From D
4
to F
4
: Color, Matter, and Charge from One Triality Twist
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com ORCID 0009-0008-8364-7155
September 2026
Abstract
Context. In the Selection–Stitch Model the vacuum is a face-centered-cubic lattice lifted
to the D
4
root lattice, whose bond algebra is so(8), and a twisted triality automorphism σ has
been proposed to select color. Problem and scope. Where can a thirds-quantized charge live,
and which su(3) can be color? No action, Hilbert space, gauge field, or dynamics is constructed.
Methods. Root systems and explicit matrix realizations of so(8) and f
4
, checked by public
scripts. Results. Within the bond algebra a fence holds: the only su(3) commuting with the
grading makes every matter sector a color octet, so triplet matter is impossible there. Adding
matter lifts the fence. Bonds and matter close into F
4
, triality becomes inner, and the fixed
subalgebra doubles to su(3)
L
⊕ su(3)
S
, of which the twisted su(3) is only the diagonal. Exactly
one factor makes the charged matter color triplets while leaving integer-charged matter color-
neutral, and under it charge modulo one follows color triality exactly as in the Standard Model.
Limitations. The integer part of the charge, which no Z
3
label can fix, and chirality, spin, and
generations do not follow; identifying the label with electric charge remains a hypothesis.
Keywords: triality automorphism; D
4
root lattice; so(8); Z
3
grading; glue classes; F
4
; exceptional
Jordan algebra; charge conjugation; Selection–Stitch Model
1 Introduction
The Standard Model quantizes electric charge in thirds and ties those thirds to color, but it takes
these facts as inputs rather than explaining them. This paper asks how far a specific discrete model
of the vacuum can account for the structure of charge — its quantization, its relation to color, and
its behavior under charge conjugation — from the geometry of the vacuum lattice alone.
1.1 The Selection–Stitch Model
The Selection–Stitch Model (SSM) is a discrete-spacetime proposal in which the vacuum is not a
smooth continuum but a physical lattice. Its pillars are stated below, since the present paper uses
their resulting geometry; a reader familiar with the series may skip to Section 2.
Growth: stitch and lift. The lattice is grown by two elementary kinematic operations on a
graph of Bell-pair entanglement bonds. A stitch adds a node within the current close-packed sheet;
a lift, an out-of-plane projection, places a node above the centroid of a triangle and so starts the
next sheet. The lift is rare for a geometric reason: a stitch is fixed by two unit-distance conditions
and admits a one-parameter family of solutions, while a lift is fixed by three and admits none, which
1
suppresses it exponentially; the published analysis fixes P
lift
= e
−3
≈ 4.98% as the amplitude for
which the growth cascade terminates at FCC [2].
The simulated crystal. Numerical simulation of these rules, with growth confined to a moving
surface front, shows that the vacuum spontaneously crystallizes into the face-centered-cubic (FCC)
packing — the densest lattice packing in three dimensions, with coordination number twelve on
every interior node. The close-packed sheets stack in the ABC order of FCC, never the ABAB
order of hexagonal close packing, with interlayer spacing h =
p
2/3 L for bond length L [2]. FCC
is therefore an output of the growth rules, not an assumption.
The vacuum as a code. The completed lattice is not inert: it carries a stabilizer quantum
error-correcting code, with checks on its vertices and voids whose commuting action maintains the
vacuum state [1]. In this reading the vacuum is a protected quantum memory, and its low-energy
excitations are constrained by the code’s structure.
Matter as a trapped node. Matter is not an added field but a defect of the vacuum: a node
trapped in one of the lattice’s tetrahedral voids, which cannot relax into the twelve-coordinated bulk
and so persists as a localized object. From this single picture the companion papers derive particle
properties: the charge −1/3 from the regular-tetrahedron bond-angle cosine and +2/3 from integer
winding under the lattice’s Bravais translations, three colors from the defect’s valence directions,
and the mass spectrum from the code’s verification costs [2, 3].
Color from the D
4
lift. The structure this paper uses is algebraic. The strong-sector program
of the series lifts the FCC spatial slice, together with the growth-front axis, to the four-dimensional
D
4
root lattice, whose bond algebra is so(8), and proposes that color is fixed by a twisted triality
automorphism of that algebra. That construction is not assumed here: Section 2 rebuilds it in full
— the root system, its automorphism group, the twist, and the fixed subalgebra — and verifies
every property this paper uses. The fixed subalgebra on the bond algebra alone is an su(3) we call
the twisted su(3); which fixed su(3) can be physical color is settled only once matter is included, in
Section 8.
1.2 Where charge could live
In the Standard Model the hypercharge assignments that fix the electric charges are inputs, and
charge conjugation relating matter to antimatter is likewise imposed. The SSM obtains color from
the geometry of its lattice, so the same geometry is where charge should be looked for. This paper
works out what the root-system structure gives and where that structure stops.
That geometry and group theory can organize charge and color is an old theme. SU(5) and
SO(10) unification place hypercharge inside a simple group [7, 11]; E
6
and its trinification sub-
group SU(3)
3
carry the fermion generation in the 27 [12, 13]; triality of so(8) has long been used
to relate the three 8-dimensional representations [6]; and the exceptional Jordan algebra, whose
automorphism group is F
4
, has been used to obtain Standard Model color from the maximal-rank
subgroup (SU(3)×SU(3))/Z
3
of F
4
[19, 20]. The present paper reaches the same su(3) ⊕su(3) by a
different route, from a triality twist of a vacuum lattice, and adds three things: which factor must be
color, why the bond algebra alone sees only their diagonal, and that, with the twist as the grading,
color then combines bond and matter generators. On the lattice side, discrete and lattice-gauge
realizations of continuum symmetry are standard [17, 18]. What is absent from that literature is
a treatment of electric charge as a grading of a twisted triality automorphism acting on a physical
vacuum lattice. Such a treatment must say explicitly what the grading can and cannot be: whether
it is an internal gauge charge or an external label, how it is normalized, how it behaves under dis-
2
crete spacetime symmetries, and where it stops short of the physical electromagnetic charge. This
paper fills that gap at the level of root-system structure, and is careful to claim nothing beyond it.
