SU(3) Triality on D_4: G_2 as the Unique Alternative

su(3)

from triality on the

lattice:

as the only alternative

Raghu Kulkarni

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

raghu@idrive.com

August 2026

Abstract

In standard lattice gauge constructions the gauge group is an input. Wilson's ac-

tion, quantum link models, and emergent-gauge-eld constructions all x it beforehand.

We work on the

root lattice, the densest lattice packing in four dimensions, whose

constant-time slices are face-centered-cubic, and dene the color of a bond to be the pair

of coordinate axes it spans. A three-state label makes

su(3)

automatic, since the traceless

operators on any three-state space close on it, and so excludes nothing. We supply a selec-

tion. Two obstructions come rst: every transporter built from a spatial lattice symmetry

acts on color through

, so no continuous color rotation is reachable, and the

-cell

straddling two time slices identies successive cages by the skew-pair exchange

⊗ σ

which is the identity on color. Triality resolves this. Of the

order-three automorphisms

, exactly

x only spatial roots, and each selects one antipodal root pair per color

class: the twelve spatial roots form

, whose four

subsystems are the four possible

selections, related by the point group. Twisting triality by

t = exp(2πiρ/3)

, with

the

Weyl vector of the selected

, deletes the six xed roots and leaves an eight-dimensional

xed subalgebra; the classication of order-three outer automorphisms of

then excludes

and leaves

su(3)

. On this lattice the algebra is therefore not put in but xed by the

lattice's own symmetry, and the argument could have failed:

survives to the last step

and is removed by a dimension count. Its Cartan subalgebra is the bond-phase torus, its

roots are the normalized orbit sums

, and

so(8)

branches to

8⊕10⊕

. The surviving

generators mix one spatial and two temporal roots, so color rotation here is necessarily

spacetime-mixing. Realizing the

root generators as link operators remains open.

Contents

1 Introduction 2

2 Framework and notation 3

3 The lattice 4

4 First obstruction: color is a nite label 5

5 Second obstruction: the temporal identication is color-blind 6

so(8)

from the lattice 8

7 Triality and the color classes 9

8 The twist that selects

su(3)

9 The xed subalgebra: analytic proof 13

10 Independent verication 14

11 Representation content 15

12 Relation to prior work 16

13 What is not established 17

14 Conclusion 18

A The anchor is xed by the stacking axis 18

1 Introduction

In standard lattice constructions the gauge group is an input.

Quantum chromo-

dynamics is built on

SU(3)

, and lattice formulations of it take the group as given. Wilson's

plaquette action assigns an

SU(3)

element to each link [2]. The KogutSusskind Hamiltonian

does the same in the temporal gauge [3]. Quantum link models replace the continuous link

variable by a nite-dimensional operator, but the algebra whose representation is used must

still be chosen [4]. The same is true of the emergent-gauge-eld program in condensed matter

physics, which is the closest existing work to a derivation. Wen constructed a bosonic model on

a cubic lattice whose low-energy theory contains quarks and gluons [5], and the construction

is explicit and correct, but the color index is introduced when the boson is split into partons.

String-net models produce non-abelian gauge elds whose group is set by the input fusion

category [6]. In every case the lattice supplies locality and a constraint structure. It does not

supply the group.

The substrate.

We work on the

root lattice, the densest lattice packing in four di-

mensions, with kissing number

[7, 9]. Section 3 derives the three facts we need from the

denition: every constant-time slice is the face-centered-cubic (FCC) lattice, the

nearest

neighbors split as twelve spatial and twelve temporal, and the structure tensor is isotropic in

all four directions. Each is a short computation and is proved where stated, so the paper is

self-contained.

A three-state space is not a selection.

Color in this framework is a three-valued label,

and it is tempting to conclude from that alone that the gauge algebra is

su(3)

: the traceless

operators on a three-dimensional space do close on

su(3)

. The inference is valid and uninfor-

mative. It holds for any three-state space whatsoever, so it excludes no competing algebra

and cannot fail. Establishing that a lattice selects a gauge algebra requires an argument with

a candidate that survives to the end and is then removed by computation. That is what is

missing, and what this paper supplies.

What this paper adds.

We supply an argument that could have failed, and record what it

costs.



Two obstruction theorems (Sections 4 and 5). No transporter built from a spatial lattice

symmetry rotates color, and the identication of successive cages supplied by the

-cell

honeycomb acts as the identity on the three-state color space. Both are negative results

about what the geometry provides, and together they show that neither the spatial nor

the temporal structure of the lattice supplies a color rotation of its own.



The selection, by a slicing-compatible triality, of one antipodal root pair from each of

the three color classes, whose positive representatives form an

system and satisfy

+ α

= α

as vectors (Section 7).



The selection of

su(3)

as the xed subalgebra of triality twisted by an order-three torus

element, with

as the only alternative and

su(3)

occurring twice as often in the scan

(Sections 8 and 9), proved analytically and veried independently by explicit construction

so(8)

(Section 10).



The identication of the surviving roots with the triality orbit sums, of the deleted color

classes with the long roots of

, and the branching

so(8) → 8 ⊕ 10 ⊕

(Section 11).

What distinguishes this from an algebraic closure is the presence of a competitor.

available at every stage and is excluded by a dimension count together with the classication

of order-three automorphisms of

, not by assumption.

Layout.

Section 2 gives the background framework and xes notation. Section 3 describes

the

lattice, its

-cell honeycomb, and the face-centered-cubic slice. Sections 4 and 5 give

the two obstruction theorems. Section 6 attaches

so(8)

to the lattice. Section 7 shows that

triality selects one root pair from each color class. Section 8 gives the twist and the dimension

count. Section 9 proves the main theorem and Section 10 veries it independently. Section

11 treats the representation content, and Section 12 places the result against related work.

Section 13 states what is not established, and Appendix A records one further consequence of

the same projection.

2 Framework and notation

Color.

A spatial bond of the lattice has a displacement vector with exactly two nonzero

entries among

, x

. Its

color class

is the pair of axes it spans, written

, or

Equivalently, the twelve bonds at a site of the FCC slice split

[4, 4, 4]

by coordinate plane, and

the class records which block a bond belongs to. This is a denition, and what happens to it

is the subject of the paper.

Why this label is called color.

Nothing said so far makes a coordinate-plane class a color

charge. The identication comes from outside this paper. In the framework we adopt, a node

trapped in a tetrahedral void of the FCC lattice is a baryon, and the three skew-edge pairs of

its bounding tetrahedron are the three color charges [1]. Under that identication the classes

dened above are the color charges of the strong sector, and the algebra selected in Section 8

is the color algebra.

