A Lattice Realization of Fermionic Matter: Topological Spin of FCC/D4 Vacuum-Code Defects and its Place in the Triality-Spin(10) Program

A lattice realization of fermionic matter: topological spin of

FCC/D4 vacuum-code defects and its place in the

triality–Spin(10) program

R. Kulkarni

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

Abstract

The Selection–Stitch Model represents the physical vacuum as a Calderbank–Shor–Steane (CSS)

stabilizer code on the face-centered-cubic (FCC) lattice, and in four dimensions the D4 lattice, with

matter as a localized defect. Combinatorial invariants of these defects reproduce light mass ratios,

fractional electric charge, three color charges, and three generations. We ask a sharper question: are

the matter defects fermions? Our result is that a matter defect is a fermion, forced by the homology

of its bond complex. The complex has H

= 3 and H

= 1: the defect carries an odd electric parity

sector supported by the H

cycles and an odd magnetic parity sector supported by the enclosing H

surface, and the charge-carrying string links the enclosing Gauss membrane exactly once mod 2,

placing the defect in the fermionic e × m sector with θ = (−1)

= −1 in the charge–ﬂux linking

sense appropriate to a three/four-dimensional code. The computation is carried out on the FCC code

veriﬁed to be a valid CSS code, and H

= 3 is simultaneously the color multiplicity, so color charge

and fermionic statistics are two readings of the same homology. We place this within the established

program—going back to Ramond and developed recently by Boyle, Furey, Gresnigt, Lisi, Todorov,

and others—in which so(8) triality organizes three fermion generations and the Spin(10) spinor

is one generation. To this program the present paper contributes a lattice-code realization of the

fermionic statistics, not an origin of the triality structure. As supporting structure we show that the

FCC lattice cannot host a local internal SU(2)

(a representation-invariant obstruction), while the

four-dimensional D4 lift supplies the internal pseudoreal SU(2) the triality program uses. We are

explicit about scope: θ = −1 establishes fermionic, not chiral, topological order. Identifying the

defect with chiral electroweak structure would require evading the chiral-central-charge obstruction

for commuting-projector codes; the construction lies outside that obstruction’s two-dimensional lo-

cal hypotheses—it is three/four-dimensional, where chiral fermionic stabilizer codes are known to

exist, and the matter defect’s logical operator is nonlocal (on FCC a ﬁve-edge, two-region object;

on D4 a four-edge object giving the Spin(8) spinors 8

⊕ 8

that match the bulk so(8) lattice)—but

whether it realizes genuinely chiral (c

−

= 0) order is left open. No parameters are ﬁtted.

1 Introduction

The Selection–Stitch Model (SSM) treats baryonic matter as a defect in a crystalline vacuum whose

ground state is the FCC lattice, and in the model’s relativistic completion the four-dimensional root

lattice D4 [1, 2]. The vacuum is a CSS stabilizer code on this lattice, and a particle is a localized

violation of the local coordination the code protects [2, 3]. From this premise the framework recovers

several dimensionless properties of matter without adjustable parameters: the proton-to-electron mass

ratio as a structural disruption count, the quark charges from tetrahedral geometry and integer winding,

three color charges from the skew-edge pairs of the bounding tetrahedron, and three fermion generations

from the order-three triality of the D4 cell. Each is a combinatorial invariant—a count of edges, pairs,

windings, or symmetry orbits.

This paper isolates one question that is not combinatorial: are the matter defects fermions? Fermionic

statistics is a topological property of an excitation in a code, and asking whether a given defect pos-

sesses it is distinct from asking what charge or winding it carries. Our main result (Section 6) is that the

FCC/D4 matter defect is a fermion, and that this is forced by the homology of its bond complex rather

than assumed. The homological derivation and its identiﬁcation H

= 3 = color multiplicity is, to our

knowledge, new; the surrounding structure is not, and we are careful to separate the two.

