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
August 2026
Abstract
In the Selection–Stitch Model the vacuum is a face-centered-cubic lattice, its four-dimensional
home is the D
4
root lattice, and color is the fixed subalgebra of a twisted triality automorphism
σ of so(8). This paper asks where electric charge lives in that structure and answers in two
movements, every step verified by exhaustive computation. First, a no-go and its consequences at
the level of the bond algebra: the commutant of the twisted color in so(8) is trivial, so no internal
hypercharge exists there; the twisted color roots carry an exact index-3 normalization; the twist
that selects color cyclically permutes the three glue classes of D
∗
4
/D
4
, with the three 8-dimensional
matter representations sitting one per class; the eigenvalue grading of the twist defines a conserved
charge quantized exactly in thirds, with every charge sector carrying the identical color content
3 ⊕
¯
3 ⊕ 1 ⊕ 1; and spatial inversion conjugates the twist to its inverse, so on the grading charge
conjugation is spatial inversion. Second, the completion that removes the one weakness of that
construction — the charge’s externality: the 24 bond roots and the 24 matter weights together
form exactly the 48-root system of F
4
, so vacuum and matter close into f
4
= so(8)⊕M, dimension
52 = 28 + 24, with the Lie bracket reproducing the interaction table. Triality is inner in f
4
, and
the charge becomes a Cartan coweight: the charge of any root vector is the coefficient of the
mark-3 simple root, mod 3. Charge is external to the bond algebra and internal to the vacuum–
matter algebra, which is as it should be: charge is a property of matter. The fixed subalgebra is
su(3) ⊕ su(3) — color and an emergent second su(3) — and its first loaded test is passed here:
the 26 of f
4
branches as (3,
¯
3) ⊕ (
¯
3, 3) ⊕ (1, 8), three families of colored matter at electric charge
∓1/3. Because the grading is discrete and exact, millicharged particles are forbidden outright.
Renormalization-group running excludes the two simplest coupling identifications. Every claim
carries a status tag.
1 Introduction
The question. The Standard Model quantizes electric charge in thirds, correlates those thirds with
color, and relates matter to antimatter by charge conjugation. None of this is explained there; the
hypercharge assignments are inputs. A framework that derives color from geometry owes an account
of where charge lives in the same geometry. This paper gives that account in two movements: a
no-go at the level of the bond algebra, and the completion that turns the no-go into structure.
The starting point. The Selection–Stitch Model (SSM) treats the vacuum as a face-centered-cubic
(FCC) lattice carrying a stabilizer code [1]; matter is a defect of the lattice [2, 3]. The strong-sector
program of the series lifts the FCC time slice to the D
4
root lattice and obtains color as the fixed
subalgebra of a twisted triality automorphism of so(8); a journal version of that construction is
in preparation. Nothing here depends on it: Section 2 builds the construction in full and verifies
every property this paper uses. A script reproducing every check is linked in the Data Availability
1
statement.
The two movements. Movement one (Sections 3–5): within the bond algebra there is no room
for charge — the commutant of the twisted color is trivial — so charge is constructed as the grading
of the twist, external to so(8) by definition, and its consequences are derived: thirds quantization,
identical charge sectors, and charge conjugation as spatial inversion. The candidate examined is
structural: charge lives in the diamond-type extension of the lattice — the glue classes of D
∗
4
/D
4
,
which is where the matter representations sit and where the companion papers place physical defects.
Movement two (Sections 6–8): the externality is an artifact of stopping at so(8). Bonds and matter
weights together are the long and short roots of F
4
; in the vacuum–matter algebra f
4
the same charge
is a Cartan coweight, triality is inner, and a second su(3) emerges whose first loaded test is passed
on the page.
Protocol. As throughout the series, the structure was computed first and compared with known
physics afterward; comparisons are labeled as comparisons. Standard background on D
4
, F
4
, triality,
and the octonions is in Refs. [4, 7, 6]; the automorphism classification is in Ref. [5]; the exceptional
Jordan route to related structures is in Ref. [11].