The mathematics below does not depend on the identication. A reader who sets it aside

may read

color

throughout as a name for a three-element label, and Theorems 1, 2, 7, and 13

are unchanged. The identication is used only in the readings oered in Sections 11 and 13,

and in the motivating discussion of Section 1.

Cages and their three states.

A tetrahedral cell of the honeycomb has four vertices and

six edges. Its edges partition uniquely into three pairs of skew edges, those sharing no vertex,

= {AB, CD}, M

= {AC, BD}, M

= {AD, BC},

(1)

by the identity





/2 = 3

. We call such a cell a

cage

and the three matchings its three states.

The three matchings are bond disjoint, and each is a single color class: for the cage at the

origin,

and

both lie in the

plane,

and

, and

and

(0,0,0)

(1,1,0)

(1,0,1)

(0,1,1)

(a) a cage of the FCC slice

= {AB, CD}

±(1, 1, 0), ± (1, − 1, 0)

= {AC, BD}

±(1, 0, 1), ± (1, 0, − 1)

= {AD, BC}

±(0, 1, 1), ± (0, 1, − 1)

the class is unoriented: both bonds share a label

(b) each matching is one coordinate plane

Figure 1: The cage and its three states. (a) A tetrahedral cell of the face-centered-cubic slice,

with its six edges colored by skew pair. (b) Each skew pair consists of two bonds lying in the

same coordinate plane, so the three matchings of the cage are exactly the three color classes.

The identication is used throughout and is proved by the displacement vectors shown.

The cage therefore sees the three classes, but only as unoriented labels:

and

are both

, and nothing in the cage distinguishes them. Section 7 shows that triality does distinguish

them, and that the distinction is what turns a label into a root. Figure 1 shows the classes.

Conventions.

Root systems, root spaces, and root vectors

always refer to the complexi-

cation

so(8, C) = h

⊕

, where

is the complexication of a Cartan subalgebra of the

compact real form. Statements about an automorphism are made on the complexication, and

its xed subalgebra is then intersected with the compact real form

so(8)

at the end of Section

9; that intersection is what carries the compactness argument. Dimensions are complex dimen-

sions on the complexication and real dimensions on the real form, and the two agree for the

objects counted here because each xed subspace is dened by real conditions. Roots are nor-

malized to length squared two. The inner product is the standard one on

. Torus elements

are written

t = exp(2πih)

with

in the Cartan subalgebra, so that

Ad(t)E

= e

2πi⟨α,h⟩

Every numerical statement in this paper is reproduced by the scripts described in the data

availability section.

3 The lattice

Denition.

The

lattice is the set of integer points in four dimensions whose coordinates

sum to an even number,

= {x ∈ Z

: x

+ x

even

(2)

Two points are neighbors when they dier by a root, which is a vector with two entries equal

±1

and two entries zero. There are

roots, and

is the maximum kissing number in four

dimensions [9]. Filling the gaps gives a honeycomb of regular

-cells, the Delaunay decompo-

sition of

[7]. This is the regular honeycomb

{3, 3, 4, 3}

of four-dimensional Euclidean space

[8], and all of its cells are alike, a property special to

points edges triangles tetrahedra

-cells

honeycomb

/2 6L

16L

12L

slice

const

/2 3L



slice at

L = 4 32 192 256 64



Table 1: Cells of the

-cell honeycomb and of its equal-time slice. The slice counts at

L = 4

are those of the FCC lattice and its tetrahedronoctahedron honeycomb.

Cell counts.

On a four-torus of even side

the honeycomb has the counts in Table 1. The

rst two follow immediately: half the points of

(Z/L)

have even coordinate sum, giving

and each has

neighbors, giving

24(L

/2)/2 = 6L

edges. The remaining counts are a direct

enumeration; all ve, and the four slice counts, are reproduced at

L = 4

by the rst script of

the data availability section. At

L = 4

the slice has

points,

192

edges,

256

triangles, and

tetrahedra. These are the site count, the bond count, the triangular plaquette count, and the

tetrahedral-void count of the FCC lattice. The identication is exact, not an analogy.

Slicing.

Setting

= c

for an integer

leaves the points of

with

≡ c (mod 2)

which is the FCC lattice for even

and its translate for odd

. Every constant-time slice of

is therefore FCC.

The root split.

The

roots split by whether they move in

. Twelve have

= 0

. These

are the FCC nearest neighbors, and each carries a color class. The other twelve have

= ±1

and one nonzero entry among

, x

. Their spatial part lies along a coordinate axis, so

they lie in no coordinate plane and carry no color class. We call the rst group spatial and the

second temporal. This split is used throughout.

Isotropy.

The structure tensor over all

roots is

µν

= 12 δ

µν

(3)

by direct enumeration: for xed

the contributing roots are the twelve vectors

±e

± e

with

ν = µ

, each contributing

= 1

, while for

µ = ν

the four vectors

±e

± e

contribute

+1, −1, −1, +1

and cancel. The lattice is isotropic in all four directions, not only the three

spatial ones.

4 First obstruction: color is a nite label

With the lattice xed, we ask what it supplies toward a connection on color. This section

answers for the spatial directions and Section 5 for the temporal one. Both answers are negative,

and together they set up the mechanism of Section 8.

Statement.

A gauge connection needs a transporter that rotates color continuously. The

generators that do this are the o-diagonal root generators

su(3)

. We show that no trans-

porter arising solely from the spatial lattice-symmetry action on these geometrically dened

color classes can supply them.

Theorem 1.

Let a transporter act on the color classes through a symmetry of the spatial lattice.

Then its action on the three classes lies in

. In particular no one-parameter family of color

rotations is reachable.

Proof.

The color class of a bond is determined by which pair of coordinate axes its displacement

vector spans. The point group of the FCC lattice acts on the three spatial axes by signed

permutation, and there are

such maps. Each sends a coordinate plane to a coordinate

plane. Enumerating all

and recording the induced map on the set

{xy, xz, yz}

returns

exactly

distinct maps, which is the full symmetric group

and nothing more. A group of

permutations of a three-element set has no continuous subgroup, so no one-parameter family

exists.

Reading.

Color as dened by coordinate planes takes values in a nite set. A continuous

rotation would have to move a bond through intermediate values, and there are none. The

Weyl group of

su(3)

is available and the root directions are not. Write

for a maximal torus

SU(3)

, reserving the symbol

for the triality map of Section 7. Theorem 1 then has the

following corollary, which we state as a remark because it depends on identifying the abelian

phases available on bonds with

, an identication made precise only in Section 9. Lattice

symmetries contribute the permutations

, phases contribute

, and together they reach at

most

N(T ) = T ⋊ S

⊂ SU (3),

(4)

a two-dimensional subset, up to sheets, of the eight-dimensional group. The root directions lie

outside it.

5 Second obstruction: the temporal identication is color-blind

Motivation.