Scope: structure, not dynamics. Every result below is a statement about the D
4
and F
4
root
systems, their automorphisms, and their representations. No action, Hilbert space, quantization,
gauge field, or S-matrix is constructed; no dynamics is proposed; and the recovery of continuum 4D
Lorentz invariance from the discrete lattice — a prerequisite for any “lattice = vacuum” program
— is a recognized open problem that this paper does not solve; Section 11 establishes only the
rotational part, that anisotropy is forbidden below sixth order. The grading label studied here is
therefore a discrete label with a physical motivation, not the electromagnetic charge of a constructed
U(1); the distinction is maintained throughout, and the identification of the two is listed among
the open problems. Because the label is Z
3
-valued, it can fix charge only modulo one; the integer
part, which separates +2/3 from −1/3, must come from elsewhere. The companion paper obtains
+2/3 by a translation-winding argument that lies outside the grading [2]; whether the two routes
describe the same charge is left open.
No charge commuting with the twisted su(3) can live in the bond algebra: Theorem 1 below
shows that its commutant in so(8) is trivial. The candidate examined instead is structural: the
grading label lives in the diamond-type extension of the lattice — the glue classes of D
∗
4
/D
4
, where
the matter representations and the trapped-node defects of Section 1.1 both sit. The results below
are what that proposal derives, computed blind: the structure was worked out first, and com-
parison with Standard Model assignments was made only afterward and is labeled as comparison
throughout.
Sections 2–5 work inside the bond algebra so(8): they rebuild the twist, show that it leaves no
room for a charge commuting with the twisted su(3), derive the grading label and its conjugation
rule, and end at a fence — within so(8), color-triplet matter is incompatible with the grading.
Sections 6–8 add matter to the algebra. Bonds and matter close into f
4
, the label becomes, up to
conjugation, a coweight, and the enlarged fixed subalgebra contains exactly one color under which
the charged matter is a triplet. Sections 9–12 state the coupling fence, the falsifiers, the isotropy of
the lattice, and what is and is not derived.
2 Setup
The D
4
root system is the set of 24 vectors ±e
i
± e
j
, 1 ≤ i < j ≤ 4, each of squared length 2.
Its twelve spatial roots ±e
i
± e
j
with i, j ≤ 3 are the twelve FCC nearest-neighbor bond directions
of a constant-time slice. The automorphism group of the root system, generated by reflections in
the 24 roots together with reflections in the 24 short vectors ±e
i
and (±1, ±1, ±1, ±1)/2, has order
1152, the Weyl group of F
4
; we verified this by explicit closure. It contains 80 elements of order 3.
Standard background on D
4
, its glue, and triality is in Refs. [4, 6].
Fix the stacking axis (1, 1, 1). The six spatial roots orthogonal to it span what we call the
geometric A
2
, the regular A
2
subalgebra of these bond roots, and the plane they span is the color
plane. Exactly two order-3 automorphisms of the root system fix these six roots pointwise (one pair
per stacking axis; eight in all across the four axes); call one of them τ . It acts as the identity on the
color plane and as a rotation by 2π/3 on the orthogonal plane spanned by u = (1, 1, 1, 0)/
√
3 and
e
4
. The Weyl group of D
4
has order 192, so the quotient Aut(D
4
)/W (D
4
) has order 6; τ lies outside
W (D
4
) and induces an outer automorphism of so(8), which is triality. The 18 remaining roots fall
into six τ -orbits of three, each orbit containing one spatial and two temporal roots. The twist is
σ = Ad
e
2πi ρ·H/3
◦ τ with ρ = e
1
− e
3
the Weyl vector of the geometric A
2
(simple roots e
1
− e
2
3
and e
2
− e
3
); the fixed subalgebra g
0
is the twisted su(3), of dimension 8; by the classification of
finite-order automorphisms [5] the only other candidate fixed subalgebra is G
2
, and the computed
dimension excludes it (dim G
2
= 14). We verified the following. Although τ fixes the roots of the
geometric A
2
, no generator of that subalgebra survives σ individually, each acquiring a nontrivial
phase, so the geometric A
2
is not fixed by the twist; Section 8 shows that it is conjugate to physical
color rather than equal to it. The eight survivors are the two fixed Cartan directions and the six
orbit sums. Each orbit-sum root is the projection of the moved roots onto the color plane and has
squared length 2/3, against 2 for a bond root. The Z
3
grading of so(8) under σ has dimensions
(8, 10, 10). [verified]
3 The twisted su(3) has no commutant
Theorem 1 (Trivial commutant). The commutant of the twisted su(3) in so(8) is trivial. In
particular no u(1) in the bond algebra commutes with it.
Proof. An element commuting with g
0
commutes in particular with its Cartan subalgebra, the
two directions spanning the color plane, and so lies in the zero-weight space of that subalgebra. No
root of D
4
is orthogonal to the color plane — the only roots that could be lie in the span of u and
e
4
, and none does — so this zero-weight space is exactly the four-dimensional Cartan subalgebra of
so(8). It remains to find which Cartan elements h commute with g
0
. Each surviving orbit generator
has the form X =
P
3
i=1
c
i
E
α
i
with all c
i
= 0, because σ acts on the three root spaces of an orbit as
a cyclic shift with nonzero phases and the eigenvectors of such a shift have no vanishing component,
and [h, X] =
P
i
c
i
⟨h, α
i
⟩E
α
i
vanishes only if ⟨h, α
i
⟩ = 0 for all three roots. Across the six orbits
this requires h to be orthogonal to all eighteen moved roots, which span R
4
; hence h = 0. □
The geometric A
2
of Section 2 — the regular subalgebra spanned by the six in-plane roots
themselves — has commutant u(1) ⊕u(1), the two Cartan directions orthogonal to the color plane.
That algebra is not fixed by the twist (Section 2). The twisted su(3), whose root generators mix
one spatial with two temporal directions, leaves no room: any charge commuting with it must be
external to so(8). Section 7 shows where it goes instead. [derived]
The commutant of physical color. Section 8 identifies physical color as a different su(3),
and whether the conclusion above carries over to it depends on the grading. With the twist σ as the
grading, color is not contained in the bond algebra, and its commutant in so(8), computed on the
explicit algebra of Section 6, is trivial: no element of the bond algebra commutes with physical color
either. Under the conjugate coweight grading of Section 7, color lies inside so(8) and its commutant
there is u(1) ⊕u(1). In both cases the commutant in the full vacuum–matter algebra is exactly the
second factor su(3)
S
of Section 8, so any continuous charge compatible with color must lie there.
