Quantum Numbers of Color and Charge in F4 Lattice Vacuum

QUANTUM NUMBERS OF COLOR AND CHARGE FROM THE FULL F

ALGEBRA OF A BELL-PAIR LATTICE VACUUM

Raghu Kulkarni

SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA

raghu@idrive.com

Abstract. The Standard Model takes as inputs that the electric-charge quantum number comes

in thirds, tied to color. Can a quantum vacuum supply both? In the Selection–Stitch Model the

vacuum is a face-centered-cubic network of Bell-pair bonds carrying a quantum error-correcting

code, and matter is not a separate substance but a node of the same lattice trapped in a tetrahedral

void. Lifting the lattice to the D

root lattice, we study its full algebra, with public-script checks

and no dynamics. The 24 bond roots and the 24 glue-class weights, which project onto its voids and

nodes, close into the F

root system, and triality becomes inner. Its grading deﬁnes a conserved

label quantized in thirds, whose conjugation is spatial inversion, and ﬁxes su(3)

⊕su(3)

. Exactly

one factor makes charged matter color triplets while leaving integer-charged matter color-neutral,

and under it charge modulo one follows color triality as in the Standard Model. Color placed

on the bonds alone is only the diagonal, under which every sector is an octet. The point group

W (F

) forbids quartic anisotropy. The integer part of charge, chirality, spin and generations do not

follow, and identifying the label with electric charge, and this factor with physical color, remain

hypotheses.

1. Introduction

Electric charge and color are quantum numbers: eigenvalues of conserved generators that label the

states of matter. The Standard Model quantizes electric charge in thirds and ties those thirds to

color, but it takes these facts as inputs without explaining them. This paper asks how far a speciﬁc

quantum model of the vacuum, a lattice of Bell-pair bonds that carries a quantum error-correcting

code, can account for the structure of these quantum numbers — the quantization of charge, its

relation to color, and its behavior under charge conjugation — from the symmetries of the lattice

alone.

2020 Mathematics Subject Classiﬁcation. Primary 17B25, 81R05; Secondary 20F55, 17C40, 11H06.

Key words and phrases. charge quantization; color; Bell-pair lattice vacuum; quantum error-correcting code;

triality; D

root lattice; exceptional Lie algebra F

; rotational isotropy.

ORCID: 0009-0008-8364-7155.

2 RAGHU KULKARNI

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

suppresses it exponentially. The published analysis ﬁxes P

lift

= e

−3

≈ 4.98% as the amplitude for

which the growth cascade terminates at FCC [1].

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 =

2/3 L for bond length L [1]. FCC

is thus 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 [2]. 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 tetrahedron’s

three skew-edge pairs, and the mass spectrum from the code’s veriﬁcation costs [1, 3].

Color from the D

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

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. On the bond algebra alone the twist ﬁxes an su(3) we call the

twisted su(3), which that program proposed as color. This paper does not stop at the bonds. In

the SSM matter is not a separate substance: a particle is a node of the same lattice, trapped

in a void. The object studied here is the whole lattice, its bonds together with the void sites

where such nodes sit. Because the SSM takes this lattice, lifted with its growth axis to D

, to be

COLOR AND CHARGE FROM THE FULL F

ALGEBRA OF THE LATTICE 3

discrete spacetime, its full algebra is in the model’s sense one algebra of spacetime and the matter

it contains; the paper uses the neutral term lattice, since boosts and Lorentz invariance are not

established here (Section 9). Section 6 shows that the twisted su(3) is the diagonal of two factors,

only one of which can be color.

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.

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 [4, 5]; E

and its triniﬁcation subgroup SU(3)

carry the fermion generation in the 27 [6, 7]. Triality of so(8) has long been used to relate the

three 8-dimensional representations [8], and the exceptional Jordan algebra, whose automorphism

group is F

, has been used to obtain Standard Model color from the maximal-rank subgroup

(SU(3) × SU(3))/Z

of F

[9, 10]. On the lattice side, discrete and lattice-gauge realizations of

continuum symmetry are standard [11, 12].

Three gaps remain in that body of work. Uniﬁcation schemes place hypercharge inside a simple

group but take the group as given, so the origin of the charge assignment is deferred, not explained.

The exceptional-Jordan constructions obtain color from F

but begin from the algebra: the choice

of which of its two su(3) factors is color is made by an additional algebraic choice, such as a

complex structure on the octonions, and nothing connects either factor to a spatial structure.

