A Self-Correcting Discrete Vacuum: Baryon Conservation and Color, from an Emergent Qutrit to Temporal su(3) Closure, on One FCC Code Complex

A Self-Correcting Discrete Vacuum:

Baryon Conservation and Color, from an Emergent Qutrit

to Temporal su(3) Closure, on One FCC Code Complex

Raghu Kulkarni

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

raghu@idrive.com

June 2026

Abstract

A single FCC code complex on a face-centered-cubic vacuum, with a quark a node trapped

in a tetrahedral void, protects both the conserved quantities of baryonic matter and its color,

and we develop both sectors at one standard of rigor, every quantitative claim reproduced by

an accompanying script. The [[192, 130, 3]] binary CSS code with weight-twelve vertex and

octahedral-void stabilizers, together with its Z

triality reﬁnement on the same chain complex,

is the common origin. In the external sector it makes the defect’s winding charge a Gauss-law

super-selection invariant, yielding the baryon ladder ∆

−

, n, p, ∆

with charges Q = −1 + W ;

the void orientation, by contrast, is not protected (a negative result we record); the migrating

defect obeys E = γE

rest

; and migration is a code-corrected string. In the internal sector

the same complex protects color. A color qutrit emerges as the symmetric sector of the cage

kinetic term T = (J

− I

) ⊗ J

; the three skew-pair matchings carry an S

Weyl symmetry

and a rank-two torus, and among the rank-two simple Lie algebras both the Weyl order and the

existence of a three-dimensional representation independently select A

∼

su(3); anchor-induced

color/dark leakage is a weight-one correctable error. The FCC complex is then gauge-ready—

its minimal plaquettes are triangles, each carrying one bond of every color—and supports a

genuine Z

triality code, whose vertex Gauss law enforces q

+ q

≡ 0 mod 3 and which

subsumes the octahedral baryon operator; the non-abelian SU(3) Gauss law at a vertex has a 513-

dimensional gauge-invariant space gated by triality, and the triangular plaquette Hamiltonian is

gauge-invariant, so a Kogut-Susskind SU(3) Hamiltonian is well-posed on the FCC plaquettes.

Building the color connection from the vacuum’s processes gives a layered dynamical result.

The spatial channels transport color only diagonally, an N(T ) = T ⋊ S

connection short of

SU(3) by the root generators because the colors are bond-disjoint. We then test whether the

stabilizer correction cycle removes this obstruction—whether a syndrome-conditioned recovery,

dressing the bare transport, induces an oﬀ-diagonal logical map—and ﬁnd on the real code that

it does not: the minimum-weight recovery of a leakage event is single-bond and same-color, so

the QEC-dressed transporter stays in N(T ). The oﬀ-diagonal direction is instead supplied by

the temporal link, the cage time evolution with oﬀ-diagonal generator J

− I

, which with the

torus closes the full eight-dimensional su(3). Leading strong-coupling conﬁnement holds on the

triangular plaquettes with a positive string tension, and the single-plaquette spectrum has the

fundamental Casimir gap C

(3) = 4/3. The color algebra, both protected sectors, and the well-

posed Hamiltonian are thus attained on one code complex, with the full color algebra appearing

only when temporal qutrit evolution is included; the continuum dynamics—conﬁnement in the

continuum limit, the running coupling, Λ

QCD

—are not addressed and remain open.

Contents

1 Introduction 3

2 The vacuum code 3

2.1 Lattice, qubits, and stabilizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Commutation and the CSS property . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Logical count and distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4 The [4, 4, 4] vertex split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 External sector I: orientation is not protected 5

4 External sector II: baryon charge as a Gauss-law super-selection rule 5

4.1 Charge as enclosed ﬂux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4.2 The winding and the baryon ladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5 External sector III: relativistic kinematics 6

6 External sector IV: migration as code-corrected transport 7

7 Internal sector I: the emergent color qutrit 8

7.1 Three colors from K

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

7.2 The kinetic-term factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

8 Internal sector II: Weyl group, torus, and the selection of A

8.1 The Weyl group from tetrahedral symmetry . . . . . . . . . . . . . . . . . . . . . . . 9

8.2 The maximal torus from bond phases . . . . . . . . . . . . . . . . . . . . . . . . . . 9

8.3 Selection of A

among the rank-two simple algebras . . . . . . . . . . . . . . . . . . 9

9 Internal sector III: code stabilization of color 9

10 Internal sector IV: the gauge-ready geometry 10

10.1 Triangular plaquettes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

10.2 Color-completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

11 Internal sector V: the Z

triality code 11

11.1 The chain complex and the CSS condition . . . . . . . . . . . . . . . . . . . . . . . . 11

11.2 Logical count and homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

11.3 Triality, correctability, and subsumption . . . . . . . . . . . . . . . . . . . . . . . . . 11

12 Internal sector VI: a well-posed Kogut-Susskind SU(3) Hamiltonian 12

12.1 The non-abelian Gauss law and the 513-dimensional vertex space . . . . . . . . . . . 12

12.2 The magnetic term is gauge-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . 12

13 Internal sector VII: the dynamical connection—spatial obstruction 12

13.1 The migration channel realizes the Weyl group . . . . . . . . . . . . . . . . . . . . . 12

13.2 Spatial channels reach only N(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

13.3 The other spatial channels are color-diagonal . . . . . . . . . . . . . . . . . . . . . . 13

14 Internal sector VIII: QEC-dressed logical transport is tested and remains diag-

onal 13

14.1 The dressed transporter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

14.2 The test on the real code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

14.3 Consequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

15 Internal sector IX: temporal resolution—the root direction from cage evolution 14

15.1 The temporal link is color-mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

15.2 The temporal generator closes su(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

16 Internal sector X: dynamical footholds 15

16.1 Strong-coupling conﬁnement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

16.2 The single-plaquette spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

17 Uniﬁcation: one code complex, two sectors 17

18 Scope and status 17

1 Introduction

The strong sector of the Standard Model is speciﬁed by ingredients taken as input: the gauge group

SU(3), the color number N

= 3, baryon-number conservation as an accidental global symmetry,

and the conﬁning dynamics. The Selection-Stitch Model proposes a common origin for these in a

discrete vacuum—an FCC entanglement network carrying a stabilizer code, in which a quark is a

node trapped at the centroid of a tetrahedral void [1]. This paper assembles, in one place and at

one standard of rigor, what that proposal establishes about the strong sector, organized around

a single claim: that one FCC code complex—the same chain complex in its binary distance-three

realization and its Z

triality reﬁnement—protects two sectors.