[verified]
Level-3 normalization. The twisted roots have squared length 2/3; the ambient so(8) roots
have squared length 2. The ratio is exactly 3: relative to the bond form, the twisted generators
carry an index-3 normalization — the embedding index of the twisted su(3) in so(8) is 3. Section 8
shows that the twisted su(3) is not physical color, so this normalization does not fix a physical
coupling. [derived]
4
4 Matter is the glue structure
The dual quotient D
∗
4
/D
4
is Z
2
× Z
2
, with three nontrivial classes represented by [1] = (
1
2
,
1
2
,
1
2
,
1
2
),
[2] = (0, 0, 0, 1), and [3] = (
1
2
,
1
2
,
1
2
, −
1
2
) [4]. The three 8-dimensional representations of so(8) sit one
per class: the weights of 8
v
lie in [2], those of 8
s
in [1], those of 8
c
in [3].
The void correspondence. The link between these classes and the lattice’s tetrahedral voids
is not an analogy but a computable projection. With the time slice spanned by e
1
, e
2
, e
3
and FCC
bond vectors (±1, ±1, 0) and permutations, project each weight onto the slice. The eight weights
of 8
s
land bijectively on the eight tetrahedral voids (±
1
2
, ±
1
2
, ±
1
2
) that surround a lattice node, and
so do the eight weights of 8
c
; at each void the two representations differ only in the sign of their
temporal component. The eight voids form the lattice’s two tetrahedral void classes, four voids
each, and each class receives four weights of 8
s
and four of 8
c
. The six spatial weights ±e
i
of 8
v
land on the six octahedral voids, and the two temporal weights ±e
4
on the node itself. FCC with
one tetrahedral void class occupied is the diamond structure (Figure 1). The companion papers
identify a particle as an extra node in exactly such a void [2, 3]; the correspondence above places
the spinor matter representations on those same sites. [derived: the class assignments and the void
correspondence; imported: the physical identification of such defects with matter]
extra node
Figure 1: The diamond extension. Black nodes are the FCC vacuum sites of one conventional
cell (corners and face centers). Orange sites are the four tetrahedral voids of one class; FCC plus
one void class is the diamond structure. Each void class carries four weights of 8
s
and four of 8
c
,
distinguished by their temporal component. Gray bonds are the nearest-neighbor bonds from each
face center to its four corners, all of length L; the bonds of the same length between adjacent face
centers are omitted for clarity, except the three thin orange ones that close the tetrahedron around
the occupied void. The thick orange bonds tie that occupied void, the extra node of the companion
matter papers, to its four cage neighbors. The dashed cube is the cell outline, not a bond.
Theorem 2 (The twist rotates the glue classes). The twist permutes the glue classes cyclically,
τ[1] → [2] → [3] → [1], and permutes the matter representations in lockstep, 8
v
→ 8
c
→ 8
s
→ 8
v
.
The operator whose fixed subalgebra is the twisted su(3) and the operator that rotates the glue classes
are the same operator.
Proof. Apply τ to each class representative and reduce modulo D
4
: the images of [1], [2], [3]
lie in [2], [3], [1] respectively. Since each 8-dimensional representation occupies a single class, the
representations are carried along the same cycle, 8
s
→ 8
v
→ 8
c
→ 8
s
. The other order-3 element
fixing the geometric A
2
, namely τ
2
, runs the cycle in reverse. □
5
5 The grading label and a color–charge fence
On the matter sector M = 8
v
⊕8
s
⊕8
c
the twist acts with order 3: the twist phases are well defined
because ρ · µ ∈ Z for all 24 weights, exactly. The eigenvalue grading of σ on M defines
ˆ
Q =
q
3
, σ
M
q
= ω
q
, q ∈ {0, +1, −1}, ω = e
2πi/3
. (1)
ˆ
Q is not an element of so(8): it is the grading derivation of the automorphism, external by con-
struction — exactly where Theorem 1 says a charge commuting with the twisted su(3) must live.
It commutes with all of g
0
because the grading respects brackets. [derived]
Theorem 3 (Uniform sectors and thirds quantization). Every τ-orbit of weight lines in M has
twist-phase sum ρ · (µ + τµ + τ
2
µ) ≡ 0 (mod 3). Hence σ
3
= 1 on M and each orbit contributes
exactly one state to each label q ∈ {0, +1, −1}. Under the twisted su(3) each of the three 8’s is
irreducible: it is the adjoint 8. Therefore every charge sector carries the identical content under the
twisted su(3)
M
q
∼
=
8, q = 0, +1, −1. (2)
The label is quantized in thirds, and the three sectors are three copies of one octet of the twisted
su(3) (Figure 2).
Proof. The phase sums are direct. For the branching, the weights of each 8 restricted to the
Cartan subalgebra of g
0
are the orthogonal projections onto the color plane: six of squared length
2/3, which are exactly the six roots of g
0
, and two zeros. A weight multiset consisting of the roots
and two zeros is the character of the adjoint. Weights of a 3 or
¯
3 would instead have squared length
1
3
·
2
3
=
2
9
. An independent check on the explicit matrix algebra confirms it: the commutant of g
0
in
End(C
8
) is one-dimensional, so 8
v
is irreducible, whereas for the geometric A
2
the same commutant
is six-dimensional, as 3 ⊕
¯
3 ⊕ 1 ⊕ 1 requires. Since σ commutes with g
0
and permutes the three 8’s
cyclically, each eigenspace M
q
is one copy of the octet. □
Theorem 4 (Charge conjugation is spatial inversion). Spatial inversion I = diag(−1, −1, −1, 1)
conjugates the twist’s root map to its inverse, Iτ I
−1
= τ
2
. Lifted to the algebra as an involution
that preserves the bond and matter sectors, it conjugates σ to σ
−1
exactly, and therefore maps
M
q
→ M
−q
. On the grading, C = P .