Lattice-gauge treatments put a continuum symmetry on a lattice by hand instead of deriving an

internal symmetry from the lattice’s own automorphisms. What is absent from all three is a

treatment of electric charge as a grading of a twisted triality automorphism acting on a physical

vacuum lattice.

The problem this paper addresses is the following. In the SSM, matter is a defect of the lattice,

and the D

lift assigns its sites to the glue classes of D

∗

. The object to study is the whole

lattice, bonds and void sites together, not the bonds alone. Three objectives follow. The ﬁrst

is to determine the algebra of the whole lattice, its bond roots and glue-class weights together,

and where in it a charge quantized in thirds can live. The second is to determine which su(3) in

that algebra can be color. The third is to separate, claim by claim, what is proved from what is

assumed or left open. Meeting them requires saying 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 discrete spacetime symmetries, and where it stops short of the physical electromagnetic

charge. The present paper reaches the same su(3) ⊕su(3) as the Jordan-algebra route, but from a

triality twist of a vacuum lattice and its void sites, and adds which factor must be color, why the

bonds alone see only their diagonal, and that, with the twist as the grading, color combines bond

and matter generators.

4 RAGHU KULKARNI

Scope: structure, not dynamics. Every result below is a statement about the D

and F

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 9 establishes only the

rotational part, that anisotropy is forbidden below sixth order. The grading label studied here is

thus 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

-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 [1]; whether the two routes

describe the same charge is left open.

Section 2 rebuilds the twist on the D

lift. Section 3 places the matter representations on the glue

classes of D

∗

, where the trapped-node defects of Section 1.1 also sit. Section 4 shows that the

lattice’s bond roots and glue-class weights close into f

, in which triality is inner. Section 5 deﬁnes

the charge label as the twist’s grading of f

, shows that it is quantized in thirds, and identiﬁes

its conjugation with spatial inversion. Section 6 ﬁnds the color factor and, as a corollary, shows

why color cannot be placed on the bonds alone. Sections 7–10 state the coupling constraint, the

falsiﬁers, the isotropy of the lattice, and what is and is not derived.

2. The lattice, the twist, and the verification protocol

The root system and its automorphisms. The D

root system is the set of 24 vectors ±e

±e

1 ≤ i < j ≤ 4, each of squared length 2. Its twelve spatial roots ±e

± e

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

and (±1, ±1, ±1, ±1)/2, has order 1152, the Weyl group of F

; we veriﬁed this by

explicit closure. It contains 80 elements of order 3. The Weyl group of D

has order 192, so the

quotient Aut(D

)/W (D

) has order 6, and its elements act on so(8) as the triality automorphisms.

Standard background on D

, its glue, and triality is in Refs. [13, 8].

The twist. Fix the stacking axis (1, 1, 1). The six spatial roots orthogonal to it span what we

call the geometric A

, the regular A

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

. It lies outside W (D

), so it 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.

COLOR AND CHARGE FROM THE FULL F

ALGEBRA OF THE LATTICE 5

The twist is σ = Ad



2πi ρ·H/3



◦ τ , with ρ = e

− e

the Weyl vector of the geometric A

(simple

roots e

− e

and e

− e

). Its ﬁxed subalgebra g

is the twisted su(3), of dimension 8. By the

classiﬁcation of ﬁnite-order automorphisms [14] the only other candidate ﬁxed subalgebra is G

and the computed dimension excludes it (dim G

= 14). Although τ ﬁxes the roots of the geometric

, no generator of that subalgebra survives σ individually, each acquiring a nontrivial phase, so

the geometric A

is not ﬁxed by the twist; Section 6 shows that it is conjugate to the color factor

identiﬁed there, not 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

grading of so(8) under σ has dimensions

(8, 10, 10). [veriﬁed]

Veriﬁcation protocol. Every claim in the paper is checked in one of three ways, and each is

labeled accordingly. Root-system statements — closures, orbit decompositions, group orders — are

computed exactly on the root vectors. Statements about the algebras are checked on explicit matrix

realizations: so(8) as 8×8 matrices, and f

as so(9)⊕16 built from nine real gamma matrices, with

its single free coupling ﬁxed by the Jacobi identity and its semisimplicity and root system conﬁrmed

directly (Section 4). Branching and representation claims are decided by two independent tests

where possible: the dimension of the commutant of a subalgebra on a representation, which is

1 exactly when the representation is irreducible, and a character decomposition of the weights

by highest weight. Invariant-theory claims (Section 9) use the Molien series, cross-checked by

averaging the group action over the monomials of each degree. Two public Python scripts, requiring

NumPy only, reproduce every check with labeled output keyed to the sections and theorems. The

structure was worked out before any comparison with Standard Model assignments, and every

such comparison is labeled as comparison.