The idea that so(8) triality underlies three fermion generations, and that the Spin(10) spinor 16 is a

single generation [14], is an active research program with a long history. It was proposed by Ramond [4]

and has been developed through the exceptional Jordan algebra and division-algebra constructions by

Boyle [5], Furey and collaborators, Gresnigt and co-authors [8], and others; it appears in Silagadze’s [6]

SO(8)-color proposal and in Lisi’s E

[7] construction. That program works at the level of representa-

tion theory. Separately, lattice stabilizer codes realize topological order, homological charges, and defect

excitations concretely—the toric and surface codes, the three-colorable topological color code [18], and

the broader family of lattice CSS codes [17, 19]—and emergent-matter programs realize gauge ﬁelds and

fermions on physical substrates [20, 21]. Neither the algebraic generation program nor the lattice-code

program has, to our knowledge, been joined at the point this paper occupies.

Our contribution is threefold, and we are careful to claim only what is new. First, we give a homolog-

ical derivation of fermionic statistics for a lattice defect: the topological spin θ = (−1)

is ﬁxed by the

homology H

, H

of the defect’s bond complex, so the defect is a fermion as a consequence of geometry

rather than assignment. Second, we realize this on the FCC/D4 vacuum stabilizer code speciﬁcally—this

lattice, this tetrahedral-void defect, on a code we verify to be valid CSS—where the computation gives

= 3, H

= 1 and hence θ = −1. Third, on this defect the electric homology H

= 3 coincides

with the color multiplicity, so color charge and fermionic statistics are the same topological datum; and

the minimal logical operator completing the defect closes on four edges on the D4 code, yielding the

Spin(8) spinors 8

⊕ 8

that the so(8) triality program uses for a fermion, matching the bulk lattice with

no larger group forced. We claim the realization and the homological mechanism; we do not claim the

D4↔ so(8) correspondence, the triality organization of generations, or homological charges in codes,

all of which are established. The role of this paper is to supply a concrete lattice substrate on which the

algebraic program’s matter content acquires fermionic statistics as a geometric fact.

Sections 3–5 assemble the supporting structure: an FCC no-go for local internal SU(2)

, the

D4/so(8) representation content of the triality program, and the topological origin of spin one-half.

Section 6 is the main result. Section 7 addresses, honestly, the chiral-central-charge obstruction and

delimits what is and is not established.

2 The combinatorial baseline

We recall, without rederiving, the quantities the framework computes, because their common type

frames the paper. The proton mass follows from the disruption count of the tetrahedral defect, (K+1)K

−

skew

K = 13 · 144 − 3 · 12 = 1836, with K = 12. The down-type charge −

is the cosine of the

regular-tetrahedron bond angle; the up-type charge +

is the singly-wound state under the bulk Bravais

translation group. Three colors are the three skew-edge pairs of the bounding K

; three generations

are the three inscribed 16-cells permuted by triality (the decomposition of the D4 root system into three

16-cells is standard [11, 12]; their identiﬁcation with generations is the framework’s, in line with the

triality program of Section 4). Every one of these is an integer read off a graph. The present paper asks

a question of a different type—the statistics of the defect—and answers it by the same combinatorial

means.

3 The FCC no-go for a local internal SU (2)

3.1 The residual defect doublet is real

Anchor selection breaks the local permutation symmetry S

of a tetrahedral defect’s four bonds to S

The zero-mode content of the electron defect, the star graph K

1,4

on the bipartite diamond sublattice,

is ﬁxed by the bipartite zero-mode theorem: the sublattice imbalance |n

− n

| = |4 − 1| = 3 yields

three protected chiral zero modes, decomposing under S

as 3 = 2 ⊕ 1, the multiplet count of a

Standard-Model generation. The reality type, however, is wrong: the Frobenius–Schur indicator ν(R) =

|G|

) of the S

standard representation is +1, and all S

irreducibles are real, whereas the weak

doublet is the pseudoreal SU(2)

fundamental (ν = −1). A real representation cannot be identiﬁed with

a pseudoreal one.

3.2 The rotation doublet is spatial

The alternative candidate, the su(2) of the six square faces of the cuboctahedral shell, is pseudoreal but

is a spatial rotation of the ⟨100⟩ lattice axes; weak isospin is internal, and identifying a spatial with

an internal symmetry is forbidden by Coleman–Mandula [10]. No Kaluza–Klein escape exists in three

dimensions.

