0
0 0 0
0 0 1
0 0 0
0 0 0
= 0, E
13
E
12
= 0, (45)
giving [E
12
, E
13
] = 0, consistent with the fact that α
12
+ α
13
= (3,
√
3) has length-squared 9 + 3 =
12 = 4 and is not in the A
2
root system. The full pattern of vanishing and non-vanishing brackets
is given in Appendix A and matches the A
2
adjacency rule exactly.
8 Fundamental representation
The eight generators act on H
C
= C
3
as 3 × 3 matrices by construction. To verify that this is the
fundamental representation 3 of su(3), we identify the three weights (eigenvalues of H
1
, H
2
on the
basis vectors |i⟩):
H
1
|1⟩ = +1|1⟩, H
2
|1⟩ =
1
√
3
|1⟩, µ
1
= (1,
1
√
3
), (46)
H
1
|2⟩ = −1|2⟩, H
2
|2⟩ =
1
√
3
|2⟩, µ
2
= (−1,
1
√
3
), (47)
H
1
|3⟩ = 0, H
2
|3⟩ = −
2
√
3
|3⟩, µ
3
= (0, −
2
√
3
). (48)
The three weights µ
1
, µ
2
, µ
3
form an equilateral triangle of side |µ
i
−µ
j
|
2
= 4 centered at the origin
(since µ
1
+µ
2
+µ
3
= 0). This is the weight diagram of the fundamental representation 3 of su(3) [3,
Section 13]. The matter paper’s identification of the three skew pairs with QCD color charges is
therefore the statement that the trapped node, viewed as a localized state in H
C
, transforms in the
fundamental representation of the geometrically constructed su(3).
9 Scope
The defect geometry alone gives, without further input:
1. Three colors, from the three skew-edge pairs of K
4
(Section 3).
2. The Weyl group S
3
, from the quotient S
4
/V
4
(Section 4).
3. The rank-2 maximal torus T
2
, from the three valence-bond U(1) phases modulo the diagonal
U(1) (Section 5).
9
4. Two Cartan and six root generators, totaling 8 = dim su(3) (Sections 5–6).
5. The A
2
root system at the correct lengths and angles (Section 7).
6. The Cartan-Weyl structure constants of su(3) (Appendix A).
7. The fundamental representation on the three colors (Section 8).
The construction does not deliver:
1. The QCD coupling g
s
and its running. These come from the bond energetics of the lattice,
which the present construction does not address.
2. Asymptotic freedom and Λ
QCD
, both dynamical scales.
3. Confinement as a Wilson-loop area law. The matter paper Section 6 derives a static topo-
logical version of confinement at the metric wall L/
√
3; the dynamical area law of continuum
QCD is a separate matter.
4. Higher generations. The construction is a first-shell defect classification; second-shell exten-
sions and heavier flavors require the broader spectrum analysis of [2].
10 Conclusions
The trapped tetrahedral defect of the SSM contains the data of su(3). Combinatorics gives the
colors, S
4
/V
4
gives the Weyl group, and Bell-pair phases give the maximal torus. The eight resulting
operators reproduce the Cartan-Weyl structure of su(3), with the A
2
root system at its standard
geometry. None of the inputs are adjusted to match the gauge group; they are read off from a
single trapped node in the FCC vacuum.
What this does not deliver is the dynamical content of QCD — the coupling, the running,
Λ
QCD
. Whether the SSM bond dynamics realizes those quantities is the obvious next question, and
one we do not answer here.
Acknowledgments
This work builds on the matter paper [1] of the SSM.
Data availability
The Python script that verifies the closure of all 28 commutators on su(3) and reproduces the
structure-constant table of Appendix A is publicly available at https://github.com/raghu91302/
ssmtheory/blob/main/verify_su3.py. No other data were generated or analyzed in this study.
A Commutator table
All
8
2
= 28 commutators among the eight operators close on the 8-dimensional space spanned by
{H
1
, H
2
, E
12
, F
12
, E
13
, F
13
, E
23
, F
23
}. Each entry below is computed by direct matrix multiplication
of the form [A, B] = AB − BA; the result is then expressed as a linear combination of the basis
operators using the explicit matrix forms in Eqs. (14)–(20). Three sample calculations were shown
in Section 7; the remaining 25 follow the same pattern.
10
[H
1
, H
2
] = 0 [H
1
, E
12
] = +2E
12
[H
1
, F
12
] = −2F
12
[H
1
, E
13
] = +1E
13
[H
1
, F
13
] = −1F
13
[H
1
, E
23
] = −1E
23
[H
1
, F
23
] = +1F
23
[H
2
, E
12
] = 0
[H
2
, F
12
] = 0 [H
2
, E
13
] = +
√
3 E
13
[H
2
, F
13
] = −
√
3 F
13
[H
2
, E
23
] = +
√
3 E
23
[H
2
, F
23
] = −
√
3 F
23
[E
12
, F
12
] = H
1
[E
12
, E
13
] = 0 [E
12
, F
13
] = −F
23
[E
12
, E
23
] = E
13
[E
12
, F
23
] = 0
[F
12
, E
13
] = E
23
[F
12
, F
13
] = 0
[F
12
, E
23
] = 0 [F
12
, F
23
] = −F
13
[E
13
, F
13
] =
1
2
H
1
+
√
3
2
H
2
[E
13
, E
23
] = 0
[E
13
, F
23
] = E
12
[F
13
, E
23
] = −F
12
[F
13
, F
23
] = 0 [E
23
, F
23
] = −
1
2
H
1
+
√
3
2
H
2
The simple roots in the (H
1
, H
2
) basis are α
1
= α
12
= (2, 0) and α
2
= α
23
= (−1,
√
3), with
inner product α
1
· α
2
= −2 and lengths |α
1
|
2
= |α
2
|
2
= 4. The Cartan matrix is computed as
A
ij
=
2α
i
· α
j
α
j
· α
j
, A =
2 −1
−1 2
!
, (49)
the standard Cartan matrix of the A
2
Dynkin diagram, confirming the algebra as su(3).
References
[1] R. Kulkarni, Matter as Incomplete Crystallization: Quark Charges, Color Confinement, and the
Proton Mass from a Single Extra Node in the Vacuum Lattice, SSMTheory Group, IDrive Inc.,
Calabasas, CA (2026). Manuscript number PHYSO-D-26-00185 (in revision at Phys. Open);
preprint: doi:10.5281/zenodo.18917946.
[2] R. Kulkarni, The mass-energy-information equivalence: A bottom-up identification of
the particle spectrum via FCC lattice error correction, Phys. Open (in press, 2026).
doi:10.1016/j.physo.2026.100414.
[3] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer (1972).
[4] H. Georgi, Lie Algebras in Particle Physics, 2nd ed., Westview (1999).
[5] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th an-
niversary ed., Cambridge University Press (2010).
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