Proof. The root identity is direct. At the level of the algebra the lift matters, because a root
automorphism lifts only up to a phase in the Cartan torus. The lift through Pin(9), acting on the
spinor directions by a product of three gamma matrices, is not an involution, and it conjugates σ to
σ
−1
only up to such a phase. Composing it with a suitable Cartan phase gives an involution that
acts on the roots as I, preserves both sectors, and satisfies IσI
−1
= σ
−1
exactly; on the explicit
algebra of Section 6 it maps each eigenspace h
q
onto h
−q
. □
Because I conjugates σ to σ
−1
, it also exchanges the adjoint grading spaces g
1
and g
2
— the
10 and 10 — so the statement covers vacuum and matter alike. It ties the conjugation of the label
to the handedness of the crystal. The two close-packed stacking senses, ABC and ACB, are mirror
images, and on the present grading the same inversion that reverses the stacking sense reverses
all grading labels. Whether this connection yields a crystallization account of the matter excess is
open. [derived; cosmological reading open]
Theorem 5 (A color–charge fence). (i) The subalgebra of so(8) that commutes with the grading
label is exactly g
0
, the twisted su(3). (ii) Every su(3) subalgebra of so(8) has embedding index 1 or
6
I
[1]
8
s
[2]
8
v
[3]
8
c
τ
one twist: fixed subalgebra and class rotation
q = −1
color 8
Q = −1/3
q = 0
color 8
Q = 0
q = +1
color 8
Q = +1/3
I : C = P
three identical octet sectors: thirds quantization
Figure 2: The charge structure. Left: the three glue classes of D
∗
4
/D
4
, one matter representation
per class; the twist τ cycles them, and spatial inversion I reflects the wheel, fixing the vector class
[2] and exchanging 8
s
with 8
c
. Right: the grading label. Each sector q ∈ {0, ±1} is one copy
of the adjoint octet of the twisted su(3), with Q = q/3; inversion maps M
q
to M
−q
, so charge
conjugation is spatial inversion. Within the bond algebra every sector is an octet rather than a
triplet (Theorem 5); Section 8 lifts this.
3: for index 1 each of the three 8’s branches as 3⊕
¯
3⊕1⊕1, and for index 3 each is the adjoint 8; g
0
has index 3. (iii) The geometric A
2
of Section 2, spanned by the six in-plane bond roots, has index
1, so matter is a color triplet under it; but its root generators are eigenvectors of σ with eigenvalues
ω
±1
, so they carry grading label ±1/3. Hence, within the bond algebra, if the grading label is electric
charge, color is forced to be g
0
, and no matter representation is a color triplet.
Proof. (i) An element commutes with
ˆ
Q exactly when it is fixed by σ, and the fixed subalgebra
is g
0
, of dimension 8. (ii) An su(3) in so(8) acts on the 8-dimensional real orthogonal representation
8
v
. Its nontrivial real 8-dimensional orthogonal representations are the adjoint and 3 ⊕
¯
3 ⊕ 1 ⊕ 1,
since a 6 would require its conjugate and exceed dimension 8; their Dynkin indices are 3 and 1.
Because 8
v
, 8
s
, and 8
c
each have index 1 in so(8), the embedding index, and hence the branching
type, is the same on all three. Section 3 gives index 3 for g
0
. (iii) The in-plane roots have squared
length 2, the ambient value, so their index is 1; the eigenvalues of σ on their root generators are
computed on the explicit algebra. □
The one su(3) that commutes with the grading makes every matter sector an octet, while the
su(3) under which matter is a triplet carries the grading label itself. Within so(8), therefore, the
grading cannot be identified with electric charge while matter remains a color triplet. The fence is
a statement about the bond algebra alone, and it is lifted one level up: once matter is included, the
fixed subalgebra doubles, and Section 8 shows that it contains exactly one su(3) that commutes with
the grading, makes the charged matter a triplet, and leaves the integer-charged matter color-neutral.
[derived (theorem)]
6 Vacuum plus matter closes into F
4
The bond algebra so(8) has the 24 roots ±e
i
± e
j
, of squared length 2. The matter sector M =
8
v
⊕8
s
⊕8
c
has 24 weights: ±e
i
for 8
v
and (±
1
2
, ±
1
2
, ±
1
2
, ±
1
2
) for 8
s
⊕8
c
(even and odd sign count),
all of squared length 1.
7
Theorem 6 (Closure). The 48 vectors — 24 long (bonds) and 24 short (matter weights) — form
the F
4
root system: the set is closed under the reflection in every one of its members. Consequently
vacuum and matter together span a single Lie algebra f
4
, which decomposes under the bond algebra
as so(8) ⊕ M, of dimension 52 = 28 + 24: 4 Cartan directions, 24 long-root generators, and 24
short-root generators.
All 48
times48 reflection images land in the set, and the short-root set equals the matter weight set element
by element. The relative scale is not free: once the bond roots are fixed, the weights of the
mathfrakso(8) representations are fixed with them, and the resulting ratio of squared lengths, 2
to 1, is exactly the long-to-short ratio of F
4
. We also built the algebra explicitly, as f
4
= so(9) ⊕16
with the spinor bracket fixed by the Jacobi identity; it is semisimple, and its roots are exactly these
48 vectors. The Lie algebra with a given root system is unique up to isomorphism, so no choice
enters once the bond roots and matter weights are fixed. [verified]
No new substrate is introduced. The long roots are the bond directions of the lift, and the
short roots are the matter weights of Section 4, whose spinor part sits on the tetrahedral voids;
both inventories were fixed before this section, for independent reasons. Theorem 6 adds no input:
it observes that the two close jointly under their own reflections, an identity with no adjustable
content that a generic pair of sets would fail. What remains a physical hypothesis is dynamical —
that the code’s interactions realize this bracket on physical operators. [derived: the closure; open:
its dynamical realization]
The bracket and its sectors. At root level, [E
α
, E
β
] ∝ E
α+β
when α + β is a root. Checking
all pairs, two matter weights from the same representation sum only to long roots or to zero, two
from different representations sum only to weights of the third, and bond-plus-matter sums land
only in matter. Read physically, two matter states of the same type combine into a bond (a state
and its conjugate weight into the Cartan), two of different types fuse into the third by the triality
product, and bonds act on matter without creating it. [derived: the sector structure; open: the
interaction reading]
7 Triality is inner, and the label is a coweight
Theorem 7 (Triality is inner). The reflection group generated by the 48 roots of F
4
has order
1152 and contains the twist’s rotation τ . What is an outer automorphism of so(8) is an inner
automorphism of f
4
.