2 Setup
The lattice and its symmetries. 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].
Color and the twist. Fix the stacking axis (1, 1, 1). The six spatial roots orthogonal to it form
the color A
2
. Exactly two order-3 automorphisms fix this A
2
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 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 the Weyl vector of the color A
2
; 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:
no color-A
2
generator survives individually (each acquires a nontrivial phase); the eight survivors are
the two fixed Cartan directions and the six orbit sums; the orbit-sum roots are the projections of
the moved roots onto the color plane; and the Z
3
grading of so(8) under σ has dimensions (8, 10, 10).
[all reconstructed and verified]
3 No internal hypercharge, and level-3 color
Theorem 1 (No internal hypercharge) The commutant of the twisted su(3) in so(8) is trivial.
In particular no u(1) in the bond algebra commutes with color.
Proof. A Cartan direction h commutes with an orbit sum
P
i
E
α
i
only if ⟨h, α
i
⟩ is the same for
all three roots of the orbit. For h in the fixed plane this holds and h lies inside g
0
; for h with
any component in the rotated (u, e
4
) plane it fails, verified directly against all six surviving orbits.
Non-Cartan candidates are excluded by the grading: so(8) = g
0
⊕ g
1
⊕ g
2
with g
1,2
the irreducible
10 and
¯
10 of g
0
, which contain no singlets. □
2
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, and these sites represent one glue class of D
∗
4
/D
4
in the time
slice. Gray bonds are the lattice’s nearest-neighbor bonds, the face diagonals, all of length L; the
dashed cube is the cell outline, not a bond. The highlighted orange bonds tie one occupied void, the
extra node of the companion matter papers, to its four cage neighbors. Axonometric projection.
One clarification guards against a natural error. The regular A
2
subalgebra — the one 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 the model’s color. The twisted su(3), whose root generators
mix one spatial with two temporal directions, leaves no room: hypercharge must be external to so(8).
[derived]
Level-3 color. 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 color generators carry an index-3
normalization. This is the first structure-fixed coupling normalization in the series, with no adjustable
content. [derived]
4 Matter is the glue structure
Glue classes. 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]. In three
dimensions the analog of occupying a glue class is occupying a tetrahedral void: FCC plus one void
class is the diamond structure. The companion papers place matter exactly there — a particle is an
extra node in a tetrahedral void [2, 3] — so the matter representations and the physical defects live
on the same extension of the vacuum lattice (Figure 1). [structure derived; the physical identification
of defects with glue-class occupancy is imported]
Theorem 2 (Color selection rotates the charge 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 that selects color and the operator that rotates the charge classes are the same
operator.
Proof. Direct computation of τ on the class representatives modulo D
4
, and of τ on the highest
weights. □
3
[1]
8
s
[2]
8
v
[3]
8
c
τ
I
one twist: color and class rotation
q = −1
3 ⊕
¯
3
1 ⊕ 1
Q = −1/3
q = 0
3 ⊕
¯
3
1 ⊕ 1
Q = 0
q = +1
3 ⊕
¯
3
1 ⊕ 1
Q = +1/3
I : C = P
three identical 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 that selects color cycles them, and spatial inversion I reflects the wheel. Right:
the grading charge. Each sector q ∈ {0, ±1} carries the identical color content 3 ⊕
¯
3 ⊕ 1 ⊕ 1, with
Q = q/3; inversion maps M
q
to M
−q
, so charge conjugation is spatial inversion. Both panels are
drawn to true aspect ratio.