Theorem 1 concerns spatial symmetries. A natural hope is that the missing

direction is temporal, and that a four-dimensional formulation supplies it. The

honeycomb

makes one version of this testable. It furnishes a canonical identication of a cage with its

successor in the next slice, through the

-cell that straddles them, and one can ask what that

identication does to color.

We x the scope at the outset, since it is easily overread. What follows is a statement

about the identication the geometry provides, not about a dynamical link variable. In a

gauge theory the temporal link is a eld, free to take any value in the group, and no theorem

about the honeycomb constrains it. The question here is the same one Theorem 1 answers in

the spatial directions: what does the lattice itself supply?

The straddling cell.

-cell centered at a half-integer point meets each of two adjacent

slices in a tetrahedron. Let the center be

h = (a +

, b +

, c +

, d +

)

. Its eight vertices are

h+s/2

for sign vectors

s ∈ {±1}

, and such a vertex lies in

exactly when

≡ 0 (mod 4)

which selects the all-plus vector, the all-minus vector, and the six vectors with two entries of

each sign: eight in all, as required. The fourth coordinate of the vertex is

d + (1 + s

)/2

, so

it equals

when

= −1

and

d + 1

when

= +1

. Among the eight admissible sign vectors,

four have

= −1

(the all-minus vector and the three two-plus vectors with

= −1

) and four

have

= +1

. The cell therefore meets each of the two slices in four vertices, and those four

are mutually adjacent, hence a tetrahedron.

Take the center at

(

)

. The four vertices at

= 0

have spatial coordinates

A = (0, 0, 0), B = (1, 1, 0), C = (1, 0, 1), D = (0, 1, 1),

(5)

which is a cage of the FCC slice in the sense of Section 2. The four at

= 1

form a cage in

the next slice. The cell is the world tube of the cage.

The induced map.

-cell is the four-dimensional cross-polytope. Its

edges are all

pairs of vertices except the four antipodal pairs, and the antipodal pairing gives a canonical

identication of the two cages. That identication respects color. The antipode of a vertex

in a cell centered at

2h − v

, so for any two vertices

X −

Y = −(X − Y )

: displacements are

negated. A color class is closed under negation, so each cage bond and its antipodal image lie

in the same class.

Each cage bond lies in exactly one straddling tetrahedron, that is, one tetrahedron of the

honeycomb with two vertices in each slice. For the bond

such a tetrahedron needs two

vertices of the far cage adjacent to

, to

, and to each other. Adjacency excludes only

antipodes, so

and

are unavailable, leaving

and

; these are adjacent to one another,

being non-antipodal. The tetrahedron is therefore

{A, B,

, and it is unique. The transport

therefore sends

to the image of

, its skew partner, and similarly for every bond.

Theorem 2.

The temporal transport induced by the straddling

-cell is the skew-pair exchange,

= I

⊗ σ

(6)

On the symmetric sector, which is the three-state color space, it acts as

; on the antisym-

metric sector it acts as

−1

. Counting straddling tetrahedra by class gives the matrix

, the

factor two being the two bonds each class contributes to a cage; after normalizing by that factor,

the induced operator on the color space is

. This holds for all

straddling cells between two

adjacent slices at

L = 4

A second weighting, counting crossing edges between pairs of cage vertices rather than

straddling tetrahedra, gives the uniform matrix

12J

. Whatever normalization is applied, this

has rank one and carries no color information either. Figure 2 shows the mechanism.

Remark 3.

Theorem 2 says that the canonical frame supplied by the honeycomb is color-

aligned: in that frame, transport from one slice to the next acts trivially on color. It does not

say that a dynamical temporal link must be trivial, and no such claim is made. The content

is that the geometry oers no color rotation of its own in the temporal direction, exactly as

Theorem 1 says it oers none in the spatial ones.

Reading.

The geometric identication transports color by doing nothing to it. The reason

is structural and is visible in the root split of Section 3: temporal roots carry no color class, so

they cannot mix classes for the same reason that they cannot be assigned one. Passing to four

dimensions therefore does not by itself supply what Theorem 1 shows the spatial directions

lack.

A remark on time evolution.

One might hope to recover an o-diagonal generator from

the cage time evolution

exp(−iH

cage

dt)

, whose generator on the three-state space is

− I

Theorem 2 shows that this operator is not the identication the lattice supplies. It is also worth

separating two things that are easily conated. The operator

− I

arises from a kinetic term

hopping amplitude between bonds sharing a vertex, and those bonds lie in dierent color

classes, so the operator is spatial and o-diagonal at once. The relevant distinction is not

spatial against temporal. It is between weight-one operators, which are color diagonal, and

weight-two hopping operators, which are not. Section 8 supplies a mechanism that relies on

neither.

= 1

= 0

AB →



(the image of its

skew partner CD)

(a) the straddling 16-cell

A B C D



xy xy

xz xz

yz yz

= 0 x

= 1

= I

⊗ σ

identity on the color space

(b) the induced map on color

Figure 2: The temporal identication is color-blind. (a) The

-cell straddling two slices,

drawn as its two tetrahedra. Every pair of vertices is joined except the four antipodal pairs

(dotted). The unique straddling tetrahedron containing the bond

is shaded; it carries

, the image of the skew partner

rather than of

itself. (b) Because the exchange

happens within each skew pair, every color class returns to itself, and the induced map on the

three-state color space is the identity.

so(8)

from the lattice

Sections 4 and 5 show that the geometry supplies no color rotation of its own, in the spatial

directions or the temporal one. A mechanism outside the transporters considered so far is

therefore needed. The lattice already carries one, and this section identies it.

The algebra is already there.

The

roots of

are the roots of

so(8)

. Assigning a

generator

to each root and taking a rank-four Cartan subalgebra

gives

dim so(8) = 24 + 4 = 28,

(7)

with the

generators in bijection with the nearest-neighbor bond directions of the lattice.

We stress what is and is not claimed. This is a statement about the root system. It says the

algebra is naturally indexed by the lattice geometry. It does not yet say that the algebra acts

on any Hilbert space of the model. That gap is the subject of Section 13.

Why

and not another lattice.

Root lattices exist in every dimension, so the observation

above is not by itself special. What is special is the symmetry. For a root system

the quotient

Aut(Φ)/W (Φ)

is the diagram automorphism group, and it is trivial or of order two in every

case except

, where it is

[10]. This

is triality, and it is the extra structure that makes

the selection in Section 8 possible. The lattice that is singled out by four-dimensional packing

is the same lattice that is singled out by diagram symmetry.

7 Triality and the color classes

Attaching

so(8)

to the lattice does not by itself single out any subalgebra, and

so(8)

is far

larger than the eight dimensions wanted. The structure that distinguishes a subalgebra is the

triality of Section 6, and this section works out what it does to the color classes.