The order-1152 group regenerated from the 48 reflections is the group of Section 2: the auto-
morphism group of the D
4
root system is the Weyl group of F
4
, acting by inner automorphisms on
the completed algebra. [verified]
Theorem 8 (Grading and coweight label). The twist σ grades f
4
with dimensions (16, 18, 18)
and restricts on so(8) to the grading (8, 10, 10) of Section 2. In the simple system α
1
= e
2
− e
3
,
α
2
= e
3
−e
4
, α
3
= e
4
, α
4
= (e
1
−e
2
−e
3
−e
4
)/2, the highest root is e
1
+ e
2
= 2α
1
+ 3α
2
+ 4α
3
+ 2α
4
,
with marks (2, 3, 4, 2), and the roots with α
2
-coefficient ≡ 0, 1, 2 (mod 3) number 12, 18, 18. The
grading of σ is conjugate, by an inner automorphism of f
4
, to the grading by the coweight of the
mark-3 node. In that conjugated frame every root vector is homogeneous, with label
q =
coefficient of α
2
mod 3, Q = q/3, (3)
where q ≡ 2 is read as −1.
8
Proof. The σ-grading is computed on the explicit algebra, where σ is an automorphism of order
3, and the coweight grading is counted on the roots; they agree dimension by dimension. The inner
automorphisms of order 3 of f
4
fall into three classes [5], obtained by deleting from the affine F
4
diagram either the mark-3 node, which leaves the semisimple A
2
× A
2
of dimension 16, or one of
two pairs of nodes, which leave fixed algebras of dimension 22 with a u(1) center. The fixed algebra
of σ is semisimple of dimension 16, so σ lies in the mark-3 class. In the paper’s own frame, by
contrast, σ mixes the root vectors of each τ -orbit, so a moved root vector is not homogeneous there.
□ [verified; conjugacy by the classification]
Section 3 found no room for a charge inside so(8) that commutes with the twisted su(3). That
finding stands and is now completed: the grading label is external to the bond algebra and internal
to the vacuum–matter algebra: σ is inner there, and in the conjugated frame the label counts how
many times the mark-3 simple root appears in a state’s root expansion, mod 3 (Figure 3). The
algebra had to grow to contain matter before it could contain the label. [derived]
h
0
(16)
su(3)
L
⊕ su(3)
S
h
−1
(18)
10 ⊕ M
−1
h
+1
(18)
10 ⊕ M
+1
[ ·, ·] [ ·, ·]
[h
−1
, h
−1
] ⊂ h
+1
f
4
= h
−1
⊕ h
0
⊕ h
+1
; the label Q = q/3 indexes the columns
Figure 3: The grading of the vacuum–matter algebra. Each sector combines a bond part (10, 10, or
the twisted su(3)) with a matter part M
q
, and the bracket respects the grading. The fixed sector
h
0
is su(3)
L
⊕su(3)
S
; the twisted su(3) of the bond algebra is its diagonal, and physical color is the
factor su(3)
L
(Section 8).
8 The color that commutes with the grading
The zero-label sector h
0
has dimension 16 and rank 4. On the explicit algebra it splits into two
commuting simple ideals of dimension 8, and deleting the mark-3 node from the affine F
4
diagram
identifies it as A
2
× A
2
[5]: h
0
= su(3)
L
⊕ su(3)
S
, where the subscripts record that the two factors
correspond to F
4
’s long and short A
2
root subsystems. Their embedding indices in f
4
, computed
from the ratio of the f
4
trace form to each factor’s own, are exactly 1 and 2. [verified ]
Theorem 9 (The twisted su(3) is the diagonal). The fixed subalgebra of the bond algebra, g
0
=
h
0
∩ so(8), is neither factor: it projects isomorphically onto both, so it is a diagonal su(3) of
su(3)
L
⊕ su(3)
S
. Its index on the 26 of f
4
is 9 = 3 + 6, the sum of the indices of the two factors.
Proof. On the explicit algebra h
0
∩ so(8) is eight-dimensional and its projections onto the two
ideals both have rank 8. The indices on the 26, whose weights are the 24 short roots and a two-fold
zero, are 3 for su(3)
L
, 6 for su(3)
S
, and 9 for g
0
, and indices add over a diagonal. □
The carrier of matter states. The matter weights appear twice in f
4
. They are the short-
root directions of the algebra (Section 6), and with a two-fold zero weight they are the weights of
the 26, the smallest nontrivial representation of f
4
and the traceless part of the exceptional Jordan
9
algebra [19, 20]. A space of matter states must carry an action of the whole algebra, and the
short-root directions alone do not form one: the bracket of two matter directions contains bond
directions. We therefore take the matter states to be the 26. Both carriers give the same color
content where it matters: in the algebra itself the charged sectors are (3, 6) at Q = −1/3 and (
¯
3,
¯
6)
at Q = +1/3, color triplets combining bond and matter directions, while the neutral sector is the
fixed subalgebra (8, 1) ⊕ (1, 8). [verified: both branchings]
Theorem 10 (Color is the long factor). Every su(3) in f
4
that commutes with the grading is su(3)
L
,
su(3)
S
, or a diagonal. The 26 decomposes by label as
26 = (3,
¯
3)
|{z }
Q=−1/3
⊕ (
¯
3, 3)
|{z }
Q=+1/3
⊕ (1, 8)
|{z }
Q=0
(4)
under su(3)
L
× su(3)
S
. Hence only su(3)
L
makes the charged matter a color triplet while leaving
the q = 0 matter, whose charge is an integer modulo one, color-neutral. Under su(3)
S
the neutral
sector is a color octet, and under any diagonal every sector contains an octet.
Proof. An su(3) commuting with the grading is fixed by σ, so it lies in h
0
. Its projection onto
each simple factor is either injective or zero, so it is a factor or the graph of an isomorphism between
them. The 26 is the traceless part of the exceptional Jordan algebra [19, 20], and its decomposition
is the full character computation: the weights in each label class, paired with the coroots of both
factors, reduce to the stated irreducibles. Under a diagonal, 3 ⊗
¯
3 = 8 ⊕1 and the neutral 8 remain
octets. □
The step from the theorem to an identification uses one physical requirement, imported from
the Standard Model rather than derived here: that integer-charged matter be color-neutral. With
it, physical color is su(3)
L
, and three facts characterize it.
First, with σ as the grading, it is not a symmetry of the bond algebra alone: on the explicit
algebra its generators have full-rank components in both the bond algebra and the matter sector,
so color is a combination of the two.
Second, it is not the twisted su(3), which is its diagonal with su(3)
S
. The identification of
color with the twisted su(3), proposed on the bond algebra alone (Section 1.1), is the shadow that
physical color casts there.