The thesis is that the [[192, 130, 3]] FCC stabilizer code [2] is simultaneously the origin of baryon-

number conservation (the external sector: charge, kinematics, migration) and of color (the internal

sector: the qutrit, the algebra, the gauge connection). Figure 1 shows the architecture. The external

sector occupies Sections 2–6; the internal sector occupies Sections 7–16, ascending through three

levels—representation, kinematics, and dynamics. Section 17 states the uniﬁcation, and Section 18

the boundary: the color algebra, both sectors, and the well-posed theory are reached; the conﬁning

continuum dynamics are not.

The discipline throughout is to state, at each step, exactly what is computed, and to mark negative

results and open questions as plainly as positive ones. Every quantitative claim is reproduced by a

script in the data-availability archive.

2 The vacuum code

2.1 Lattice, qubits, and stabilizers

The vacuum is the FCC lattice [12], with the four-atom conventional basis (0, 0, 0), (

, 0), (

, 0,

), (0,

We work on a periodic 2 × 2 × 2 block, which contains N = 32 sites. The nearest neighbors of a

site are the twelve ⟨110⟩/

√

2 directions at distance a/

√

2 (the FCC kissing number 12), and on the

block there are E = 192 distinct nearest-neighbor bonds. A qubit is placed on each bond, giving a

192-qubit code space.

FCC code complex

binary [[192, 130, 3]] code + Z

triality reﬁnement

External sector (baryon number) Internal sector (color)

charge Q = −1 + W (Gauss law)

baryon ladder ∆

−

, n, p, ∆

E = γE

rest

; migration string

qutrit, S

, torus, A

=su(3)

triality; well-posed KS Hamiltonian

N(T ) spatially → su(3) with time

Figure 1: One FCC code complex, two protected sectors. The same chain complex—in its binary

[[192, 130, 3]] realization and its Z

triality reﬁnement—makes the defect’s winding charge a Gauss-

law invariant (external) and protects the emergent color qutrit (internal); the internal sector ascends

from representation through kinematics to dynamics.

Two stabilizer families are deﬁned. To each vertex v associate the star operator

e∈star(v)

, |star(v)| = 12, (1)

the product of Z over the twelve bonds incident to v. The octahedral voids of the FCC lattice

(the octahedral interstitial sites) number 32 on the block; the six atoms nearest a void o form an

octahedron whose twelve edges are nearest-neighbor bonds, and to each void associate the cage

operator

e∈oct(o)

, |oct(o)| = 12, (2)

the product of X over those twelve bonds. Both families are weight-twelve.

2.2 Commutation and the CSS property

is a product of Z and B

a product of X, so they commute if and only if their supports overlap

in an even number of bonds. A vertex v lying on the octahedron of void o contributes the bonds

incident to v within that octahedron; a direct enumeration shows this overlap is either 0 (when v is

not a vertex of the octahedron) or exactly 4 (when it is), both even. Hence [A

, B

] = 0 for all v, o,

and the code is a Calderbank-Shor-Steane (CSS) code [5, 6], with H

the vertex–bond incidence

and H

the void–bond incidence over F

2.3 Logical count and distance

Each bond is incident to exactly two vertices, so every bond appears in exactly two star operators;

therefore

= ⊮ (each Z

appears twice and Z

= ⊮), one relation among the 32 Z-stabilizers,

leaving 31 independent. Each bond borders exactly two octahedral voids, so likewise

= ⊮,

leaving 31 independent X-stabilizers. The independent stabilizer count is 31 + 31 = 62, and the

logical-qubit count is

k = E − 62 = 192 −62 = 130, (3)

the code [[192, 130, 3]]. For the distance: a single-bond Z error on bond e anticommutes with

exactly the cage operators of the two voids bordering e, a weight-two X-syndrome that is distinct

for each bond, so single-bond Z errors are detectable and correctable; symmetrically a single-bond

X error ﬂags the two vertices of e. Weight-one errors are thus correctable, consistent with distance

three.

2.4 The [4, 4, 4] vertex split

The twelve bonds incident to a vertex lie in the three coordinate planes—four in the xy plane, four

in xz, four in yz—giving a [4, 4, 4] partition of the star. This partition is invisible in the external

sector but is the seed of color: in Section 8 the three classes are the three colors, and in Section 11

the Z

vertex Gauss law factorizes through it into the triality condition. The two stabilizer types

are the two halves of a discrete gauge structure, in the manner of the toric code [7]: the vertex

Z-stars are Gauss-law generators, the octahedral X-cages are magnetic operators, and the same

FCC chain complex below carries both sectors, through its binary distance-three realization and

its Z

triality reﬁnement.