Proposition 1. On the three-dimensional FCC lattice a local internal SU(2)

is not realizable: the

available two-dimensional structures are real (permutation symmetry) or spatial (square-face rotation),

whereas weak isospin is pseudoreal and internal.

The obstruction is a representation invariant. It is the lattice statement of a general fact—that internal

chiral gauge structure is not freely available on a low-dimensional local code—and it motivates the four-

dimensional lift.

4 The D4/so(8) structure and the triality program

FCC is the root lattice A

of su(4)

∼

so(6) and is three-dimensional; D4 is the root lattice of so(8) and

is four-dimensional [11]. All D4 facts used in this paper—the 24 roots, the 16-cell deep holes, and the

triality relating the three eight-dimensional representations—are standard [11, 12]. The extra dimension

supplies the structure the no-go lacked, and it is the structure the triality program of Refs. [4, 5, 6, 7, 8, 9]

already uses.

4.1 An internal pseudoreal SU (2)

so(8) contains so(4)⊕so(4) = su(2)

⊕4

. We adopt as a stated premise the framework’s Signature Selec-

tion postulate: one so(4) factor is identiﬁed with spacetime and broken to the Lorentz algebra so(3, 1).

Its content, for the present paper, is only the choice of which so(4) is spacetime; nothing below depends

on its derivation. The second factor, su(2)

⊕ su(2)

, is orthogonal to spacetime and available as an in-

ternal symmetry—the left–right-symmetric electroweak structure of Pati–Salam type [13], which is the

structure Ref. [5] identiﬁes from the exceptional Jordan algebra. Its SU(2) fundamental is pseudoreal

(ν = −1; the quaternion subgroup Q

⊂ SU(2) conﬁrms this).

4.2 The spinor as the chiral electroweak fermion

so(8) has three real eight-dimensional representations related by triality: the vector 8

and the spinors

, 8

. Branching under (spacetime × internal) so(4) [26],

→ (2, 2; 1, 1) ⊕ (1, 1; 2, 2), 8

→ (2, 1; 2, 1) ⊕ (1, 2; 1, 2). (1)

matter = defect in the

FCC / D4 vacuum stabilizer code

FCC (

): residual doublet

is REAL & SPATIAL

D4 root lattice of

(8)

internal pseudoreal

(2)

Coleman--Mandula

spinor

→

(2)

doublet

= chiral electroweak fermion

spin-

from tethered defect

(Finkelstein--Rubinstein, isotropic)

edges

→

Spin(2

)

;

D4:

= 4

→

Spin(8)

⊕

defect bond complex homology:

= 3

(electric = color)

= 1

(magnetic)

on the verified CSS code

= (

−

: FERMION

⇒

spinor

⊂

fermionic statistics as a homological fact; chiral order left open

Figure 1: The assembled structure. FCC admits only a real, spatial doublet and is excluded by Coleman–

Mandula; the D4 lattice, root lattice of so(8), supplies an internal pseudoreal SU(2), and the spinor 8

is the chiral electroweak fermion of the triality program [4, 5]. Spin one-half follows from the tethered-

defect topology; the edge count ﬁxes the emergent group, and the defect’s electric and magnetic charges

are the homology H

= 3, H

= 1 of its bond complex (Section 6), forcing θ = −1 (fermionic

statistics; the chiral question is separate, Section 7).

The spinor 8

is the chiral electroweak fermion—a left-handed Weyl component in an internal SU(2)

doublet, correlated with spacetime chirality—while the vector 8

is not. This is the branching used

throughout the triality program [5, 8]; we reproduce it to ﬁx notation and to identify which representation

a fermionic defect must occupy. Figure 1 summarizes the assembled structure.