3. Matter on the glue classes of D

∗

The dual quotient D

∗

is Z

×Z

, with three nontrivial classes represented by [1] = (

[2] = (0, 0, 0, 1), and [3] = (

, −

) [13]. The three 8-dimensional representations of so(8) sit

one per class: the weights of 8

lie in [2], those of 8

in [1], those of 8

in [3].

The void correspondence. The link between these classes and the lattice’s tetrahedral voids is a

computable projection. With the time slice spanned by e

, e

and FCC bond vectors (±1, ±1, 0)

and permutations, project each weight onto the slice. The eight weights of 8

land bijectively on

the eight tetrahedral voids (±

, ±

) that surround a lattice node, and so do the eight weights

of 8

; 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

and four of 8

. The six spatial weights ±e

of 8

land on the six octahedral voids,

and the two temporal weights ±e

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 [1, 3]; the correspondence above places the spinor matter representations

6 RAGHU KULKARNI

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

and four of 8

, 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 1 (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

→ 8

The twist that ﬁxes the twisted su(3) is the same operator that rotates the glue classes.

Proof. Apply τ to each class representative and reduce modulo D

: 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

→ 8

. The other order-3 element

ﬁxing the geometric A

, namely τ

, runs the cycle in reverse. □

4. The full lattice algebra is F

The bond algebra so(8) has the 24 roots ±e

± e

, of squared length 2. The matter sector M =

⊕8

has 24 weights: ±e

for 8

and (±

, ±

) for 8

⊕8

(even and odd sign count),

all of squared length 1.

Theorem 2 (Closure). The 48 vectors — 24 long (bonds) and 24 short (matter weights) — form

the F

root system: the set is closed under the reﬂection in every one of its members. Consequently

the bonds and sites of the one lattice span a single Lie algebra f

, 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.

COLOR AND CHARGE FROM THE FULL F

ALGEBRA OF THE LATTICE 7

All 48 × 48 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 so(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

. We also built the algebra explicitly, as f

= 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 glue-class weights of Section 3, which project onto the same lattice’s voids and nodes;

the matter of the SSM, a trapped node, sits on those sites, so the short-root sector M is called the

matter sector without being a second substance. Both inventories were ﬁxed before this section, for

independent reasons. Theorem 2 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

, E

] ∝ 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]

Theorem 3 (Triality is inner). The reﬂection group generated by the 48 roots of F

has order

1152 and contains the twist’s rotation τ . What is an outer automorphism of so(8) is an inner

automorphism of f

The order-1152 group regenerated from the 48 reﬂections is the group of Section 2: the automor-

phism group of the D

root system is the Weyl group of F

, acting by inner automorphisms on

the completed algebra. [veriﬁed]

5. The charge label

The twist σ is an automorphism of order 3 of f

, and its eigenvalue grading deﬁnes a label on the

whole lattice algebra:

(1)

Q =

, σ



= ω

, q ∈ {0, +1, −1}, ω = e

2πi/3

On the matter sector M = 8

⊕ 8

the twist phases are well deﬁned because ρ · µ ∈ Z for

all 24 weights. Because the grading respects brackets, the label is additive under the bracket and

8 RAGHU KULKARNI

conserved by every element of the zero-label sector h

. We write M

for the matter part of h

[derived]

Theorem 4 (Grading and coweight label). The twist σ grades f

with dimensions (16, 18, 18)

and restricts on so(8) to the grading (8, 10, 10) of Section 2. In the simple system α

= e

− e

= e

−e

, α

= e

, α

= (e

−e

)/2, the highest root is e

= 2α

+3α

+4α

+2α

with marks (2, 3, 4, 2), and the roots with α

-coeﬃcient ≡ 0, 1, 2 (mod 3) number 12, 18, 18. The

grading of σ is conjugate, by an inner automorphism of f

, to the grading by the coweight of the

mark-3 node. In that conjugated frame every root vector is homogeneous, with label

(2) q =



coeﬃcient of α



mod 3, Q = q/3,

where q ≡ 2 is read as −1.