3 External sector I: orientation is not protected

Before the positive external results we record a natural candidate invariant that fails; the failure

sharpens what the code protects. A tetrahedral void has an orientation—the sense in which its

trapped node binds its four cage vertices—and one might expect it to be topologically protected. It

is not. Consider the two ways a relocation can treat the orientation. Preserving it keeps the node

bound through the six second-neighbor bonds of length a/

√

2 that maintain the original binding

sense; ﬂipping it rebonds the node through the four nearest cage bonds of length a/2. The elastic

cost scales with total bond length, so

cost(ﬂip)

cost(preserve)

4 · (a/2)

6 · (a/

√

6a/

√

≈ 0.667. (4)

The orientation-ﬂipping relocation is therefore cheaper, by a factor 2/3, than the orientation-

preserving one. Orientation is not a protected quantity. The lesson is that protection in this

vacuum is not conferred by local geometric labels, which can be relaxed by cheaper rebondings, but

by global Gauss-law ﬂuxes, which cannot; the protected external invariant is the winding charge of

the next section.

4 External sector II: baryon charge as a Gauss-law super-selection

rule

4.1 Charge as enclosed ﬂux

The protected external quantity is the defect’s winding charge, and it is protected because it is

a Gauss-law ﬂux. The vertex stars A

generate a discrete Gauss law: on a state with a trapped

defect, the product of A

over a closed surface Σ enclosing the defect equals the net Z-ﬂux through

Σ, and this is a stabilizer-measured, conserved quantity. It is conserved because the boundary

map’s rows sum to zero:

= ⊮ globally (Section 2), so the total enclosed ﬂux cannot change

under any local operation that does not cross Σ. The charge is therefore a super-selection label,

invariant under every local rebonding interior to Σ—unlike the orientation, which is a local label

and is not invariant.

4.2 The winding and the baryon ladder

A trapped node bonds to four cage vertices; three carry the valence structure and one is the anchor

ﬁxing the baseline. Each valence bond carries a binary winding w

∈ {0, 1} (whether its Bell-pair

phase has wound), and the net winding is W =

i=1

∈ {0, 1, 2, 3}. The enclosed ﬂux is the

anchor baseline plus the valence winding,

Q = −1 + W, (5)

the −1 from the anchor and +W from the valence bonds. The per-quark charge is correspondingly

= −

+ w

, fractional because the anchor baseline −1 is shared democratically across the three

colors. Summing,

= −1 + W = Q, consistent with Eq. (5). The four winding values give

the baryon ladder of Figure 2 and Table 1: W = 0 gives ∆

−

with all quarks at −

and Q = −1;

W = 1 the neutron, one quark at +

and two at −

, Q = 0; W = 2 the proton, Q = +1; W = 3

the ∆

, all quarks at +

, Q = +2. The integer quark charges and the baryon ladder are thus

super-selection consequences of Eq. (5), not separate inputs.

baryon W quark charges (−

+ w

) Q = −1 + W

∆

−

0 (−

, −

) −1

n 1 (+

, −

) 0

p 2 (+

, +

, −

) +1

∆

3 (+

, +

) +2

Table 1: The baryon ladder generated by the winding W . Each unit of winding ﬂips one valence

quark from −

to +

and raises Q by one.

0 1 2 3

winding W =

−1

electric charge Q = ¡1 + W

(¡

; ¡

)

; ¡

)

; +

; ¡

)

+ +

; +

)

Figure 2: The baryon spectrum as a Gauss-law super-selection ladder; Q = −1+W , with per-quark

charges shown.

5 External sector III: relativistic kinematics

A defect set in motion is a soliton of the bond dynamics. In the long-wavelength limit the bond-

phase ﬁeld obeys a sine-Gordon-type equation, whose static kink is the defect; in lattice units its

rest energy is E

rest

= 8. The sine-Gordon kink is a Bogomolnyi (topological) soliton [11], for which

the boosted energy is exactly the Lorentz-transformed rest energy,

E(v) = γ E

rest

, γ =

1 − v

, (6)

with c the lattice signal speed. We verify this on the lattice: boosting the discrete kink proﬁle

by velocity v and evaluating the lattice energy functional reproduces γE

rest

to six decimal places

across v/c ∈ [0, 0.95] (Figure 3). The relation is exact for the one-dimensional Bogomolnyi soliton;

a fully three-dimensional defect inherits E = γE

rest

by Lorentz covariance of the long-wavelength

eﬀective theory, which is the level at which the claim is made, and which we do not strengthen to

an exact three-dimensional statement.

0.0 0.2 0.4 0.6 0.8

defect velocity v=c

1.0

1.5

2.0

2.5

3.0

E=E

rest

= °

boosted lattice soliton

matches °E

rest

to 6 d.p.

Figure 3: The migrating defect’s energy follows E = γE

rest

with E

rest

= 8, veriﬁed against the

lattice soliton boost to six decimal places.

6 External sector IV: migration as code-corrected transport

Baryon transport is defect migration, and migration is realized as a string of single-bond distur-

bances that the code corrects en route. Translating the defect by one lattice step toggles the bonds

along its path. By Section 2 each toggled bond is a weight-one error with a deﬁnite two-void (or

two-vertex) syndrome, hence correctable; the correction repairs the local disturbance while leaving

the enclosed ﬂux—the charge of Eq. (5)—unchanged, since the migration string does not cross the

surface deﬁning Q. Migration is therefore code-assisted string transport: the external charge is

conserved not by a continuous symmetry imposed by hand but by the error-correcting structure of

the vacuum, which detects and undoes every local step that would otherwise leak it. The string

transports the charge as a logical invariant, in direct analogy with the transport of a logical op-

erator along a corrected path in a stabilizer code. This is the external half of the “one code, two

sectors” thesis; the internal half follows.