Third, it is conjugate in f
4
to the geometric A
2
of Section 2. All sixteen long-root A
2
subsystems
of F
4
form a single Weyl orbit; the geometric A
2
is one of them, and in the conjugated frame of
Theorem 8 so is the long factor. If the three colors that the companion papers read off the defect’s
valence directions are the geometric A
2
of the bond plane, as the construction of Section 2 takes
them to be, then they are recovered, up to an inner automorphism, as the one color that commutes
with the grading. [derived (theorem); verified on the explicit algebra]
The twist and the coweight grading. The twist σ and the mark-3 coweight grading of
Theorem 8 are conjugate in f
4
, so they assign identical color and charge quantum numbers to
the 26. They are not the same grading of the geometric bond–matter split, and on the explicit
algebra they differ in two ways. The twist grades so(8) as (8, 10, 10) and cycles the three matter
types, 8
v
→ 8
c
→ 8
s
→ 8
v
, which is the glue-class rotation of Theorem 2; the coweight grading
grades so(8) as (10, 9, 9) and leaves each 8 in place. Under the twist, color mixes bond and matter
generators; under the coweight grading, color lies entirely inside so(8). This paper takes σ as the
grading because it is the lattice’s triality twist, the operator that rotates the glue classes. The
author’s extension to E
6
, accepted for publication in Symmetry, works in the coweight frame, where
color is built from bond roots. Which grading the dynamics realizes is not settled here. [verified]
10
Under su(3)
L
the (3,
¯
3) block is three color triplets with label Q ≡ −1/3 ≡ 2/3 (mod 1), the
class shared by the down-type charge −1/3 and the up-type charge +2/3; they form one antitriplet
of su(3)
S
rather than three singlets. The conjugate block carries Q ≡ 1/3, and the color singlets
carry 0. All three colors of a triplet share one label, because su(3)
L
commutes with σ. With triality
t = 1 for 3, 2 for
¯
3, and 0 for singlets, every block obeys Q ≡ −t/3 (mod 1), the relation between
charge and triality that the Standard Model obeys [9]. [derived]
Three residues bound this reading. The label fixes charge only modulo one, so it places up-
and down-type quarks in the same class and cannot supply the integer part that separates them
(Section 12). The color singlets form one octet of su(3)
S
rather than three lepton families. And
the horizontal triplet has the shape of three families without their content — chirality, spin, and a
mass hierarchy do not yet attach to it. [open]
9 A fence: the simplest coupling identification fails
The bond algebra suggests two candidate ratios for the hypercharge-to-color coupling at the lattice
scale: 3, from the index-3 normalization of the twisted su(3), and 1/3, from the Z
3
charge quantum.
Section 8 already removes the first on structural grounds, since the twisted su(3) is not physical
color and its normalization was therefore never a color coupling. The running excludes both.
One-loop Standard Model running [10] from M
Z
to the Planck mass gives α
−1
3
≈ 52.4, α
−1
2
≈
49.5, and α
−1
1
≈ 33.3 in GUT normalization [7]. Converting to the hypercharge coupling itself,
α
−1
Y
=
5
3
α
−1
1
≈ 55.4, so α
3
/α
Y
≈ 1.06 at the lattice scale; in GUT normalization the corresponding
ratio is α
3
/α
1
≈ 0.64. Either way the ratio is far from 3 and from 1/3, so both identifications are
excluded independently of the normalization convention. The exclusion does not depend on the
precise matching scale: the lattice scale sits somewhat below the Planck mass, and lowering the
matching scale by a factor of up to 3 moves the two ratios only to 1.06–1.11 and 0.64–0.66. The
comparison uses unmodified Standard Model running and is therefore a bare-SM consistency check
rather than a model-specific prediction: any states the framework adds below the lattice scale would
alter it.
The normalization of su(3)
L
relative to the grading is not fixed here, and the constraint that
remains is sharp: any SSM derivation of coupling magnitudes must produce near-equal couplings
at the lattice scale, so bare geometric ratios are not the whole story. [excluded]
10 Predictions and falsifiers
Two falsifiable statements follow from the structure.
No millicharged states. The label of Theorem 3 is the eigenvalue grading of an order-3
automorphism. It is discrete and exact: there is no small parameter in the construction, so nothing
can shift a label away from a multiple of 1/3 by any amount. If all matter is a lattice defect, as
the framework asserts, then every particle — including any dark-sector state on the same lattice
— carries a grading label that is an exact multiple of 1/3, with zero deviation. Consequently, if
the label is realized as physical electric charge modulo one by the dynamics left open here, no
millicharged state is possible. A single confirmed millicharged particle would refute the conjunction
of the two premises: that all matter is a lattice defect, and that the grading label is the physical
charge modulo one. Dedicated searches are active [8, 10]. [derived, conditional on the framework
premise that all matter is a lattice defect]
11
Charge and triality. In the Standard Model, charge modulo one is fixed by color triality t:
Q ≡ −t/3 (mod 1), so that both quark charges +2/3 and −1/3 are ≡ 2/3, antiquarks are ≡ 1/3,
and color singlets carry integer charge [9]. Within the bond algebra the grading violates this, placing
its thirds on octets of triality 0 (Theorem 5). With color identified as su(3)
L
, the completed algebra
obeys it exactly (Theorem 10). This is a falsifiable statement: a color-singlet state with fractional
charge, or a colored state whose charge violates the relation, would refute the identification of the
label with charge modulo one. Searches for free fractional charges bear on it directly [10]. What
the label cannot test is the integer part of the charge, which it does not fix. [derived: the relation;
open: the integer part]
The thirds themselves are a postdiction: quark charges have been known for sixty years. What
is new is a structural source for them — an automorphism grading rather than an assignment —
and what that source would forbid if it is identified with physical charge.
11 Rotational isotropy and Lorentz invariance
A discrete vacuum cannot carry exact rotational symmetry, so a “lattice = vacuum” program owes
an account of why the world looks isotropic. This section gives the part that follows from the
lattice alone, and states what it does not reach. Nothing here depends on the grading or on the
identification of color; it is a property of D
4
.
The point group and the second moment. An effective description built on a lattice
is constrained by the point group: every term must be invariant under it, so the order at which
anisotropy may first appear is fixed by the invariant polynomials of that group, not by the dynamics.
Two facts about D
4
then do the work. Its 24 roots have the isotropic second moment
S
µν
=
X
α
α
µ
α
ν
= 12 δ
µν
, (5)
so no quadratic anisotropy exists, in all four directions and not only the three spatial ones. And its
point group is Aut(D
4
) = W (F
4
) of order 1152 (Section 2), which is larger than the point group of
any other four-dimensional lattice.