An explicit triality of the lattice.

Let

be the normalized Hadamard matrix on

(Z/2)

and let

negate the fourth coordinate. The composite

T = Φ

(8)

has order three and preserves the root system. Its orbits on the

roots are six singletons and

six triples.

The triality is not freely chosen.

Not every order-three lattice automorphism is com-

patible with the time slicing. The compatible ones are constrained, and the constraint is a

statement about root subsystems rather than a numerical accident.

Lemma 4.

The twelve spatial roots form a root subsystem of type

. Each color class is an

× A

subsystem of it, consisting of two orthogonal antipodal pairs.

Proof.

The spatial roots are the vectors

±e

± e

with

1 ≤ i < j ≤ 3

, twelve in all, each of

length squared two, spanning a three-dimensional space and closed under addition whenever

the sum is a root. This is the root system

, which is isomorphic to

. The class

consists

±(e

+ e

)

and

±(e

− e

)

, and

⟨e

+ e

, e

− e

⟩ = 0

, so the class is a pair of orthogonal

systems; likewise for

and

Lemma 5.

contains exactly four subsystems of type

. Each of them meets each color class

in exactly one antipodal pair, and the Weyl group

W (A

)

∼

permutes the four transitively.

Proof.

In the model

− e

: 1 ≤ i = j ≤ 4}

, a closed rank-two subsystem is either

× A

, with four roots, or

, with six. The

subsystems are the sets

− e

}

with

i, j

ranging over a three-element subset of

{1, 2, 3, 4}

, and there are





= 4

such subsets. The

symmetric group

= W (A

)

permutes the four subsets transitively.

For the second claim, let

α, β, α+β

be the positive roots of an

subsystem, so

⟨α, β⟩ = −1

No two of them lie in a common color class: by Lemma 4 two distinct roots of one class are

either orthogonal, giving inner product

, or antipodal, giving

−2

, and neither is

−1

. Three

positive roots distributed among three classes with no two sharing a class must occupy one

class each, and each brings its negative with it.

What the four subsystems are.

The count four in Lemma 5 has a direct reading on the

lattice. For each of the four

⟨111⟩

directions, the six spatial roots orthogonal to it form an

subsystem, and these four exhaust the list. The

⟨111⟩

directions are the four body diagonals of

the cubic cell, which are the four close-packed stacking axes of the face-centered-cubic lattice.

Choosing an

subsystem is therefore choosing a stacking axis, and the point group permutes

the four axes, which is why Proposition 6 nds the four selections in a single orbit. The

subsystem xed by the triality

dened above is the one orthogonal to

(1, −1, −1)

Proposition 6.

Let

be an automorphism of

of order three whose xed roots are six roots,

all spatial. Then those six roots form one of the four

subsystems of Lemma 5; in particular

they comprise exactly one antipodal pair from each color class, and any two such automorphisms

have xed sets carried into one another by the point group of the slice.

Proof.

The xed set of

is closed: if

xes

and

α + β

is a root, it xes

α + β

It is contained in the spatial

of Lemma 4 by hypothesis. A closed subsystem of

with

six roots has rank two, since

contains no three mutually orthogonal roots, and a closed

rank-two subsystem with six roots is of type

. Lemma 5 then gives both remaining claims,

the last because

= W (A

)

is realized by signed permutations of the spatial axes.

The counts, by exact enumeration.

Proposition 6 does not say how many such automor-

phisms exist. That is settled by enumeration rather than by argument, and we label it as such.

The group

Aut(D

)

is generated by the signed permutations of the four coordinates together

with the Hadamard map

; closing under multiplication in exact rational arithmetic gives

1152

elements, in agreement with

Aut(D

) = W (F

)

. Testing each element for order three,

and computing its xed roots by direct substitution, gives the counts in Table 2. The eight

compatible automorphisms realize the four

subsystems, two automorphisms to each, an el-

ement and its inverse. We x one of them, namely

, for the rest of the paper; by Proposition

6 the choice is immaterial.

elements of

Aut(D

) = W (F

) 1152

of order three

xing no root

xing six roots, not all spatial

xing six roots, all spatial

Table 2: Exact enumeration of

Aut(D

)

, by closure of the generating set in rational arithmetic.

The eight automorphisms in the last row realize the four

subsystems of Lemma 5, two

apiece.

Theorem 7.

The six roots xed by

are all spatial, and they comprise exactly one antipodal

pair drawn from each color class,

±(1, 1, 0) ⊂ xy, ±(1, 0, 1) ⊂ xz, ±(0, −1, 1) ⊂ yz.

(9)

Fixing the representatives

= (1, 1, 0), α

= (1, 0, 1), α

= (0, −1, 1),

(10)

one from each antipodal pair, these three form the positive roots of an

system and satisfy

+ α

= α

(11)

Each of the six triality orbits of size three contains exactly one spatial root and two temporal

roots.

Corollary 8.

Among the three selected roots and their negatives there are exactly two triples

summing to zero. Each takes one root from each color class, has pairwise angles of

120

◦

, and

lies in the close-packed plane of the selected subsystem. The two are exchanged by inversion.

Proof.

The three selected roots span a two-dimensional space and satisfy the single relation

(11), so the space of linear relations among them is one-dimensional. Rewriting (11) as

−

+ α

= 0

exhibits one triple with coecients

±1

, and negating all three gives the other;

no further triple with coecients in

{±1}

exists, since any such relation is a multiple of (11).

Each triple contains one root per class by Theorem 7. The pairwise inner products are

−1

between roots of length squared two, so the angles are

120

◦

, and all three lie in the plane of

the subsystem, which is orthogonal to

(1, −1, −1)

by Lemma 5.

(1, 1, 0)

(1, 0, 1)

(0, − 1, 1)

−α

+ α

= α

one antipodal pair per class

(a) the roots triality selects

−

orbit sums / 3: the surviving roots

norm 2/3; the deleted roots have norm 2

(b) the roots that survive the twist

Figure 3: Triality selects a root from each color class. (a) The six triality-xed roots of

Each color class contains four roots; triality xes one antipodal pair from each, shown here

with its class named beneath. The three positive representatives form an

system and satisfy

+ α

= α

, the shaded parallelogram. The class alone does not carry the orientation; the

selection does. (b) The six triality orbit sums, divided by three, are the roots of the algebra

that survives the twist, of length squared

2/3

against

for the deleted roots. The two triples,

marked

and

−

, are exchanged by spatial inversion, which acts on the color algebra as its

conjugation automorphism. The deleted roots are drawn faint for comparison; both sets lie in

the same plane.

What the theorem does and does not say.

A color class contains four roots, not two.