7 Internal sector I: the emergent color qutrit

7.1 Three colors from K

The same defect carries color. It is bounded by four sites through four valence bonds, and the

complete graph K

on the bonds admits exactly three perfect matchings into skew pairs,

= {AB, CD}, M

= {AC, BD}, M

= {AD, BC}, (7)

so the color number is the rigid combinatorial count





= 3, (8)

with no fourth pairing, and none for an odd cage (K

has no perfect matching). The dimension of

the color Hilbert space is ﬁxed before any algebra is written (Figure 4).

node

— M

= {AB, CD}

– – M

= {AC, BD}

··· M

= {AD, BC}

three skew pairings

= three colors

Figure 4: The trapped node bonds to the four cage vertices A, B, C, D; K

has exactly three perfect

matchings into skew pairs (the three colors), and the pairs are bond-disjoint.

7.2 The kinetic-term factorization

Color is not merely a label; it emerges as a dynamical sector. The cage degrees of freedom factor as

the three-matching space Z

tensored with the two-state space of each skew pair (the two members

of a pair). On this 3 × 2 space the nearest-neighbor kinetic term, which hops amplitude between

bonds sharing a vertex, takes the factorized form

T = (J

− I

) ⊗ J

, (9)

where J

is the 3×3 all-ones matrix (hopping between distinct matchings), I

the identity (removing

the diagonal), and J



1 1



the all-ones matrix on the two members of a skew pair. The

eigenstates of J

are the symmetric and antisymmetric combinations

⟩ =

√



⟩ + |e

⟩



, |−

⟩ =

√



⟩ − |e

⟩



, (10)

with J

⟩ = +2|+

⟩ and J

|−

⟩ = 0, so the antisymmetric states are annihilated by the hopping.

Since T = (J

− I

) ⊗ J

, the symmetric sector and antisymmetric sector decouple, and on the

symmetric (color) sector T restricts (absorbing the factor 2 into the hopping rate Γ) to



color

= J

− I





0 1 1

1 0 1

1 1 0





, (11)

a Hamiltonian on the qutrit H

= span

{|+

⟩, |+

⟩} with six color transitions |+

⟩ ↔ |+

⟩.

The antisymmetric states |−

⟩ are dark: they carry eigenvalue 0 of J

and decouple from the color

dynamics entirely. The qutrit is therefore the symmetric sector of the cage hopping, derived from

Eq. (9) rather than assumed. The structural fact used repeatedly below is that the three matchings

are bond-disjoint: each cage bond belongs to exactly one skew pair, hence to exactly one color.

8 Internal sector II: Weyl group, torus, and the selection of A

8.1 The Weyl group from tetrahedral symmetry

The symmetric group S

acts on the four bond labels. The Klein four-group V

⊂ S

of double

transpositions {e, (AB)(CD), (AC)(BD), (AD)(BC)} ﬁxes each matching as a set, so the induced

action on the three matchings is through the quotient

∼

, (12)

which permutes {M

, M

} as the full symmetric group on three objects. This S

is the Weyl

group of A

. Direct enumeration of the 24 bond permutations conﬁrms exactly the six color

permutations are realized.

8.2 The maximal torus from bond phases

The three valence bonds carry Bell-pair phases ϕ

, ϕ

. The diagonal combination ϕ

+ ϕ

is the overall U(1) phase, which is not color and is identiﬁed with electromagnetism elsewhere in

the program; the two independent traceless combinations form the rank-two maximal torus T

the color algebra, with the Cartan generators λ

, λ

as their conjugate charges.

8.3 Selection of A

among the rank-two simple algebras

The data—a three-dimensional representation with Weyl group S

and a rank-two torus—select

∼

su(3), and the selection is sharp. There are exactly three rank-two simple Lie algebras [9],

and the geometry supplies two independent data, each of which excludes the other two (Table 2).

The Weyl-group orders are 6, 8, 12, so the order-six S

is the Weyl group of A

alone. The smallest

nontrivial representation dimensions are 3, 4, 7, so a three-color (three-dimensional) representation

exists only for A

. Either datum excludes B

and G

algebra Weyl-group order smallest nontrivial rep selected by

= su(3) 6 (S

) 3 both

∼

so(5) 8 4 neither

12 7 neither

Table 2: Among the rank-two simple Lie algebras, both the S

Weyl symmetry (order 6) and the

existence of a three-dimensional representation independently select A

∼

su(3).

This is a selection of the rank-two representation and Weyl structure, not a derivation of the

continuous gauge dynamics. Once a genuine qutrit exists, the traceless operators on it close on

su(3) automatically; that closure is not independent evidence and is not oﬀered as such.

9 Internal sector III: code stabilization of color

The color qutrit is a protected logical subspace of the code of Section 2. The anchor, needed to ﬁx

the charge baseline and the Weyl structure, breaks the cage symmetry between the two members

of a skew pair and so couples the color qutrit to the dark sector. Modeling the anchor as an on-site

asymmetry between the bonds of a pair, the leakage operator connects |+

⟩ to |−

⟩ at a rate set by

the anchor strength,

leak

− Γ

, (13)

with Γ

the anchored and Γ the symmetric hopping rate; the leakage vanishes in the symmetric

limit Γ

= Γ. The color/dark exchange is the exchange parity of a single skew pair, which is

a single-bond Z operator, hence a weight-one error of the code—exactly the correctable class at

distance three, ﬂagged by the bordering-void syndrome of Section 2.

The protection operates in two regimes. Without active correction, the steady-state dark population

is perturbatively bounded by ∼ (Γ

leak

/Γ)

: below one percent for a weak anchor (Γ

/Γ ≲ 1.2, giving

0.7%), rising to about ﬁfteen percent only at the strong value Γ

/Γ = 2. With periodic stabilizer

projection, the leakage is suppressed by the quantum Zeno eﬀect when the correction rate exceeds

the leakage rate, γ

corr

≳ Γ, the dark population then falling as γ

−2

corr

. Color is, in this precise and

quantiﬁed sense, a protected logical degree of freedom, on the same code that protects the external

charge—but the protection is conditional on the anchor being weak or the correction fast, a caveat

we state rather than suppress.