Theorem 11 (Isotropy to sixth order). The spaces of W (F
4
)-invariant polynomials in degrees 2,
4 and 6 have dimensions 1, 1 and 2. The unique quartic invariant is (k
2
)
2
. Hence no quartic
anisotropy exists on the D
4
lattice, and the first anisotropic invariant appears at degree 6.
Proof. The dimensions are the coefficients of the Molien series |G|
−1
P
g
det(1−tg)
−1
, computed
here by closing the reflection group in exact arithmetic and evaluating the series, and independently
by averaging the action of all 1152 elements on the monomials of each degree; the two agree. A
single invariant in degree 2 and a single one in degree 4 leave (k
2
) and (k
2
)
2
as the only possibilities,
both isotropic. □
Comparison with the hypercubic lattice. For the hypercubic lattice the same computation
gives 1, 2 and 3. The extra quartic invariant is
P
i
k
4
i
, the familiar lowest-order anisotropy of cubic
lattices, and it is absent on D
4
. Table 1 collects the comparison.
Triality is what protects it. The protection comes from the part of the point group that the
Weyl group does not contain. Using only W (D
4
), of order 192, the invariant dimensions in degrees
2, 4, 6 are 1, 3, 4: a quartic anisotropy exists, exactly as on a cubic lattice. Adjoining the triality
automorphisms, which is what raises W (D
4
) to Aut(D
4
) = W (F
4
), removes it. The same S
3
that
grades the charge in Section 7 is the one that pushes anisotropy from degree 4 to degree 6.
12
Table 1: Dimensions of the spaces of invariant polynomials of the two point groups. The D
4
lattice
has no quartic invariant beyond (k
2
)
2
, so its leading anisotropy is pushed from degree 4 to degree
6.
degree D
4
: W (F
4
), order 1152 hypercubic: W (B
4
), order 384 isotropic only
2 1 1 1
4 1 2 1
6 2 3 1
Degree 6 is the best a crystal can do. No finite group gives exact isotropy. The invariant
ring of a rank-4 reflection group is generated by four basic invariants whose degrees multiply to
the group order; for W (F
4
) these are 2, 6, 8, 12, with 2 · 6 · 8 · 12 = 1152. The degree-2 generator
is k
2
, and the next generator is by construction not a power of it, so anisotropy is unavoidable
at that degree. Only an infinite group, O(4) itself, has an invariant ring generated by k
2
alone.
Among rank-4 reflection groups just one does better than W (F
4
), the 600-cell group H
4
with degrees
2, 12, 20, 30; it is not crystallographic, so no lattice can carry it. Degree 6 is therefore optimal for a
four-dimensional crystal, and D
4
attains it. [verified]
Isotropy is a four-dimensional prop erty. The result is a property of the four-dimensional
lattice and not of its spatial slice. The fourth moment of the twelve FCC bond directions of a
constant-time slice, taken alone, is cubic-anisotropic: writing T
µνλρ
= α(δδ+δδ+δδ)+β
P
i
e
µ
i
e
ν
i
e
λ
i
e
ρ
i
gives α = 1 and β = −1, and a companion paper uses exactly this to bound the leading Lorentz-
violating correction on the slice [14]. Over all 24 roots of D
4
, the twelve temporal bonds cancel
that term exactly: the same decomposition gives α = 1 and β = 0. The anisotropy of the slice
is removed by the directions the slice does not contain, which is the tensor form of Theorem 11.
[verified]
The permitted dispersion relation. Any dispersion relation on this vacuum has the form
ω
2
= c
2
k
2
1 + b
1
(ka)
2
+ O
(ka)
4
, with a the lattice spacing. The term b
1
(ka)
2
comes from the
unique quartic invariant (k
2
)
2
and is isotropic, so it rescales the speed without selecting a direction;
the first direction-dependent term is O
(ka)
4
, two orders beyond a cubic lattice, where anisotropy
already enters at (ka)
2
. At a spacing near the Planck length both terms are minute. For an optical
photon (ka)
2
≈ 10
−55
; even for a cosmic ray at 10
20
eV, the largest available momentum, the
isotropic term is ≈ 2 × 10
−16
and the anisotropic one ≈ 3 × 10
−32
. The anisotropy is therefore not
measurable by any foreseeable experiment, and this section should be read as a consistency result
rather than a falsifier: what it establishes is that the lattice’s own symmetry forbids the leading
anisotropy, not that an experiment could see it if it did not. The distinction matters because tree-
level smallness is not by itself protection: loop corrections can promote a Planck-suppressed Lorentz-
violating operator to an O(1) effect unless a symmetry forbids it [15]. A forbidden invariant cannot
be generated at any order, which is what invariant theory delivers here and numerical suppression
does not. [verified]
Boosts, speeds, and experiment. Three things remain, and none follows from the point
group.
First, boosts. The lattice has a rest frame, and its symmetry group is finite, so it contains no
boost. Rotational isotropy to any finite order says nothing about the non-compact directions of the
Lorentz group.
Second, a universal limiting speed. Emergent relativistic behavior requires every species to
share one speed c. Nothing here forces that, and radiative corrections generically split the speeds
13
of different species unless a symmetry protects them, which is a known fine-tuning problem for
emergent Lorentz invariance [15]. The companion paper that derives a laboratory threshold on this
vacuum assumes the single speed as a stated postulate rather than deriving it [14], and the same
gap is left open here.
Third, the experimental confrontation. Tests of Lorentz invariance constrain preferred-frame
effects tightly across many sectors [16], and comparing this vacuum against them requires the
dynamics the paper does not construct: a dispersion relation derived from a Hamiltonian rather
than permitted by a symmetry.
The summary is narrow. Rotational isotropy is protected to sixth order, as a theorem about
the lattice. Lorentz invariance is not established, and the boost sector is untouched. [open]
12 What is derived and what is not
Table 2 gives the status of every claim. Two of the open items need comment.