The class

, for instance, is

±(1, 1, 0)

together with

±(1, −1, 0)

, and triality xes only the

rst pair. So the classes are not the roots. What Theorem 7 gives is a

selection

: triality picks

one antipodal pair out of each class, and the three picks form

. This induces a bijection

{xy, xz, yz} ←→ {

positive roots of

}, c 7→ α

(12)

and it is through this bijection, not through an identity, that every later statement relating

color to roots should be read. The orientation carried by a root is supplied by the selection;

the class alone does not carry it.

The selection is sharper than a choice of representatives from each class. It also resolves

each skew pair of a cage. For the cage of Section 2, the two bonds of each matching are split

class xed by

not xed

xy AB = (1, 1, 0) CD = (−1, 1, 0)

xz AC = (1, 0, 1) BD = (−1, 0, 1)

yz BC = (0, −1, 1) AD = (0, 1, 1)

so exactly one bond of each skew pair is triality-xed. The cage sees three unoriented classes;

the triality orients them.

Reading.

This is the central geometric fact of the paper. The three color classes are not an

arbitrary labeling that happens to number three. Each supplies, through (12), one positive

root of an

system sitting inside the lattice and singled out as the triality-invariant plane,

and the three supplied roots obey the

relation (11) as vectors. The count three is matched

by the root system, not asserted by a separate combinatorial argument.

The second half of Theorem 7 is what evades the obstructions. The generators associated

with the size-three orbits are combinations of one spatial and two temporal roots. They are

neither purely spatial nor purely temporal. Theorem 1 constrains spatial transporters, and

Theorem 2 constrains temporal ones, and neither applies to a mixed object.

8 The twist that selects

su(3)

Theorem 7 produces a root subsystem, not yet a subalgebra, and untwisted triality xes too

much: its xed subalgebra is

, of dimension fourteen. This section introduces a torus element

that cuts the xed subalgebra down to eight dimensions, and Section 9 proves that the result

su(3)

Setup.

Let

be an order-three automorphism of

so(8)

lifting

, and let

t = exp(2πih)

with

h ∈ h

. Consider

σ = Ad(t) ◦ τ.

(13)

For

to have order three, the phase accumulated around each triality orbit must be trivial.

Writing

= α + T α + T

for the sum over a size-three orbit, this is

⟨s

, h⟩ ∈ Z

for all six orbits

(14)

Under condition (14) each size-three orbit contributes exactly one generator to the xed sub-

algebra, and each xed root

contributes one if and only if

⟨α, h⟩ ∈ Z

. The dimension is

therefore

dim Fix(σ) = 2 + #{

xed roots with

⟨α, h⟩ ∈ Z} + 6.

(15)

With

h = 0

this is

2 + 6 + 6 = 14

, the algebra

, in agreement with the classical fact that the

xed subgroup of untwisted triality is the automorphism group of the octonions [11].

The torus element.

Let

be the Weyl vector of the

system of Theorem 7 and set

h =



, 0,

, 0



(16)

This is one third of a lattice bond, namely the

bond, so the twist is a

phase on the

bonds. Direct evaluation gives

⟨α, h⟩ ∈ {±

, ±

}

on all six xed roots, none of them integers,

and

⟨s

, h⟩ ∈ {0, ±1}

on all six orbit sums, all of them integers. Equation (15) then gives

dim Fix(σ) = 2 + 0 + 6 = 8.

(17)

The scan.

Scanning

over a grid of sixths of the four-torus and keeping only the elements

for which

has order three, the dimension of the xed subalgebra takes the value

points and

points, and no other value anywhere. This reproduces from scratch the

classical dichotomy for order-three outer automorphisms of type

, whose xed groups are of

type

or of type

[13, 14]. The dimension-eight case is twice as common in the scan. We

state this as a count and not as a statement about genericity: the scan is over a nite grid of

the torus, and no claim is made about density, measure, or conjugacy classes. What the count

does show is that the

case is not an isolated coincidence of one torus element, and that

both cases occur in quantity. Figure 4 shows where the eight dimensions come from and how

the two cases are distributed.

2 6 6

untwisted, h = 0

14 = dim G

2 6

twisted, h = ρ/3

8 = dim (3)

Cartan fixed roots orbits

(a) where the eight dimensions come from

(3)

(dim 8)

(dim 14)

order-3 torus elements

(b) the torus scan

Figure 4: The selection. (a) The dimension of the xed subalgebra, by source. Untwisted

triality keeps the Cartan subalgebra, the six xed roots, and one generator per triality orbit,

giving

. The twist by

h = ρ/3

gives each xed root a primitive cube-root phase and so

removes all six, leaving

su(3)

. (b) Over the order-three torus elements of the scan, only these

two dimensions occur, and the

case occurs twice as often. The counts are over a nite grid

and are not a claim about genericity.

9 The xed subalgebra: analytic proof

The counting argument of Section 8 assumed that the structure constants cooperate. This

section removes the assumption and proves Theorem 13 outright. Section 10 then veries it

independently by explicit computation on

so(8)

Setup and notation.

Let

be a

-stable Cartan subalgebra of

so(8)

and let

be a lift of

to an automorphism of

so(8)

of order three whose xed subalgebra is

. Such a lift is the

triality automorphism, and the identity

Fix(τ) = G

is classical [11]. Write

τ(E

) = ε

T α

, ε

= 1,

(18)

for the induced action on root vectors, and set

σ = Ad(t) ◦ τ, t = exp(2πi ρ/3),

(19)

with

ρ = α

+ α

= α

the Weyl vector of the

system of Theorem 7. Since

Ad(t)

trivial on

and

-invariant,

restricts to

Lemma 9

(Order three)

= 1

Proof.

Because

= 1

and the torus is abelian,

= Ad



t τ(t) τ

(t)



. As

is xed by

have

τ(t) = τ

(t) = t

, so the argument is

exp(2πiρ)

. Now

ρ = (1, 0, 1, 0)

is itself a root of

and all roots have integer coordinates, so

⟨α, ρ⟩ ∈ Z

for every root

. Hence

Ad(exp(2πiρ))

acts as the identity on every root space and on

, and

= 1

Lemma 10

(The signs are trivial on xed roots)

= 1

for each of the six

-xed roots

Proof.

Decompose

so(8) = h⊕

and count the

-xed vectors. Write

for the subspace

xed by

. The Cartan contributes

dim h

= 2

, since

has order three on a four-

dimensional space and its xed subspace is the plane spanned by the six xed roots. A

xed root

contributes one dimension exactly when

= 1

. On a

-orbit of size three,

permutes the three one-dimensional root spaces cyclically with phases, so the xed subspace

is one-dimensional when the product of the phases around the orbit is

and zero-dimensional

otherwise. There are six xed roots and six orbits of size three, so

14 = dim Fix(τ) = 2 + a + b, 0 ≤ a ≤ 6, 0 ≤ b ≤ 6,

(20)

where

counts xed roots with

= 1

. This forces

a = b = 6

, so every

with

T α = α

equals

one.