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

fall into three classes [14]. They are obtained by deleting from the

aﬃne F

diagram either the mark-3 node, which leaves the semisimple A

×A

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]

Theorem 5 (Thirds quantization). Every τ -orbit of weight lines in M has twist-phase sum ρ ·

(µ + τ µ + τ

µ) ≡ 0 (mod 3). Hence σ

= 1 on M, the label takes only the values

Q ∈ {0, ±1/3},

and each orbit contributes exactly one state to each label.

Proof. The phase sums are direct. A τ-orbit of three weight lines on which σ acts as a cyclic shift

with phases whose sum is ≡ 0 (mod 3) has σ

= 1 on its span, and the three eigenvalues of such

a shift are 1, ω, ω

, one each. □

On the bonds alone,

Q is not an element of so(8): there σ is outer (Section 2), and its grading

derivation is external. In f

the twist is inner (Theorem 3), so the label is internal to the full lattice

algebra; in the conjugated frame of Theorem 4 it counts how many times the mark-3 simple root

appears in a state’s root expansion, mod 3 (Figure 3). The algebra had to include the void sites

before it could contain the label. [derived]

Theorem 6 (Charge conjugation is spatial inversion). Spatial inversion I = diag(−1, −1, −1, 1)

conjugates the twist’s root map to its inverse, IτI

−1

= τ

. Lifted to the algebra as an involution

that preserves the bond and matter sectors, it conjugates σ to σ

−1

exactly, and so maps M

→ 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

COLOR AND CHARGE FROM THE FULL F

ALGEBRA OF THE LATTICE 9

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 4 it maps each eigenspace h

onto h

−q

. □

Because I conjugates σ to σ

−1

, it also exchanges the adjoint grading spaces g

and g

— the

10 and 10 — so the statement covers the bond and matter sectors alike. [derived] Here charge

conjugation means the map q → −q of the grading label; it is the charge conjugation of electric

charge only under the identiﬁcation of the label with electric charge, which is open (Section 10).

[1]

[2]

[3]

one twist: ﬁxed subalgebra and class rotation

q = −1

(3,

Q = −1/3

q = 0

(1, 8)

Q = 0

q = +1

(

3, 3)

Q = +1/3

I : C = P

the three label sectors of the 26: thirds quantization

Figure 2. The charge structure. Left: the three glue classes of D

∗

, 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

with 8

. Right: the three

label sectors of the 26 under su(3)

×su(3)

(Theorem 8), with Q = q/3; inversion,

lifted as an involution, maps the sector q to −q, so conjugation of the label is spatial

inversion (Theorem 6).

(16)

su(3)

⊕ su(3)

−1

(18)

10 ⊕ M

−1

(18)

10 ⊕ M

[ ·, ·] [ ·, ·]

−1

, h

−1

] ⊂ h

= h

−1

⊕ h

; the label Q = q/3 indexes the columns

Figure 3. The grading of the full lattice algebra. Each sector combines a bond

part (10, 10, or the twisted su(3)) with a matter part M

, and the bracket respects

the grading. The ﬁxed sector h

is su(3)

⊕ su(3)

; the twisted su(3) of the bond

algebra is its diagonal, and Section 6 identiﬁes su(3)

as the color factor.

10 RAGHU KULKARNI

6. The color that commutes with the grading

The zero-label sector h

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

diagram

identiﬁes it as A

× A

[14]. Thus h

= su(3)

⊕ su(3)

, where the subscripts record that the

two factors correspond to F

’s long and short A

root subsystems. Their embedding indices in f

computed from the ratio of the f

trace form to each factor’s own, are exactly 1 and 2. [veriﬁed]

Theorem 7 (The twisted su(3) is the diagonal). The ﬁxed subalgebra of the bond algebra, g

∩ so(8), is neither factor: it projects isomorphically onto both, so it is a diagonal su(3) of

su(3)

⊕ su(3)

. Its index on the 26 of f

is 9 = 3 + 6, the sum of the indices of the two factors.