5 The grading charge and the periodic table
The charge operator. 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, verified directly. 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 charge 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), verified for all orbits. Hence σ
3
= 1 on M and each
orbit contributes exactly one state to each charge q ∈ {0, +1, −1}. Under the twisted color, each of
the three 8’s branches as 3 ⊕
¯
3 ⊕ 1 ⊕ 1 (projected weight norms 2/3 six times and 0 twice, in each
rep); therefore every charge sector M
q
carries the identical content
M
q
∼
=
3 ⊕
¯
3 ⊕ 1 ⊕ 1, q = 0, +1, −1. (2)
Charge is quantized in thirds, and matter organizes into a three-column periodic table (Figure 2):
three charge sectors, each holding one color triplet, one antitriplet, and two singlets.
Theorem 4 (Charge conjugation is spatial inversion) Spatial inversion I = diag(−1, −1, −1,
1) conjugates the twist to its inverse: IτI
−1
= τ
2
, verified directly. Therefore I maps M
q
→ M
−q
.
On the grading, C = P .
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 matter–antimatter distinction
to the handedness of the crystal. The two close-packed stacking senses, ABC and ACB, are mirror
images; a companion analysis in preparation finds that a defect worldline converts this handedness
4
into parity violation. On the present grading the same inversion that reverses the stacking sense
reverses all charges. Whether this connection yields a crystallization account of the matter excess is
an open question we flag and do not pursue. [derived; cosmological reading open]
6 Vacuum plus matter closes into F
4
The two root sets. 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.
Theorem 5 (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
= so(8) ⊕ M, of dimension 52 = 28 + 24:
4 Cartan directions, 24 long-root generators, 24 short-root generators.
Verified exhaustively: all 48 × 48 reflection images land in the set, and the short-root set equals
the matter weight set element by element. The identification is not an embedding chosen among
options; with the bond roots and matter weights the model already fixed, the F
4
system is the unique
completion. [verified]
What is assumed and what is discovered. No new substrate is introduced in this paper. The
model’s substrate is unchanged: the FCC lattice, its code, and matter as defects, with the lattice
itself the series’ root postulate. The two vector sets above were fixed before this paper and for
independent reasons: the long roots are the bond directions of the lift, and the short roots are the
matter weights forced by the glue classes where the matter papers place physical defects. Theorem 5
adds no input; it observes that these two inventories, already on the books, close jointly under their
own reflections — an identity with no adjustable content that a generic pair of sets would fail. F
4
is
the name of what the model already contained, not a structure reached for. What remains a physical
hypothesis is dynamical: that the code’s interactions realize this bracket on physical operators. That
reading is tagged as interpretation below, and its tests are stated in Section 10. [scope statement]
The bracket is the interaction table. At root level, [E
α
, E
β
] ∝ E
α+β
when α + β is a root.
Checking all pairs: same-representation matter pairs sum only to long roots; matter pairs from two
different representations sum only to weights of the third; and gauge-plus-matter sums land only in
matter. In physical language: matter–antimatter pairs annihilate into gauge, two different matter
types fuse into the third by the triality (octonion) product, and gauge acts on matter without creating
it. What a field theory writes as its vertex list, f
4
writes as its Lie bracket. [sector structure verified;
the interaction reading is interpretation]
7 Triality is inner, and charge is a coweight
Theorem 6 (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
.
Verified by regenerating the order-1152 group from the 48 reflections and locating τ inside it. The
1152 elements are exactly the automorphisms used in Ref.; the group there called the automorphisms
of the D
4
root system is the Weyl group of F
4
, acting by inner automorphisms on the completed
algebra. [verified]
5
g
0
(16)
su(3)
color
⊕ su(3)
new
g
−1
(18)
¯
10 ⊕ M
−1
g
+1
(18)
10 ⊕ M
+1
[ ·, ·] [ ·, ·]
[g
−1
, g
−1
] ⊂ g
+1
f
4
= g
−1
⊕ g
0
⊕ g
+1
; charge Q = q/3 labels the columns
Figure 3: The charge grading of the vacuum–matter algebra. Each sector mixes a bond part (10,
¯
10, or the two su(3)’s) with a matter part M
q
; the bracket respects the grading, and charge is the
coweight of the mark-3 node of F
4
. Drawn to true aspect ratio.