Lemma 11

(Each large orbit contributes one generator)

For every

-orbit

of size three,

dim



α∈O



= 1

Proof.

On the three-dimensional space spanned by

, E

T α

, E

, the map

is a cyclic shift

with nonzero phases. Its cube is multiplication by the product of those phases, and this equals

by Lemma 9. A cyclic shift of a three-dimensional space whose cube is the identity has the

three cube roots of unity as eigenvalues, each with multiplicity one. Exactly one eigenvalue is

Lemma 12

(No xed root survives the twist)

For each of the six

-xed roots

, the eigenvalue

diers from

Proof.

By Lemma 10 and

T α = α

, the eigenvalue is

2πi⟨α,ρ⟩/3

. The three positive xed roots

give

⟨α

, ρ⟩ = 2 − 1 = 1, ⟨α

, ρ⟩ = −1 + 2 = 1, ⟨α

, ρ⟩ = ⟨ρ, ρ⟩ = 2,

(21)

using

⟨α

, α

⟩ = −1

and the normalization

⟨α, α⟩ = 2

. So

⟨α, ρ⟩/3 ∈ {±

, ±

}

over the six

xed roots, never an integer, and the eigenvalue is a primitive cube root of unity.

Theorem 13.

Fix(σ)

∼

su(3)

Proof.

Summing the contributions: the Cartan gives

dim h

= 2

; the six xed roots give

nothing, by Lemma 12; the six orbits of size three give one generator each, by Lemma 11.

Hence

dim Fix(σ) = 2 + 0 + 6 = 8.

(22)

The automorphism

has order three by Lemma 9, and it is outer, since

Ad(t)

is inner and

maps to an element of order three in

Out(D

)

∼

. By the classication of order-three

outer automorphisms of type

, the xed subgroup is of type

or of type

[13], a case

of Kac's classication of automorphisms of nite order [14]. Since

dim G

= 14 = 8

, the

xed subalgebra is of type

. By the conventions of Section 2 this is a statement about the

complexication; since

is dened by real conditions, its xed subalgebra meets the compact

real form

so(8)

in a real form of

, and a subalgebra of a compact algebra is compact. The

compact real form of

su(3)

Where the twist does its work.

Setting

t = 1

changes only Lemma 12: all six xed roots

then have eigenvalue one and the count reads

2 + 6 + 6 = 14

, which is

. The entire eect

of the twist is to give the six xed roots a primitive cube-root phase and so delete them. The

generators that survive are those attached to the mixed orbits, each containing one spatial and

two temporal roots.

10 Independent verication

The proof above rests on one imported fact,

dim Fix(τ) = 14

, and on the classication. We

check the conclusion independently by constructing the triality automorphism explicitly and

computing the xed subspace as a linear algebra problem, with no root combinatorics and no

assumed structure constants.

Quantity Predicted Computed

Octonion algebra normed, alternative, non-assoc. conrmed

Local triality solution space

28 28

Order of

A 7→ B

, xed dim.

21 = dim so(7) 2

Order of composite, xed dim.

14 = dim G

Bracket preserved by

yes yes

-stable Cartan, dimension

4 4

, residual

1.8 × 10

−14

Torus scan, dimensions reached

and

only

(

)

= 1

(Lemma 9) yes yes

Closure of the xed subspace subalgebra residual

7.5 × 10

−15

Trace form negative denite negative denite

Rank, number of roots

6 2

Root relation

a + b = c

satised

-xed Cartan inside

su(3)

yes yes

Table 3: Independent verication of Theorem 13 by explicit construction on

so(8)

. These are

numerical results, reported as a check on the analytic argument of Section 9 rather than as its

substitute. All entries are produced by the third script of the data availability section.

Construction.

The octonions are built by CayleyDickson doubling of the quaternions and

veried to be a normed alternative non-associative algebra. The principle of local triality states

that for each

A ∈ so(8)

there are

B, C ∈ so(8)

with

A(xy) = (Bx)y + x(Cy)

for all

x, y ∈ O,

(23)

and that the triple is determined by any one of its members [12, 11]. Treating (23) as a linear

system in the

unknown coecients of

(A, B, C)

, with one equation per pair of basis octonions

and output component, gives

512

equations whose solution space is

dimensional, with the

projection onto the

block invertible. So

A 7→ B

and

A 7→ C

are well-dened linear maps on

so(8)

. Both have order two with

-dimensional xed subalgebra

so(7)

; their composite has

order three, preserves the bracket, and has

-dimensional xed subalgebra. That composite

, and the value

is an independent conrmation of the fact imported in Lemma 10.

Result.

A regular element of

Fix(τ)

has a four-dimensional centralizer in

so(8)

, which is

therefore a Cartan subalgebra, and it is

-stable because the element is

-xed. Scanning

the order-three elements of that torus, the xed subalgebras of

have dimension

and

no other value, in the ratio

2 : 1

, matching Section 8. At a dimension-eight point the xed

subspace is closed under the bracket, has negative denite trace form, rank two, and six roots

satisfying

a + b = c

. Table 3 lists the checks. These are numerical results and are reported as

verication; the proof is the analytic argument of Section 9.

11 Representation content

Which roots the surviving algebra has.

The twist deletes the six triality-xed roots

and keeps one generator from each orbit of size three (Lemmas 12 and 11). The roots of the

surviving

su(3)

are therefore not the deleted ones. A generator attached to an orbit

has

weight, under the Cartan

, equal to the projection of any

α ∈ O

onto the invariant plane,

and that projection is the same for all three members:

pr(α) =



α + T α + T



(24)

So the six vectors

are the roots of the algebra that survives.

Long and short roots of

The deleted roots have length squared

and the surviving

roots length squared

2/3

, a ratio of three. This is the long-to-short ratio of

, and it identies

the mechanism cleanly. Untwisted triality gives

, whose long roots are the six triality-xed

roots and whose short roots are the six vectors

. The twist by

ρ/3

removes the long roots

and leaves the short ones, and the short roots of

form an

system. The color classes

select the long roots of

; the algebra that survives is built on the short ones.

The branching.

Under the surviving

su(3)

, the weights of the adjoint of

so(8)

are the zero

weight with multiplicity four, each of the six roots with multiplicity three, and the six deleted

roots with multiplicity one, the last at three times the root length in the normalization where

roots have length squared two. This multiset is exactly that of

so(8) −→ 8 ⊕ 10 ⊕

10, 8 + 10 + 10 = 28.