10 Internal sector IV: the gauge-ready geometry

10.1 Triangular plaquettes

A lattice gauge theory is built on plaquettes, the minimal closed loops of the bond graph. The

FCC nearest-neighbor graph is close-packed, and its minimal loops are triangles: three mutually

adjacent sites. On the block there are 256 such triangular plaquettes, and each nearest-neighbor

bond lies in exactly four of them, so the triangles tile the two-skeleton uniformly. A gauge action

on the FCC vacuum is therefore intrinsically triangular rather than square; this is unusual but not

pathological.

10.2 Color-completeness

The triangles are color-complete: every one of the 256 carries exactly one bond of each of the three

coordinate-plane color classes. To see this, take a triangle on sites 0, (

, 0), (

, 0,

); its three edges

have displacement vectors (

, 0), (

, 0,

), and (0, −

), which lie in the xy, xz, and yz planes

respectively—one of each class. The same holds for all 256 triangles. Consequently a plaquette

term Re Tr(U

) on an FCC triangle couples one bond of each color, and the elementary

loops are not color-blind (Figure 5)—the geometric prerequisite for color-mixing dynamics.

each of the 256 triangular plaquettes

carries one bond of every color

Figure 5: A color-complete triangular plaquette: every FCC triangle has one bond from each

coordinate-plane color class (R, G, B), so a Wilson plaquette term mixes the three colors by con-

struction.

11 Internal sector V: the Z

triality code

11.1 The chain complex and the CSS condition

The binary code of Section 2 protects only the Z

reduction of triality. The triangular two-cells lift

it to genuine Z

. We build the cellular chain complex over Z

with 0-cells the vertices, 1-cells the

bonds (now carrying qutrits with the Z

clock and shift operators), and 2-cells the 256 triangles.

The boundary maps are ∂

(bond → its two endpoints, with +1 at the head and −1 at the tail)

and ∂

(triangle → its three bonds, with cyclic orientation). The vertex Z-stabilizers are the rows

of ∂

and the triangular X-stabilizers the rows of ∂

⊤

, so the CSS condition is

⊤

= ∂

⊤

∂

⊤

= (∂

∂

)

⊤

= 0 mod 3, (14)

which holds automatically, being the topological identity ∂

∂

= 0. (By contrast the octahedral-

void operators do not survive the Z

lift: the four-bond vertex–void overlap is 4 ≡ 1 mod 3, and

the naive qutrit generalization fails to commute, with 152 violations on the block.)

11.2 Logical count and homology

A rank computation over GF(3) gives rank H

= 31 (the boundary rank of a connected graph,

N −1) and rank H

= 158, so

k = 192 − 31 − 158 = 3, (15)

equal to the ﬁrst Betti number b

) = 3 of the three-torus: the three logical qutrits are the three

independent noncontractible cycles, the correct homological count rather than an artifact. The

integer complex closes as well, ∂

∂

= 0 over Z, so an integer-charge code exists with triality as its

mod-three reduction; comparing dim H

over Q, F

, and F

gives 3 in every case, so H

= Z

free with no torsion at any prime. Triality here is therefore the mod-three reduction of the integer

charge, not a topologically forced class—a negative we record rather than overstate.

11.3 Triality, correctability, and subsumption

At each vertex the twelve incident bonds split [4, 4, 4] by color (Section 2), and the Z

vertex Gauss

law factorizes through this split into the three class charges q

, q

, imposing

+ q

≡ 0 mod 3, (16)

genuine triality—the center Z

⊂ SU(3) governing color superselection—rather than the Z

shadow

of the binary code. Single-bond Z errors have weight-four triangular syndromes (the four triangles

containing the bond), all 192 distinct, hence correctable; the total triality charge is conserved since

the boundary rows sum to zero. Finally, the weight-twelve octahedral baryon operator, suitably

-oriented, is a boundary in the triangular complex—it lies in the image of ∂

, hence is a product

of triangular stabilizers and is already implied by them. The triangular Z

code therefore subsumes

the octahedral structure: one Z

code carries both the triality (internal) and the baryon (external)

sectors.

12 Internal sector VI: a well-posed Kogut-Susskind SU(3) Hamil-

tonian

12.1 The non-abelian Gauss law and the 513-dimensional vertex space

Triality is the abelian center; the non-abelian question is whether SU(3) link variables admit a

consistent gauge-invariant Hamiltonian in the Kogut-Susskind formulation [4] of Wilson’s lattice

gauge theory [3]. Place a fundamental on each bond, oriented so a vertex sees 3 on outgoing and

on incoming links. A twelve-link vertex with six of each orientation has gauge-invariant (physical,

uncharged) states given by the singlets in 3

⊗6

⊗

⊗6

. Decomposing 3

⊗6

into irreps by iterated

fusion and summing the squared multiplicities (equivalently, integrating |χ

over SU(3)) gives a

singlet multiplicity of

dim



singlets in 3

⊗6

⊗

⊗6



= 513. (17)

The local gauge-invariant Hilbert space at a vertex is 513-dimensional. It is nonzero precisely

because the net triality is 6 − 6 ≡ 0 mod 3; an unbalanced vertex such as 3

⊗5

(triality 2) has no

singlet, and gauge invariance is forbidden. The solvability of the non-abelian Gauss law is thus

controlled, at the center, by the same triality condition (16) that the abelian code enforces: the

qutrit code is the center-reduction of the non-abelian Gauss law.