Proof. On the explicit algebra h

∩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)

, 6 for su(3)

, and 9 for g

, and indices add over a diagonal. □

The carrier of matter states. The matter weights appear twice in f

. They are the short-root

directions of the algebra (Section 4), and with a two-fold zero weight they are the weights of the

26, the smallest nontrivial representation of f

and the traceless part of the exceptional Jordan

algebra [9, 10]. 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 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 (

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 8 (Color is the long factor). Every su(3) in f

that commutes with the grading is su(3)

su(3)

, or a diagonal. The 26 decomposes by label as

(3) 26 = (3,

|{z}

Q=−1/3

⊕ (

3, 3)

|{z}

Q=+1/3

⊕ (1, 8)

|{z}

Q=0

under su(3)

× su(3)

. Hence only su(3)

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)

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

. 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 decomposition of the 26 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,

COLOR AND CHARGE FROM THE FULL F

ALGEBRA OF THE LATTICE 11

the color factor is su(3)

. Its identiﬁcation with the physical color of the strong interaction is, like

the identiﬁcation of the grading label with electric charge, a hypothesis (Section 10). Three facts

characterize the color factor.

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)

. The identiﬁcation of color

with the twisted su(3), proposed on the bond algebra alone (Section 1.1), is the shadow that the

color factor casts there (Section 6.1).

Third, it is conjugate in f

to the geometric A

of Section 2. All sixteen long-root A

subsystems

of F

form a single Weyl orbit; the geometric A

is one of them, and in the conjugated frame

of Theorem 4 so is the long factor. If the three colors that the companion papers read oﬀ the

tetrahedron’s skew-edge pairs are the geometric A

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 commutant of the color factor. Whether color leaves room for a continuous charge on

the bonds 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 4, is trivial.

Under the conjugate coweight grading, color lies inside so(8) and its commutant there is u(1)⊕u(1).

In both cases the commutant in f

is exactly the second factor su(3)

, so any continuous charge

compatible with color must lie there. [veriﬁed]

Under su(3)

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)

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)

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 [15]. [derived]

Three limits remain. 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 10). The

color singlets form one octet of su(3)

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]

The twist and the coweight grading. The twist σ and the mark-3 coweight grading of The-

orem 4 are conjugate in f

, 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,

12 RAGHU KULKARNI

→ 8

, which is the glue-class rotation of Theorem 1; the coweight grading grades

so(8) as (10, 9, 9) and leaves each 8 in place. Under the twist, color mixes bond and matter genera-

tors; 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. Which grading

the dynamics realizes is not settled here. [veriﬁed]

6.1. Why the bonds alone are not enough. The strong-sector program placed color on the

bonds alone, as the twisted su(3), g

= h

∩so(8). Theorem 7 shows that it is the diagonal of the

two factors. The following corollary records what this means on the bonds, and so why the void

sites are needed.

Theorem 9 (The bonds alone cannot carry color and charge). (i) Under the twisted su(3) each

of the three 8’s is the adjoint 8, so every label sector of the matter weights is an octet. (ii) The

subalgebra of so(8) that commutes with the label is exactly g

; its commutant in so(8) is trivial,

and its embedding index in so(8) is 3. (iii) Every su(3) subalgebra of so(8) has embedding index 1

or 3: for index 1 each of the three 8’s branches as 3 ⊕

3 ⊕1 ⊕1, and for index 3 each is the adjoint.

The geometric A

of Section 2, spanned by the six in-plane bond roots, has index 1, but its root

generators are eigenvectors of σ with eigenvalues ω

±1

, so they carry label ±1/3. Hence, on the

bonds alone, if the label is electric charge, color is forced to be g

, and no matter representation is

a color triplet.

Proof. (i) The weights of each 8 restricted to the Cartan subalgebra of g

are their orthogonal

projections onto the color plane: six of squared length 2/3, which are exactly the six roots of g

and two zeros. That is the character of the adjoint; weights of a 3 or

3 would have squared length

. On the explicit matrix algebra the commutant of g

in End(C

) is one-dimensional,

so 8

is irreducible, whereas for the geometric A

it is six-dimensional, as 3 ⊕

3 ⊕ 1 ⊕ 1 requires.