(25)

The adjoint

is the color algebra itself. The decuplets are the remaining twenty generators of

so(8)

, and they are not color gauge elds.

No fundamental appears, and this is as it should be.

The branching (25) contains no

. The lattice algebra supplies the adjoint, not the fundamental, so nothing here delivers quark

color as a representation carried by

so(8)

. That is the expected structure rather than a defect:

in any gauge theory the fundamental is carried by matter elds and not by the gauge algebra.

What the lattice would have to supply, in a construction of the kind discussed in Section 13,

is a matter eld on the sites or the cages transforming in

. We do not construct one.

Spatial inversion is the conjugation automorphism.

Inversion

v 7→ −v

preserves

and commutes with

, so it acts on the xed subalgebra. On the invariant plane it acts as

−1

, and an automorphism of

su(3)

acting as

−1

on a Cartan subalgebra is the conjugation

automorphism, which carries every representation to its conjugate and in particular exchanges

and

. The two tetrahedral-void orientations of the face-centered-cubic cell are related by

inversion, so the two orientations are related by conjugation of the color algebra.

Relation to the skew-pair count.

The cage count of Section 2 also gives three, and it

agrees with the present one for a concrete reason: the three skew pairs of a tetrahedral cage

are the three coordinate-plane classes, so both count the same three planes. Their roles dier.

The matching count gives a three-state label and nothing more. The root system gives a Weyl

group, a Cartan subalgebra, and a weight lattice, and it places the three classes as the long

roots of

, which the twist removes. The matching count is therefore a corollary of the root

structure rather than an independent origin of color.

12 Relation to prior work

We place the result against three bodies of work: the argument it is meant to improve on, the

literature on

su(3)

inside

so(8)

, and earlier proposals that put an algebra on a four-dimensional

lattice.

The qutrit argument, and why it is not enough.

A defect bounded by four sites carries

three skew-pair states, and one may build the traceless operators on that three-dimensional

space and observe that they close on

su(3)

. The observation is correct. It is also unavailable

as a selection, because the same closure holds for any three-state space and therefore excludes

nothing. The argument given here has a competitor at every stage: untwisted triality gives

, and only the twist by an order-three torus element reduces it to

su(3)

. The two routes

also assign dierent roles to the number three. In the qutrit argument the three colors label

a representation. Here each color class is mapped by triality to a selected positive root, that

root is then deleted by the twist, and the surviving algebra is built on the triality orbit sums

instead (Section 11). The skew-pair count is then the unoriented shadow of the root structure:

the three skew pairs of a tetrahedral cage are the three coordinate-plane classes, and triality

resolves each pair into a xed bond and a non-xed one, which is exactly the orientation the

count by itself does not carry.

su(3)

inside

so(8)

That

su(3)

sits inside

so(8)

is not new, and the octonionic route to it

is a developed subject. Todorov identies

su(3)

⊕ u(1) ⊕ u(1)

as the subalgebra of

so(8)

commuting with a complex structure and a charge operator [15]. Dubois-Violette and Todorov

build the internal space of the Standard Model on the exceptional Jordan algebra

, where

triality is associated with the three generations [16, 17]. Those constructions use commutants

and Jordan structure. The present one uses the xed subalgebra of a twisted order-three

automorphism, which is a dierent mechanism, and it is tied to a lattice: the roots in question

are bond directions, and the torus element is a phase on a bond. The mathematics underlying

the dichotomy is classical in any case. Order-three outer automorphisms of type

have xed

groups of type

[13], a special case of Kac's classication of nite-order automorphisms

[14]. We claim no novelty there, and Section 9 reproduces it as a check.

so(8)

on a four-dimensional lattice.

Smith's



program places

Spin(0, 8)

on a

HyperDiamond lattice, which is

, with gauge bosons on links and fermions on vertices, and

decomposes the

generators into

Spin(5)

SU(3)

, two copies of

SU(2)

, and four copies of

U(1)

[18]. That work also invokes the triality automorphism of

Spin(0, 8)

explicitly, using it

to reduce an eight-dimensional spacetime to four and to pass from

Spin(0, 8)

to the Standard

Model group [19]. Both the lattice and the appeal to triality are therefore anticipated, and we

record the precedent plainly. The mechanism is dierent. There the reduction is dimensional

and the

SU(3)

arrives through Wolf's classication of quaternionic symmetric spaces, as the

isometry group of

. Here the spacetime dimension is four throughout, and

su(3)

is the

xed subalgebra of a triality automorphism twisted by a torus element, with

excluded by

a dimension count.

Emergent gauge elds.

The condensed-matter program produces non-abelian gauge elds

from local bosonic models, and Wen's cubic-lattice construction yields quarks and gluons [5].

The group enters when the boson is split into partons. In string-net models it is set by the

input fusion category [6]. The distinction drawn in Section 1 applies to all of these: the lattice

supplies locality and constraints, and the group is chosen.

13 What is not established

We state the boundary of the result as sharply as the result.

This is not yet a gauge theory.

The

so(8)

of Section 6 is attached to the root system

of the lattice. Nothing here shows that its generators act on states of the model. To obtain

a gauge theory, the

root generators must be realized as operators on the links, which

requires link Hilbert spaces larger than a single bond phase. This is precisely the quantum link

construction [4], and it is the natural next step. Gauge theory on non-cubic lattices is by now

an active subject in its own right, with Hamiltonians built on triamond [20] and honeycomb

[21] geometries, so the machinery for putting group generators on the links of a lattice like this

one exists. What those constructions have in common with all the rest is that the group is

chosen before the lattice is used. The dierence from existing work is the point of the paper:

in a quantum link model the group is chosen, and here the lattice selects it, with

as the

only alternative.

A conditional remark on neutral congurations.

Corollary 8 is a statement about roots

and says nothing by itself about matter. We record one conditional consequence, without

adopting its premise. The weights of the fundamental of the selected algebra are one third of

the selected roots, so a eld in the fundamental couples to the bond phases with one third the

strength of a bond and is charged under the center

SU(3)

. If color is realized on this

lattice as such a center charge, then a conguration of three charges summing to zero must

sit on one of the two triples of Corollary 8, hence on a planar star at

120

◦

in the selected

close-packed plane, with the two stars conjugate to one another. We neither construct such a

eld nor derive an energy for the conguration, and without a tension nothing here bears on

connement.

Dynamics are untouched.

Connement in the continuum limit, the running coupling, and

QCD

are not addressed. The selection of an algebra is not a derivation of a theory.

One triality, chosen concretely.

We construct one order-three automorphism and one

family of torus elements. All order-three outer automorphisms of a given type are conjugate,

so the result is representative, but we have not classied the possible choices intrinsically in

terms of the lattice.