12.2 The magnetic term is gauge-invariant

A Kogut-Susskind Hamiltonian adds the magnetic (plaquette) operator Re Tr(U

) on each

triangular plaquette, and consistency requires it to commute with the vertex Gauss-law generators

. Because a triangular plaquette is a closed oriented loop on which each of its three vertices

touches exactly two links, an inﬁnitesimal gauge rotation g

at a vertex acts on the two incident

plaquette links by left- and right-multiplication and cancels inside the trace. Explicitly, applying

the eight SU(3) generators as inﬁnitesimal vertex rotations to a random plaquette holonomy gives

|d Tr(W )| ∼ 10

−16

, zero to machine precision. A gauge-invariant Kogut-Susskind SU(3) Hamilto-

nian is therefore constructible on the triangular FCC plaquettes: the Hamiltonian construction is

well-posed (a consistent operator on the gauge-invariant space, not a derivation of QCD dynamics).

13 Internal sector VII: the dynamical connection—spatial obstruc-

tion

13.1 The migration channel realizes the Weyl group

A well-posed Hamiltonian is kinematics; a dynamical gauge structure needs a transporter U

between neighboring color ﬁbers, built from the vacuum’s processes. The elementary migration

channel, the orientation-ﬂipping hop, identiﬁes three cage vertices and swaps one, inducing a per-

mutation of the four valence-bond labels and hence a permutation of the three matchings. Enu-

merating the 24 bond permutations realizes exactly the six color permutations, the Weyl group

. The hop transports color by relabeling it—a Weyl element—not by rotating it through the

continuous group.

13.2 Spatial channels reach only N(T )

Combining the Weyl permutations with the Cartan torus T

of Section 8, the reachable transporters

are exactly the monomial matrices (one nonzero entry per row and column), which form the

normalizer of the torus,

⟨T, S

⟩ = N(T ) = T ⋊ S

⊂ SU(3). (18)

A scan of 2×10

random torus–Weyl products conﬁrms every one is monomial. The links are N(T )-

valued: a genuine real-space color connection, with U

7→ G

−1

under N (T) frame changes

and gauge-invariant closed-loop holonomy Tr U

. It is not full SU(3): the monomial matrices

are a measure-zero, two-dimensional (plus six-sheet) subset of the eight-dimensional group, and a

continuous root-direction element e

itλ

is not monomial.

13.3 The other spatial channels are color-diagonal

Neither of the remaining spatial channels supplies the missing oﬀ-diagonal direction. Single-bond

error correction: the recoveries are single-bond Pauli operators, and projected onto the qutrit they

are diagonal—a phase correction Z

acts as the identity when b is not in color i, and maps |+

⟩ to

its dark partner (projected away) when it is; generating the algebra of all single-bond recoveries

yields only diagonal maps (maximum oﬀ-diagonal element zero). This is examined in full, on the

real code, in Section 14, where the conjecture that the correction cycle promotes the transporter to

full rank is stated and tested. The anchor: H

anchor

= I

⊗σ

is oﬀ-diagonal in the color/dark axis,

but the induced color-to-color map obtained by leaking to the dark sector and recovering back,

anchor

)(P

anchor

), is the identity—oﬀ-diagonal color content zero. The common cause

is bond-disjointness: every operator built from the spatial cage bonds touches at most one color

and is therefore color-diagonal on the qutrit; the oﬀ-diagonal root generators E

cannot be built

from bond operators at all (supplying them through enlarged link Hilbert spaces is the route taken

by quantum-link models [8]). The obstruction is genuine, but it constrains operators built from the

spacelike bonds and says nothing about the temporal direction.

14 Internal sector VIII: QEC-dressed logical transport is tested

and remains diagonal

The spatial obstruction reaches only N(T ) for bare transport. A natural conjecture is that the

obstruction is an artifact of bareness—that the physical transport is not bare stitching but stitch-

ing dressed by the stabilizer correction cycle, and that the syndrome-conditioned recovery could

promote the rank-deﬁcient bare transporter to a full-rank one carrying the root generators. We

state the conjecture precisely and test it on the real code; it fails.

14.1 The dressed transporter

Let H

C,x

= span{| +

(x)⟩, | +

(x)⟩} be the protected color qutrit at defect site x,

with projector P

C,x

. Bare cage-to-cage transport is M

bare

= P

C,x

C,y

, with S

the physical

stitching/migration map; by Section 13 it is monomial. The QEC-dressed transporter inserts the

recovery R

that the code applies after detecting the stitching disturbance,

QEC

= P

C,x

C,y

, (19)

and the conjecture is that M

QEC

can be full rank, its determinant-one polar unitary U

polar(M

QEC

)/ det(···)

1/3

then serving as a candidate SU(3) link.

14.2 The test on the real code

The recovery R

is not free: it is ﬁxed by the code as the minimum-weight operator carrying the

leakage syndrome. We compute it on the actual [[192, 130, 3]] code. A color/dark leakage is the

relative-sign ﬂip within a matching, a single-bond Z on one cage edge; its real syndrome is the two

octahedral voids bordering that edge (weight two). Exhaustive search over all Z-operators returns a

unique minimum-weight recovery of weight one—the same single bond—touching exactly one color.

There is no competing lower- or equal-weight recovery, and no cross-color syndrome degeneracy:

all six cage edges have distinct syndromes (the distance-three property), so the decoder can never

be steered to a wrong-color recovery. When a leakage hits two colors at once, the minimum-

weight recovery factors into two independent single-bond ﬁxes, one per color, whose product is

block-diagonal in color. Consequently

QEC

∈ N(T ), (20)

not full SU(3): R

is color-diagonal, so P

C,x

adds only a diagonal phase to the already-monomial

bare

, and the polar unitary remains monomial. The search did ﬁnd weight-three syndrome-free

cage operators spanning all three colors (one bond from each), but these are products of three

commuting single-bond Z’s—simultaneous diagonal phases, i.e. Cartan/torus elements already in

T , not oﬀ-diagonal root generators. The correction cycle does not remove the bond-disjointness

obstruction; it reproduces it.