Since σ commutes with g

and permutes the three 8’s cyclically, each label sector is one copy of

the octet. (ii) An element commutes with

Q exactly when it is ﬁxed by σ. An element commuting

with g

lies in the zero-weight space of its Cartan subalgebra, which is the Cartan subalgebra of

so(8) because no root of D

is orthogonal to the color plane. Each surviving orbit generator has

nonzero components on all three roots of its τ -orbit, so a Cartan element commuting with it is

orthogonal to all three; across the six orbits it is orthogonal to all eighteen moved roots, which

span R

, and so vanishes. The index is the ratio of squared root lengths, 2 to 2/3. (iii) An su(3)

in so(8) acts on the real orthogonal representation 8

, whose nontrivial 8-dimensional options are

the adjoint and 3 ⊕

3 ⊕ 1 ⊕ 1, of Dynkin indices 3 and 1, since a 6 would require its conjugate.

Because 8

, 8

, and 8

each have index 1 in so(8), the branching type is the same on all three.

The in-plane roots have the ambient squared length 2, so their index is 1; the eigenvalues of σ on

their root generators are computed on the explicit algebra. □

Part (i) also follows inside f

from Theorems 7 and 8, since under any diagonal every sector contains

an octet. The corollary is not a separate result but the reason the void sites are required: the one

COLOR AND CHARGE FROM THE FULL F

ALGEBRA OF THE LATTICE 13

su(3) on the bonds that commutes with the label makes every matter sector an octet, while the

one under which matter is a triplet carries the label itself. Including the void sites resolves this,

because the ﬁxed subalgebra doubles and its long factor is conjugate, by Theorem 8 and the third

fact above, to that same geometric A

. [derived (theorem)]

7. A coupling constraint: the simplest ratios are excluded

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

charge quantum.

Section 6 already removes the ﬁrst on structural grounds, since the twisted su(3) is not the color

factor and its normalization was never a color coupling. The running excludes both.

One-loop Standard Model running [16] from M

to the Planck mass gives α

−1

≈ 52.4, α

−1

≈ 49.5,

and α

−1

≈ 33.3 in GUT normalization [4]. Converting to the hypercharge coupling itself, α

−1

≈ 55.4, so α

/α

≈ 1.06 at the lattice scale; in GUT normalization the corresponding ratio

is α

/α

≈ 0.64. In both conventions 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, so it is a bare-SM consistency check

and not a model-speciﬁc prediction: any states the framework adds below the lattice scale would

alter it.

The normalization of su(3)

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 cannot account for them alone. [excluded]

8. Predictions and falsifiers

Two falsiﬁable statements follow from the structure. Two more are stated elsewhere: the exclusion

of the simplest coupling ratios (Section 7), and the test that distinguishes the twist from the

coweight grading (Section 10).

No millicharged states. The label of Theorem 5 is the eigenvalue grading of an order-3 auto-

morphism. 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 mil-

licharged 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

14 RAGHU KULKARNI

charge modulo one. Dedicated searches are active [17, 16]. [derived, conditional on the framework

premise that all matter is a lattice defect]

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 [15]. On the bonds alone the grading violates this, placing

its thirds on octets of triality 0 (Theorem 9). With color identiﬁed as su(3)

, the completed algebra

obeys it exactly (Theorem 8). 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 [16]. 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.

9. 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

The point group and the second moment. An eﬀective description built on a lattice is con-

strained 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

then do the work. Its 24 roots have the isotropic second moment

(4) S

µν

= 12 δ

µν

so no quadratic anisotropy exists, in all four directions and not only the three spatial ones. And its

point group is Aut(D

) = W (F

) of order 1152 (Section 2), which is larger than the point group

of any other four-dimensional lattice.

Theorem 10 (Isotropy to sixth order). The spaces of W (F

)-invariant polynomials in degrees 2,

4 and 6 have dimensions 1, 1 and 2. The unique quartic invariant is (k

)

. Hence no quartic

anisotropy exists on the D

lattice, and the ﬁrst anisotropic invariant appears at degree 6.

Proof. The dimensions are the coeﬃcients of the Molien series |G|

−1

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

COLOR AND CHARGE FROM THE FULL F

ALGEBRA OF THE LATTICE 15

single invariant in degree 2 and a single one in degree 4 leave (k

) and (k

)

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

, the familiar lowest-order anisotropy of cubic

lattices, and it is absent on D

. Table 1 collects the comparison.