14 Conclusion

The

lattice is selected in four dimensions by packing, by kissing number, and by diagram

symmetry, and these turn out to be the same selection. Its nearest-neighbor directions are

the roots of

so(8)

. Its triality, constrained by the slicing to

of the

order-three lattice

automorphisms, xes one antipodal root pair in each of the three color classes of the face-

centered-cubic slice, and the three selected positive roots form an

subsystem. Twisting

triality by a

phase on the bonds leaves

su(3)

, with

as the only alternative, and

occurs twice as often as

across the torus elements scanned. The Cartan subalgebra of that

su(3)

is the bond-phase torus, its roots are the normalized orbit sums

, and

so(8)

branches

under it to

8 ⊕ 10 ⊕

, with spatial inversion acting as the conjugation automorphism.

Two obstruction theorems make the mechanism necessary rather than optional. No spatial

transporter rotates color, because color is a three-element label and the geometry acts on it

through

. The identication the honeycomb supplies between successive cages rotates color

no further, because the straddling

-cell acts by skew-pair exchange, which is the identity

on the color qutrit. The generators that do the work belong to triality orbits containing one

spatial and two temporal roots, and they lie outside the reach of both theorems.

An argument that reaches

su(3)

from the mere existence of three states could not have

reached anything else. The argument given here could have reached

, and did not. That is

the dierence between a closure and a selection, and it is the content of this paper. Turning

the selected algebra into a gauge theory is the work that remains.

A The anchor is xed by the stacking axis

The framework of Section 2 splits the four bonds of a cage into one anchor and three valence

bonds, and takes the choice of anchor to be dynamical, breaking the tetrahedral

to the

acting on the remaining three [1]. This appendix records that on the

lattice the split is not

a choice. The color plane of Section 7 makes it, and it singles out the same

The projection.

Let

A, B, C, D

be the vertices of a cage,

¯v

their centroid, and

ˆr

= K − ¯v

the four centroid-to-vertex directions. Let

denote orthogonal projection onto the color plane,

the plane of the selected

subsystem, which by Lemma 5 is orthogonal to the stacking axis

(1, −1, −1)

Proposition 14.

For every tetrahedral cage of the slice, exactly one of the four directions

ˆr

is annihilated by

, and the remaining three project to roots of the algebra of Theorem 13, of

length squared

2/3

, summing to zero.

Proof.

For the cage

A = (0, 0, 0)

B = (1, 1, 0)

C = (1, 0, 1)

D = (0, 1, 1)

the centroid

(

)

and the four directions are

ˆr

= −

(1, 1, 1)

ˆr

(1, 1, −1)

ˆr

(1, −1, 1)

ˆr

(−1, 1, 1)

. Since

ˆr

is a multiple of

(−1, 1, 1)

, it is a multiple of the stacking axis and

pr ˆr

= 0

. Projecting the other three gives

pr ˆr

(−2, −1, −1), pr ˆr

(1, 2, −1), pr ˆr

(1, −1, 2),

(26)

each of length squared

2/3

and each equal to

for a triality orbit, hence a root by Section

11. Their sum vanishes, as it must, because the four

ˆr

sum to zero and projection is linear.

For the remaining cages we do not invoke the point group, which permutes the four stacking

axes and therefore does not x the plane onto which

projects. Instead, observe that the

directions

ˆr

are unchanged by a lattice translation, so a cage enters only through its shape,

and the shapes fall into nitely many translation classes. There are exactly two: the slice

carries twice as many tetrahedral cells as sites, and the two classes are the two orientations of

the tetrahedral void, with direction sets

{(−1, −1, −1), (−1, 1, 1), (1, −1, 1), (1, 1, −1)},

{(1, 1, 1), (1, −1, −1), (−1, 1, −1), (−1, −1, 1)},

(27)

the second being the image of the rst under

v 7→ −v

. Inversion xes the stacking axis as

a line, hence xes the plane of the projection and commutes with

, and it carries roots to

roots. The displayed computation settles the rst class and inversion settles the second. A

scan over all

216

cages of the slice at

L = 6

, reported in the data availability section, conrms

the conclusion directly.

What this does and does not give.

The

1 + 3

split and the reduction

→ S

are

obtained rather than assumed, and the residual

is the Weyl group of the algebra selected

in Section 9. That is the whole of the claim.

Three things are not claimed. The three projected directions are

roots

, not weights, so

nothing here assigns a fundamental representation to the cage; by the branching (25) no

fundamental occurs in

so(8)

at all. The identication of the axial direction as an anchor is a

reading, not a derivation: the projection distinguishes that direction, and the framework calls

the distinguished bond the anchor, but no property of an anchor beyond distinguishability is

established. Finally, the pairwise inner products of the three projections are

−1/3

, numerically

the value of the regular-tetrahedron bond-angle cosine used in [1]. The agreement is one of

normalization and not of content: the projections have length squared

2/3

, so their mutual

angle is

120

◦

and their cosine is

−

. We ag this because the coincidence invites the conclusion

that the tetrahedral cosine has been derived here, and it has not.

stacking axis



lies along the axis;



do not

(a) the four centroid-to-vertex directions



= 0

three roots of the surviving (3), summing to zero

each of length squared 2/3; pairwise 120

∘

(b) projected onto the color plane

Figure 5: The anchor is xed by the geometry. (a) The four centroid-to-vertex directions of

a cage. One of them lies along the stacking axis. (b) Their images in the color plane, drawn

to scale: the axial direction projects to zero, and the other three become roots of the selected

algebra, at

120

◦

and summing to zero. The hexagon is the root system of that algebra.

Data availability

Three Python scripts reproduce every numerical statement in this paper. They are available

in the repository github.com/raghu91302/ssmtheory.

ssm_temporal_color.py

veries the root split of Section 3, the isotropy relation (3), The-

orem 2 for all

straddling cells, the

count of Theorem 1, and the cell counts of Table 1.

It uses the standard library only, its arithmetic is exact, and it runs in a few seconds.

ssm_d4_triality_su3.py

veries Theorem 7, the torus element (16), the dimension count

(15), the torus scan of Section 8, Lemmas 4 and 5 including the identication of the four

subsystems with the four close-packed plane orientations, Proposition 6, the enumeration of

Aut(D

)

reported in Table 2, Corollary 8, Proposition 14 including the scan over all

216

cages, and the weight statements of Section 11. It also uses the standard library only and its

arithmetic is exact.

so8_triality_su3_verify.py

carries out the explicit construction of Section 10 and re-

produces every line of Table 3. It requires NumPy and SciPy and runs in about a minute.

Being numerical, it veries Theorem 13 rather than proving it; the proof is analytic and is

given in Section 9.

Declaration of competing interest

The author declares no competing nancial interests or personal relationships that could have

appeared to inuence the work reported in this paper.

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