Color as a bond–matter symmetry. With the twist σ as the grading, physical color su(3)
L
combines bond and matter directions (Section 8), so a color rotation acts on vacuum and matter
together, not on the bonds alone. Under the conjugate coweight grading, color is a symmetry of
the bonds. The two gradings therefore make different claims about the same lattice: if no color
rotation can be implemented by bond operations alone, the twist is the physical grading; if one can,
the coweight grading is. Which the code dynamics realizes is open. [open]
The integer part. The grading fixes charge modulo one, with representatives {0, ±1/3}. The
integer part, which separates the up-type +2/3 from the down-type −1/3 and the electron from the
neutrino, cannot be fixed by any Z
3
label; in the Standard Model it comes from the weak isospin.
Two candidate sources exist in the series, and neither is the grading. The companion matter
paper obtains +2/3 from integer winding under the lattice’s Bravais translations [2], and a separate
analysis by the author of the F
4
→E
6
extension, accepted for publication in Symmetry, reaches a
parity-violating su(2) on defect worldlines but stops short of the chiral projection. Whether either
route supplies the integer part that this grading cannot fix, and whether the two routes agree, is
open. [open]
14
Table 2: Status of every claim in the paper, in section order, with the open items last. The label
key is given in the first row.
statement status
Label key: verified = established by the verification scripts; derived (theorem) = proved analytically; derived
= a direct consequence without separate theorem; imported = from a cited work; excluded = ruled out; open
= not settled here. A qualifier after a comma or semicolon (e.g. conditional) narrows a label without adding
a new one; the same labels are used inline in the text.
Aut of the D
4
roots has order 1152; 80 order-3 elements verified
twisted su(3) reconstructed; grading (8, 10, 10) verified
commutant of the twisted su(3) in so(8) trivial derived (theorem)
commutant of color in f
4
is su(3)
S
(both frames) verified
twisted su(3) has index 3 in so(8) derived
twist 3-cycles glue classes and matter reps together derived (theorem)
matter reps sit one per glue class derived
spinor weights project bijectively onto a node’s tetrahedral
voids
derived
defects in those voids are matter imported [2, 3]
ˆ
Q = q/3 external to so(8), conserved; thirds quantization derived
under the twisted su(3) each sector is an octet, M
q
∼
=
8 derived (theorem)
only g
0
commutes with the grading in so(8); index 3 derived (theorem)
geometric A
2
gives triplets but carries label ±1/3 derived (theorem)
triplet matter with the grading as charge, within so(8) excluded, fence
C = P on the grading, for the involutive lift of inversion derived (theorem)
bonds + matter close into F
4
; f
4
= so(8) ⊕ M derived (theorem); verified
triality outer on so(8), inner in f
4
; grading conjugate to the
mark-3 coweight
derived (theorem); verified
h
0
= su(3)
L
⊕ su(3)
S
, indices 1 and 2 verified
twisted su(3) is the diagonal (index 9 = 3 + 6) derived (theorem)
matter states form the 26, the smallest irrep of f
4
verified
26 = (3,
¯
3)
−1/3
⊕ (
¯
3, 3)
+1/3
⊕ (1, 8)
0
derived (theorem); verified
color = su(3)
L
, unique if integer-charged matter is color-
neutral
derived (theorem)
color mixes bond and matter under σ; lies in so(8) under
the coweight
verified
commutant of color in so(8): trivial under σ, u(1)
2
under
the coweight
verified
color conjugate to the geometric A
2
in both frames verified
charge mod 1 obeys Q ≡ −t/3, the Standard Model relation derived
no quartic anisotropy on D
4
(first at degree 6), due to tri-
ality; optimal for a crystal
verified
emergent Lorentz invariance (boosts, a universal speed) open
label exact in thirds; no millicharge if label = charge mod
1
derived, conditional
coupling ratios 3 and 1/3 at the lattice scale, any scale
M
P
/3–M
P
excluded, bare-SM running
label identified with electric charge modulo one open
integer part of the charge (not fixable by a Z
3
label) open
which grading the dynamics realizes (twist or coweight) open
three families; lepton sector in family form open
15
13 Conclusion
The twist that was proposed to select color does two things in the bond algebra. It grades the
matter representations, whose spinor parts sit on the lattice’s tetrahedral voids, into a conserved
label quantized in thirds, and, for the involutive lift of inversion, it identifies charge conjugation
with spatial inversion. But within so(8) the only su(3) it leaves fixed makes every matter sector an
octet, so the bond algebra alone cannot hold color-triplet matter with this label as its charge.
Adding matter to the algebra resolves this. Bonds and matter close into f
4
, the twist becomes
inner, and the label becomes, up to conjugation, a coweight. The fixed subalgebra doubles to
su(3)
L
⊕su(3)
S
, and exactly one of its factors makes the charged matter color triplets while keeping
the integer-charged matter color-neutral: su(3)
L
. With it as color, charge modulo one follows
color triality exactly as in the Standard Model. This color is not the twisted su(3), which is only
its diagonal shadow on the bonds. With the twist as the grading it combines bond and matter
generators; under the conjugate coweight grading it lies among the bonds; in either frame it is
conjugate in f
4
to the geometric color of the lattice.
Independently of the grading, the lattice also protects rotational isotropy: the point group
W (F
4
) admits no quartic anisotropy, so the first direction-dependent term in a dispersion relation
is two orders higher than on a cubic lattice. Boosts and a universal limiting speed are not addressed.
The following do not follow: the integer part of the charge, spin, chirality, the three genera-
tions, the lepton families, and the coupling magnitudes, for which the simplest geometric ratios are
excluded. The results are statements about root systems and their automorphisms; identifying the
label with electric charge modulo one, and supplying the integer part, would further require a U(1)
gauge sector and dynamics that are not constructed here. The open items are listed in Table 2.
Data availability
All computations are specified in the text. Two Python scripts reproduce every check in this paper,
with labeled output keyed to the sections and theorems; both require NumPy only. The first,
https://github.com/raghu91302/ssmtheory/raw/main/charge_scripts.zip, covers the bond-
algebra results of Sections 2–5 and the coupling fence of Section 9. The second, https://github.
com/raghu91302/ssmtheory/raw/main/f4_scripts.zip, covers the F
4
results of Sections 6–8, the
algebra-level form of Theorem 4, the commutant of color in Section 3, the two gradings of Section 8,
the charge–triality relation of Section 10, and the invariant degrees of Section 11.
Funding
This research received no external funding.
Conflicts of interest
Author Raghu Kulkarni was employed by SSMTheory Group, IDrive Inc, Calabasas, USA. The
author declares that the research was conducted in the absence of any commercial or financial
relationships that could be construed as a potential conflict of interest.
16
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