Table 1. Dimensions of the spaces of invariant polynomials of three point groups

in four dimensions, with the O(4) count for comparison. Without triality, W (D

)

admits a quartic anisotropy like the hypercubic group; with it, W (F

) does not, so

the leading anisotropy of the D

lattice is pushed from degree 4 to degree 6.

degree W (F

), order 1152 W (D

), order 192 W (B

), order 384 O(4)

lattice) (no triality) (hypercubic) (isotropic)

2 1 1 1 1

4 1 3 2 1

6 2 4 3 1

Triality as the source of the protection. The protection comes from the part of the point

group that the Weyl group does not contain. Using only W (D

), 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

) to Aut(D

) = W (F

), removes

it. The same triality that grades the charge in Section 5 is the one that pushes anisotropy from

degree 4 to degree 6.

Optimality of degree 6 among crystals. 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

) these are 2, 6, 8, 12, with 2 · 6 · 8 · 12 = 1152. The degree-2 generator

is k

, 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

alone.

Among rank-4 reﬂection groups just one does better than W (F

), the 600-cell group H

with

degrees 2, 12, 20, 30; it is not crystallographic, so no lattice can carry it. Degree 6 is therefore

optimal among four-dimensional crystals whose point group is a reﬂection group, and D

attains

it. Extending the statement to every crystallographic point group would rest on the classiﬁcation

of the maximal ﬁnite subgroups of GL(4, Z), which is not carried out here. [veriﬁed]

Isotropy is a four-dimensional property. The result is a property of the four-dimensional lat-

tice 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

gives

α = 1 and β = −1, and a companion paper uses exactly this to bound the leading Lorentz-violating

correction on the slice [18]. Over all 24 roots of D

, 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 10. [veriﬁed]

16 RAGHU KULKARNI

The permitted dispersion relation. Any dispersion relation on this vacuum has the form

= c



1 + b

(ka)

+ O



(ka)



, with a the lattice spacing. The term b

(ka)

comes from

the unique quartic invariant (k

)

and is isotropic, so it rescales the speed without selecting a

direction; the ﬁrst direction-dependent term is O



(ka)



, two orders beyond a cubic lattice, where

anisotropy already enters at (ka)

At a spacing near the Planck length both terms are minute. For an optical photon (ka)

≈

−55

; even for a cosmic ray at 10

eV, the largest available momentum, the isotropic term is

≈ 2 × 10

−16

and the anisotropic one ≈ 3 × 10

−32

. The anisotropy is not measurable by any

foreseeable experiment, and this section should be read as a consistency result, not 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 [19]. 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 of

diﬀerent species unless a symmetry protects them, which is a known ﬁne-tuning problem for

emergent Lorentz invariance [19]. The companion paper that derives a laboratory threshold on

this vacuum assumes the single speed as a stated postulate rather than deriving it [18], and the

same gap is left open here.

Third, the experimental confrontation. Tests of Lorentz invariance constrain preferred-frame eﬀects

tightly across many sectors [20], and comparing this vacuum against them requires the dynam-

ics 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]

10. 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 symmetry of the whole lattice. With the twist σ as the grading, the color factor

su(3)

combines bond and matter directions (Section 6), so a color rotation acts on bonds and

COLOR AND CHARGE FROM THE FULL F

ALGEBRA OF THE LATTICE 17

sites together, not on the bonds alone. Under the conjugate coweight grading, color is a symmetry

of the bonds. The two gradings thus 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

label; in the Standard Model it comes from the weak

isospin. One candidate source exists in the series, and it is not the grading: the companion matter

paper obtains +2/3 from integer winding under the lattice’s Bravais translations [1]. Whether it

supplies the integer part that this grading cannot ﬁx is open, and so is chirality, which this paper

does not address. [open]

18 RAGHU KULKARNI

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

roots has order 1152; 80 order-3 elements veriﬁed

twisted su(3) reconstructed; grading of so(8) (8, 10, 10) veriﬁed

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 [1, 3]

twist 3-cycles glue classes and matter reps together derived (theorem)

bond roots and glue-class weights close into F

; f

= so(8)⊕

derived (theorem); veriﬁed

triality outer on so(8), inner in f

derived (theorem); veriﬁed

grading of f

(16, 18, 18), conjugate to the mark-3 coweight derived (theorem); veriﬁed

Q = q/3 internal to f

, conserved; thirds quantization derived (theorem)

C = P on the grading, for the involutive lift of inversion derived (theorem)

= su(3)