Theorem 7 (Grading and coweight charge) The twist σ grades f
4
with dimensions (16, 18, 18),
verified orbit by orbit with all twist phases. In the standard 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). Expanding all 48 roots in this basis, the roots with α
2
-coefficient ≡ 0, 1, 2 (mod 3) number
12, 18, 18: the grading of σ is the grading by the coweight of the mark-3 node. The conserved charge
of any root vector is
q =
coefficient of α
2
mod 3, Q = q/3. (3)
Both gradings were computed independently and agree dimension by dimension; they are conjugate
realizations of the same order-3 inner class, as the deletion of the mark-3 node from the affine F
4
diagram prescribes [5]. [verified; conjugacy by the classification]
The resolution. The Section 3 finding stands and is now completed rather than contradicted:
charge is external to the bond algebra, because so(8) has no room for it, and internal to the vacuum–
matter algebra, because f
4
does. The algebra had to grow to contain matter before it could contain
charge, which is as it should be: charge is a property of matter. The charge statement itself becomes
crystallographic (Figure 3): charge counts how many times the mark-3 simple root appears in a
state’s root expansion, mod 3. [derived]
8 The fixed subalgebra is su(3) ⊕ su(3)
The count. The zero-charge sector has dimension 16 and rank 4: the two fixed Cartan directions,
two zero-weight generators built from the short-root orbits whose color-plane projections vanish,
six root vectors from the long-root orbit sums, and six from the remaining short-root orbit sums.
Deleting the mark-3 node from the affine F
4
diagram leaves A
2
× A
2
[5]: the fixed subalgebra of
the order-3 class is su(3) ⊕ su(3), and F
4
’s two A
2
subsystems are precisely its long and short root
subsystems. The long A
2
is the twisted color of the companion papers. The short A
2
is new: a
second su(3), built from matter orbit sums, present in the algebra because matter is present in the
algebra. [dimensions and rank verified; identification by the classification]
What the second su(3) is not yet. We do not know its physical interpretation, and we do not
assign one. Two disciplined remarks bound the question. First, it is a horizontal structure: it
acts on the matter sector and commutes with color, so whatever it organizes, it organizes across
matter species, not within a color multiplet. Second, an S
3
-containing horizontal symmetry ranging
over three of something is the shape that several unexplained family patterns request, including the
charged-lepton mass relation; we flag the shape and pursue nothing here. The decisive computation
6
is the branching of the 26-dimensional fundamental of f
4
— the exceptional Jordan algebra’s traceless
part [7, 11] — under the pair, and it is carried out here. The 26 has the 24 short roots as its nonzero
weights plus a two-fold zero weight. Pairing every weight with the coroots of both factors (long A
2
spanned by α
1
and −θ, short A
2
by α
3
and α
4
; conjugation-invariant by the classification) gives
26 = (3,
¯
3) ⊕ (
¯
3, 3) ⊕ (1, 8). (4)
The color-triplet sector is an irreducible bifundamental: each color weight pairs with each horizontal
weight exactly once, so the three color triplets form one horizontal antitriplet, not three horizontal
singlets. The content comes in threes; the loaded test is passed. The joint charge table is sharper
still: the entire (3,
¯
3) block carries q ≡ −1, electric charge Q = −1/3; the conjugate block carries
+1/3; the octet of color singlets is exactly neutral. The algebra’s colored matter comes in three
families at the down-quark charge. Two honest residues bound the family reading: the color singlets
form a horizontal octet, not three singlet families, so the lepton sector is not yet in family shape;
and ±2/3 remains absent, the same missing integer unit the companion papers flag. Surviving the
test is necessary for a family interpretation, not sufficient: threes are not generations until chirality
and mass attach to the horizontal index. [branching and charge table verified; family interpretation
open]
9 A fence: the simplest coupling identification fails
The structure fixes two candidate ratios for the hypercharge-to-color coupling at the lattice scale:
3, from the root-length normalization, and 1/3, from the Z
3
charge quantum. One-loop Standard
Model running [12] from M
Z
to the Planck mass gives α
−1
3
≈ 52, α
−1
2
≈ 50, and α
−1
1
≈ 33 in GUT
normalization [8], so the measured ratio α
3
/α
Y
at the lattice scale is close to 1, not 3 and not 1/3.