14.3 Consequence

The QEC recovery cycle stabilizes color—it is exactly the weight-one correctable structure of Sec-

tion 9—but it does not supply the oﬀ-diagonal direction. This is a tested-negative result, and it

sharpens the picture: the search space of code-internal spatial operations has been checked, bare

and dressed alike, and all of it lands in N (T). The root generators must come from elsewhere. The

remaining channel, the temporal one, is the subject of the next section.

15 Internal sector IX: temporal resolution—the root direction

from cage evolution

15.1 The temporal link is color-mixing

A four-dimensional Wilson formulation carries, in addition to the spatial links, temporal links

connecting a defect to itself at the next time step, and a temporal link is not a bond operator: it

is the time-evolution transporter of the color qutrit,

= exp



− iH

cage





. (21)

Its generator is the cage Hamiltonian projected onto the color sector, which by the factorization (9)

(with J

acting as +2 on the symmetric sector, absorbed into Γ) is

cage



= J

− I





0 1 1

1 0 1

1 1 0





, (22)

oﬀ-diagonal in the color index. The temporal link is therefore color-mixing—non-monomial for

any dt = 0 (oﬀ-diagonal weight 0.29 at dt = 0.3, 0.665 at dt = 1.0), connecting color i to color j

directly. This is exactly the operation the spatial channels could not perform, and it is available

here because temporal transport is generated by the cage kinetic term, which mixes the matchings,

rather than by a single-bond operator, which cannot. Bond-disjointness constrains the spacelike

bonds; it does not constrain time evolution.

15.2 The temporal generator closes su(3)

A single oﬀ-diagonal generator suﬃces. The operator J

− I

is the S

-symmetric combination of

the three root directions, and together with the two Cartan generators λ

, λ

supplied by the bond

phases it closes under commutation to the full algebra. Computing the Lie closure,

⟨λ

, λ

⟩ = 2, ⟨λ

, λ

, J

− I

⟩

Lie

= 8 = dim su(3), (23)

(a full Gell-Mann basis returns 8 by the same routine, conﬁrming the count). The commutators

[λ

, J

−I

] and [λ

, J

−I

] generate the remaining root directions that no spatial channel reached.

The four-dimensional construction therefore attains the entire color algebra, and the oﬀ-diagonal

generator is not introduced by hand: it is the cage’s own temporal evolution, the same operator

− I

) whose symmetric sector deﬁned the qutrit. Table 3 collects the logical color action of

every channel.

Channel Logical action on the qutrit Reached

Spatial migration color permutation S

(Weyl)

Bond phases (Cartan) diagonal phase T (torus)

Single-bond recovery diagonal T (torus)

Anchor (round trip) identity {1}

Spatial combined monomial N(T ) = T ⋊ S

Temporal link oﬀ-diagonal J

− I

root direction

Temporal + torus Lie closure full su(3)

Table 3: Logical color action of each vacuum channel. The spatial channels are color-diagonal and

combine to N(T ); the temporal link is oﬀ-diagonal and closes su(3) with the torus.

16 Internal sector X: dynamical footholds

With the algebra and the well-posed Hamiltonian in hand, two bounded dynamical results follow

on the triangular plaquettes; both are leading-order strong-coupling statements, not continuum

results.

16.1 Strong-coupling conﬁnement

The leading strong-coupling term of a Wilson loop [3] tiles the minimal surface the loop bounds

with plaquettes, each contributing one power of the fundamental character coeﬃcient; the character-

expansion technique and the resulting area law are standard [10]. A side-L triangular Wilson loop

in an FCC {111} plane (a triangular lattice of FCC bonds) bounds a minimal surface of exactly

triangles, with perimeter 3L bonds, so

⟨W ⟩ ∼ f

min

= f

, (24)

which decays with the enclosed area, not the perimeter: an area law, hence conﬁnement. The

per-plaquette factor is f = c

/N, with c

(β) the SU(3) fundamental character coeﬃcient and the

extra 1/N from the link integrations in the tiling. Computing c

(β) by Haar Monte Carlo gives

(β) → β/6 at small β (not β/18, which is the per-plaquette factor f after the link integration),

so the string tension is

σa

= −ln f → ln

= ln

, β → 0, (25)

a ﬁnite, positive tension per triangle (Figure 6). The leading strong-coupling expansion conﬁnes

on the triangular geometry with no obstruction. It is a small-β statement: by β ∼ 6, the regime

that would correspond to coarse-grained continuum QCD, the per-plaquette factor exceeds unity

and the expansion breaks down, so the weak-coupling/coarse-grained regime is not accessible this

way.

0 1 2 3 4 5 6 7 8

string tension ¾a

expansion

breaks down

small-¯: ln(18=¯)

¾a

= ¡ln(c

=N) (Haar MC)

Figure 6: Leading strong-coupling string tension on the triangular plaquettes (Haar Monte Carlo),

with the small-β form ln(18/β). The expansion breaks down near β ∼ 6; conﬁnement here is the

strong-coupling statement only.

16.2 The single-plaquette spectrum

The smallest gauge-invariant sector is a single plaquette, whose Kogut-Susskind Hamiltonian [4]

on the gauge-invariant (class-function) sector is

H = H

− λH

, H

|r⟩ = C

(r)|r⟩, H

(ˆχ

+ ˆχ

†

), (26)

with H

the quadratic Casimir (diagonal in irreps) and H

multiplication by the fundamental

character (oﬀ-diagonal via the SU(3) fusion rules), truncated by a Casimir cutoﬀ. The spectrum is

real (Hermitian H

) and converges fast: the gap stabilizes by cutoﬀ C

≤ 10 (dimension 15). In

the strong-coupling limit λ → 0 the ground state is the trivial irrep (C

= 0, the conﬁned vacuum)

and the gap is exactly

∆ = C

(3) =

, (27)

the single-plaquette fundamental Casimir gap, with no tuning; as λ grows the ground state spreads

over higher irreps and the levels reorganize in the known crossover (Figure 7). The full H

+ H

is thus a sensible operator with this gap. This is the universal single-plaquette problem—at one

plaquette the holonomy is a single SU(3) element regardless of plaquette shape—so it conﬁrms the

Hamiltonian but does not by itself probe triangular-speciﬁc physics or continuum conﬁnement, and

the value 4/3 is the fundamental Casimir, not a derivation of the physical QCD mass gap.