⊕ su(3)

, 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

veriﬁed

26 = (3,

−1/3

⊕ (

3, 3)

+1/3

⊕ (1, 8)

derived (theorem); veriﬁed

color = su(3)

, 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

color conjugate to the geometric A

in both frames veriﬁed

commutant of color: in f

, su(3)

; in so(8), trivial (σ) or

u(1)

(coweight)

veriﬁed

charge mod 1 obeys Q ≡ −t/3, the Standard Model relation derived

bonds alone: octets under g

; trivial commutant; index 1

or 3

derived (theorem)

triplet matter with the label as charge, on the bonds alone excluded

coupling ratios 3 and 1/3 at the lattice scale, any scale

/3–M

excluded, bare-SM running

label exact in thirds; no millicharge if label = charge mod

derived, conditional

no quartic anisotropy on D

(ﬁrst at degree 6), due to tri-

ality; optimal among reﬂection point groups

veriﬁed

label identiﬁed with electric charge modulo one open

color factor su(3)

identiﬁed with physical color open

integer part of the charge (not ﬁxable by a Z

label) open

emergent Lorentz invariance (boosts, a universal speed) open

which grading the dynamics realizes (twist or coweight) open

three families; lepton sector in family form open

COLOR AND CHARGE FROM THE FULL F

ALGEBRA OF THE LATTICE 19

11. Conclusion

In the Bell-pair vacuum of the SSM, matter is not a separate substance but a node of the same

lattice trapped in a void, and this paper has treated the whole lattice, bonds and void sites

together. Lifted to D

, the 24 bond roots and the 24 glue-class weights, whose spinor part sits on

the tetrahedral voids that trap the defects, close into the F

root system. No input is added, since

both inventories were ﬁxed before the closure was checked. In f

the triality twist is inner, and

its grading deﬁnes a conserved label quantized in thirds, internal to the full lattice algebra, whose

conjugation, for the involutive lift of inversion, is spatial inversion.

The ﬁxed subalgebra of the grading is su(3)

⊕ su(3)

, and exactly one of its factors makes the

charged matter color triplets while keeping the integer-charged matter color-neutral: su(3)

. With

it as color, charge modulo one follows color triality exactly as in the Standard Model. With the

twist as the grading, color combines bond and matter generators; under the conjugate coweight

grading it lies among the bonds; in either frame it is conjugate in f

to the geometric color of the

lattice. The twisted su(3) that the bonds alone ﬁx is the diagonal of the two factors, under which

every sector is an octet. This is why color cannot be placed on the bonds alone, and why the void

sites are needed.

Independently of the grading, the lattice also protects rotational isotropy. Its point group W (F

)

admits no quartic anisotropy, so the ﬁrst direction-dependent term in a dispersion relation is two

orders higher than on a cubic lattice. The protection comes from triality, since the Weyl group of

alone does admit a quartic anisotropy, and degree 6 is the best any four-dimensional crystal

with a reﬂection point group can reach. Boosts and a universal limiting speed are not addressed.

The three objectives set in Section 1 are thereby met. The bond roots and glue-class weights of

the one lattice form f

, and a thirds-quantized charge label lives in it as the twist’s grading. Color

must be su(3)

, given the one Standard Model requirement stated in Section 6. And Table 2

separates what is proved from what is assumed or open.

The following do not follow: the integer part of the charge, spin, chirality, the three generations,

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 the color factor with physical color, and supplying

the integer part, would further require gauge sectors and dynamics that are not constructed here.

The open items are listed in Table 2.

Statements and Declarations

Competing interests. The author is employed by IDrive Inc. and has no ﬁnancial or non-

ﬁnancial interests relevant to the content of this article.

20 RAGHU KULKARNI

Funding. No funding was received for conducting this study.

Author contributions. The author is the sole contributor to this work.

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 twist of Section 2, the glue classes of Section 3, the thirds quantization and the root-level

form of Theorem 6 in Section 5, the bond-only corollary of Section 6.1, and the coupling constraint

of Section 7. The second, https://github.com/raghu91302/ssmtheory/raw/main/f4_scripts.

zip, covers the twist geometry of Section 2, the F

results of Sections 4–6, the algebra-level form of

Theorem 6, the commutant of color and the two gradings of Section 6, the charge–triality relation

of Section 8, and the invariant degrees of Section 9.

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