Both simplest identifications are excluded. The constraint this buys is sharp: any SSM derivation of
coupling magnitudes must produce near-equal couplings at the lattice scale — the lattice needs its
own unification-like mechanism, and bare geometric ratios are not the whole story. We report this
the way the series reports every failed simplest version: as a fence that narrows the target. [derived
negative]
10 Predictions and falsifiers
A structural result should say what it forbids.
No millicharged states. The charge of Theorem 7 is a coweight of a fixed root system, and root
systems do not deform: there is no small parameter anywhere in the construction, so nothing can
shift a charge 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
electric charge that is an exact multiple of e/3, with zero deviation. A single confirmed millicharged
particle falsifies the framework outright. Dedicated searches are active [10, 12]. [derived, conditional
on the framework premise that all matter is a lattice defect]
Internal counting falsifier. The charge of every state must equal its mark-3 coefficient mod 3;
any future defect construction in the series whose computed charge violates this counting refutes the
identification. The 26-branching test of Section 8 has been run and passed; what can still kill the
family reading is named there: the octet-shaped lepton sector and the absent +2/3 unit. [falsifiers
stated]
7
The selection computation is loaded either way. The open diagonal-selection problem is not
internal bookkeeping. If the code dynamics permits the off-diagonal slots of the periodic table — color
triplets at charge 0, charged color singlets — then the framework predicts exotics that fractional-
charge searches in bulk matter already constrain severely [12]. If the dynamics forbids them, the
framework derives the charge–triality correlation that the Standard Model assumes [9], which no input
here was chosen to produce. Either outcome confronts experiment. [stated; computation queued]
One concession, for the record. The thirds themselves are a postdiction: quark charges have
been known for sixty years. What is new is where they come from — an automorphism grading that
is a coweight of the vacuum–matter algebra, not an assignment — and what that origin forbids.
11 What is derived and what is not
The diagonal selection. In the Standard Model, electric charge modulo 1 equals color triality
divided by 3 for every fermion; this correlation is tied to the Z
6
global structure of the gauge group [9].
The periodic table of Theorem 3 contains the diagonal slots the Standard Model occupies — triplets
at ±1/3, singlets at 0 — and also off-diagonal slots it does not: triplets at 0, charged singlets at ±1/3.
The grading alone does not impose the correlation. A physical selection mechanism is required, and a
candidate is visible: a localized defect occupies one glue class, which breaks σ locally, so the realizable
(color, q) pairs are set by which occupancies the code dynamics admits. Computing that selection
against the explicit defect constructions of Refs. [2, 3] is the next defined step. [open; mechanism
candidate stated ]
The integer unit. The grading yields {0, ±1/3}; the up-type charge ±2/3 requires one integer
charge unit on top of the thirds, which in the Standard Model is supplied by the weak sector. A
weak sector is not yet constructed in this framework; a companion analysis in preparation reaches
a parity-violating vector su(2) on defect worldlines and stops short of the chiral projection. The
integer unit waits on that sector. [open]
statement status
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 twisted color in so(8) trivial derived (theorem)
color carries index-3 normalization against the bond form derived
twist 3-cycles glue classes and matter reps together derived (theorem)
defects occupy glue classes (extra node in a void) imported [3]
ˆ
Q = q/3 conserved; thirds quantization; identical sectors derived (theorem)
C = P on the grading derived (theorem)