0 1 2 3 4 5 6 7 8

magnetic coupling ¸ (weaker gauge coupling ! )

−14

−12

−10

−8

−6

−4

−2

energy

gap = C

(3) =

Figure 7: Single-plaquette SU(3) Kogut-Susskind spectrum versus magnetic coupling λ. The strong-

coupling gap is C

(3) = 4/3; the ground state spreads over higher irreps as λ grows.

17 Uniﬁcation: one code complex, two sectors

The two sectors are protected by one FCC code complex through one mechanism. The external

charge of Eq. (5) is the vertex Gauss-law ﬂux; the internal triality of Eq. (16) is the Z

reﬁnement

of the same vertex condition; and the 513-dimensional non-abelian Gauss law has that triality as

its center-reduction. The binary distance-three code and the Z

triality code are not literally the

same stabilizer code—the octahedral void operators do not survive the Z

lift (Section 11)—but

they are two realizations on the same FCC chain complex, the binary one carrying the external

structure and the Z

one the internal triality. The chain is a single Gauss law reﬁned in three

steps,

Z (integer charge) −→ Z

(triality) −→ SU(3) (color), (28)

each the next reﬁnement of the vertex condition, all on the same complex. Both sectors share

the distance-three protection: a single-bond disturbance is the correctable error that protects the

migrating charge (external) and the color/dark leakage (internal) alike. And the octahedral baryon

operator, which carries the external sector at the Z

level, is subsumed as a boundary by the

triangular triality code that carries the internal sector (Section 11). One code complex, one error-

correcting mechanism, one Gauss law reﬁned from Z to Z

to SU(3): charge and color are two

readings of a single protected structure.

18 Scope and status

The results reached are deﬁnite and should be stated with their boundaries, and the boundaries

are sharp enough to list in three classes.

Reached. In the external sector: the winding charge is a Gauss-law super-selection invariant giving

the baryon ladder; the orientation is explicitly not protected; the migrating defect obeys E = γE

rest

;

migration is code-corrected transport. In the internal sector: the color qutrit, the S

Weyl group,

the rank-two torus, and the selection of A

∼

su(3) are derived; the qutrit is code-protected

(conditionally on a weak anchor or fast correction); the FCC complex supports a genuine Z

triality code and a well-posed SU(3) Hamiltonian; the bare spatial connection is N (T )-valued by

bond-disjointness; and the temporal cage evolution supplies the oﬀ-diagonal root generators and

closes the full su(3). Leading strong-coupling conﬁnement holds on the triangular plaquettes, with

single-plaquette fundamental Casimir gap C

(3) = 4/3.

Tested and negative. The QEC correction cycle was tested as a possible source of the oﬀ-diagonal

SU(3) direction, M

QEC

= P

C,x

C,y

. On the real code the minimum-weight recovery R

single-bond and same-color, hence color-diagonal, so M

QEC

∈ N(T ) and the dressed transporter

does not generate root directions (Section 14). The QEC cycle stabilizes color but does not remove

the spatial obstruction; the root direction comes from the temporal link, not from recovery.

Not reached. The continuum dynamics. That the algebra closes and the Hamiltonian is well-

posed does not establish continuum conﬁnement, asymptotic freedom, or the value of Λ

QCD

; the

strong-coupling area law and the single-plaquette gap are footholds, and the expansion delivering

them breaks down at the couplings corresponding to the coarse-grained regime. The correction-

cycle generator L

corr

, the loop stiﬀness κ

, and the eﬀective coupling g

eﬀ

are not derived from the

microscopic vacuum dynamics. The E = γE

rest

relation is exact for the one-dimensional soliton and

inherited by the three-dimensional defect by covariance, not proven directly in three dimensions.

The closure of su(3) is at the level of the color algebra; whether the temporal-link construction

is fully consistent with the baryon and triality protection of the spatial code, and whether the

four-dimensional theory reproduces QCD under coarse-graining over the physical FCC vacuum,

are open.

The central claim, stated with these boundaries, is therefore the following: one FCC code complex

protects baryon charge and color, the geometry selects the qutrit and triality, the bare spatial

transport is obstructed to N(T ), the QEC recovery cycle does not remove that obstruction, and

the temporal cage Hamiltonian supplies the missing root directions and closes su(3). The full

color algebra appears only when temporal qutrit evolution is included. The boundary this paper

draws is between that structure—two protected sectors, a closed color algebra, and a well-posed

kinematics—which is reached, and the conﬁning continuum dynamics, which are not.

Data availability

The Python scripts reproducing every result are available as a single archive, ssm strong scripts.zip,

downloadable at https://github.com/raghu91302/ssmtheory/raw/main/ssm _strong_script

s.zip. Requiring NumPy (some also SciPy/Matplotlib), they cover: the orientation cost and code

build of Sections 2–3; the baryon spectrum and kinematics of Sections 4–5; the migration string of

Section 6; the qutrit emergence, Weyl/rank-two selection, and QEC stabilization of Sections 7–9;

the gauge-ready geometry, Z

triality code, and SU(3) Gauss law and plaquette of Sections 10–

12; the spatial N (T ) obstruction, the tested-negative QEC-dressed recovery (the minimum-weight

recovery computation of Section 14), and the temporal su(3) closure of Sections 13–15; and the

strong-coupling and single-plaquette computations of Section 16. No other data were generated or

analyzed in this study.

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