48 vectors close as the F
4
root system; 52 = 28 + 24 derived (theorem)
bracket sector rules; interaction reading verified; interpretation
triality inner in f
4
verified (theorem)
charge = mark-3 coefficient mod 3, a Cartan coweight derived (theorem)
fixed subalgebra su(3) ⊕ su(3) verified + classification
branching 26 = (3,
¯
3) ⊕ (
¯
3, 3) ⊕ (1, 8); threes confirmed verified
colored block at Q = ∓1/3; singlet octet neutral verified
family interpretation of the horizontal index open
coupling ratios 3 and 1/3 at the lattice scale excluded (fence)
no millicharged states (exact e/3 quantization) derived, conditional
diagonal (charge–triality) selection open; loaded either way
integer unit for ±2/3 open (weak sector)
8
12 Conclusion
Charge in this framework lives where the matter lives, twice over. At the level of the bond algebra
it lives outside: the commutant of color is trivial, and the conserved charge is the grading of the
twist, quantized in thirds, with charge conjugation equal to spatial inversion. At the level of the
vacuum–matter algebra it lives inside: bonds and matter weights are the long and short roots of F
4
,
the same twist is inner there, and the same charge is a Cartan coweight counted from the crystal’s
own simple roots. The completion also delivers what no one ordered: a second su(3) in the fixed
subalgebra, whose first loaded test — the branching of the 26 — is passed on the page, with three
families of colored matter at the down-quark charge. What remains open is stated with equal care:
the selection of the charge–triality diagonal, the integer unit that completes the quark charges, the
family interpretation of the horizontal index, and the coupling magnitudes, for which the simplest
geometric identifications are excluded. Each open item is a defined computation, not a hope.
Declaration of competing interest
The author declares that he has no known competing financial interests or personal relationships
that could have appeared to influence the work reported in this paper.
Data availability
All computations are specified in the text. Python scripts reproducing every check in this paper,
with labeled output keyed to the sections and theorems, are available at github.com/raghu91302/
ssmtheory/raw/main/charge scripts.zip and .../f4 scripts.zip (NumPy only).
References
[1] R. Kulkarni, “A 67%-rate CSS code on the FCC lattice: [[192, 130, 3]] from weight-12 stabilizers,”
arXiv:2603.20294 (2026).
[2] R. Kulkarni, “Matter as incomplete crystallization,” Phys. Open 27, 100423 (2026).
doi:10.1016/j.physo.2026.100423.
[3] R. Kulkarni, “The mass–energy–information equivalence: a bottom-up identification
of the particle spectrum via FCC lattice error correction,” Phys. Open (2026).
doi:10.1016/j.physo.2026.100414.
[4] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer,
New York (1999). doi:10.1007/978-1-4757-6568-7.
[5] V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge
(1990). doi:10.1017/CBO9780511626234.
[6] J. C. Baez, “The octonions,” Bull. Am. Math. Soc. 39, 145 (2002). doi:10.1090/S0273-0979-01-
00934-X.
[7] J. F. Adams, Lectures on Exceptional Lie Groups, University of Chicago Press, Chicago (1996).
[8] H. Georgi and S. L. Glashow, “Unity of all elementary-particle forces,” Phys. Rev. Lett. 32, 438
(1974). doi:10.1103/PhysRevLett.32.438.
[9] D. Tong, “Line operators in the Standard Model,” J. High Energy Phys. 2017, 104 (2017).
doi:10.1007/JHEP07(2017)104.
9
[10] A. Ball et al., “Search for millicharged particles in proton–proton collisions at
√
s = 13 TeV,”
Phys. Rev. D 102, 032002 (2020). doi:10.1103/PhysRevD.102.032002.
[11] I. Todorov and M. Dubois-Violette, “Deducing the symmetry of the standard model from the
automorphism and structure groups of the exceptional Jordan algebra,” Int. J. Mod. Phys. A
33, 1850118 (2018). doi:10.1142/S0217751X1850118X.
[12] R. L. Workman et al. (Particle Data Group), “Review of particle physics,” Prog. Theor. Exp.
Phys. 2022, 083C01 (2022). doi:10.1093/ptep/ptac097.
10