5 Spin from defect topology

Integer spin is automatic on a lattice: a vector defect is spin one, a symmetric traceless tensor de-

fect spin two. Half-integer spin is not automatic—the sign −1 under 2π is the nontrivial class of

(SO(3)) = Z

, invariant under deformation—but it is available. A defect tethered to the bulk has

orientation conﬁguration space SO(3), and by the Finkelstein–Rubinstein mechanism [15], which quan-

tizes solitons such as skyrmions as fermions, the defect can be quantized in the double-valued sector,

acquiring isotropic spin one-half: exp(i 2π σ ·ˆn) = −⊮ for every ˆn, with the su(2) algebra closing. No

fundamental Dirac ﬁeld is introduced. The relation to the next section should be stated precisely: the

Finkelstein–Rubinstein mechanism shows that a double-valued spin-one-half quantization is available

to a tethered defect, but by itself it does not decide whether a given defect occupies that sector. Section 6

supplies the stronger, code-theoretic statement: the speciﬁc FCC/D4 defect actually lies in the fermionic

e × m sector. Availability comes from the conﬁguration space; the sector is ﬁxed by the homology.

6 Matter defects are fermions by homology

6.1 Topological spin from code charges

In a CSS code the two charge sectors of an excitation are its electric charge p (a Z-stabilizer violation,

made by an X-string) and its magnetic charge q (an X-stabilizer violation, made by a Z-operator).

In the two-dimensional anyon setting the elementary charge and ﬂux are bosons and mutual semions,

= −1, so for a = e

= (−1)

, (2)

the standard toric-code fact ε = e×m [17]. Our defect, however, lives in a three-dimensional (FCC) and

four-dimensional (D4) code, where excitation types and braiding have a different categorical structure,

so Eq. (2) cannot simply be imported. The correct higher-dimensional statement is the charge–monopole

mechanism [16]: a pointlike composite carrying both an electric charge and an enclosed magnetic ﬂux

acquires its statistics from the mod-2 linking of the charge-carrying string with the enclosing ﬂux surface.

We state this for the present code as a lemma, each clause of which is veriﬁed on the explicit FCC

construction.

Lemma 1. Let the trapped node be enclosed by a surface S crossing exactly its defect bonds. (i) The

membrane operator M

, deﬁned as Z on the edges crossing S, equals the product of the vertex Z-checks

enclosed by S (interior edges cancel pairwise); it is the Gauss-law operator measuring the defect’s

enclosed charge, and for the tetrahedral defect it is Z

⊗4

on the four defect bonds. (ii) Any X-string

carrying the defect’s electric charge into the bulk terminates on the trapped node and therefore contains

an odd number of its bonds; its overlap with M

is odd, giving exactly one mod-2 anticommutation —

the linking of the charge worldline with the ﬂux surface. (iii) By contrast, every closed 1-cycle has even

degree at the node and commutes with M

: the sign is invisible to the local operator algebra and is

irreducibly a linking statement. Consequently a defect carrying odd electric and odd magnetic parity

has topological spin θ = (−1)

= −1 in the sense of charge–ﬂux linking, which is the appropriate

reading of Eq. (2) in three and four dimensions.

Clause (iii) matters: the same computation that shows the single void hosts no local logical qubit

(Section 6) shows the local X- and Z-operators on the void all commute, so any claim that the fermionic

sign arises from a local anticommutation would be false. The sign lives in the linking of the extended

string with the enclosing membrane, exactly as for charge–monopole composites [16].

6.2 The charges are the defect’s homology

Taking each bond to be a stabilized pair and the defect to extend the code—its bonds contribute new

stabilizers—the two charges are not assigned but are the homology of the defect’s bond complex. The

trapped tetrahedral defect has a central node, four defect bonds, the six edges of the bounding tetrahe-

dron, and its four triangular faces as ﬁlled 2-cells. Over GF(2),

= (E − rank ∂

) − rank ∂

= (10 − 4) − 3 = 3, H

= F − rank ∂

= 4 − 3 = 1. (3)

The three classes of H

are the closed bond-loops bounding no face; they coincide with the skew-edge

pairs and hence with the three color charges. The class of H

is the tetrahedron’s enclosing surface

(Fig. 2).

In a topological CSS code the Z-checks are vertex coboundaries and the X-checks are void bound-

aries, so electric charge is measured by H

and magnetic by H

. That this dictionary is the code’s actual

structure follows from CSS validity: building the FCC code from its stated rules on a ﬁnite patch—

qubits on nearest-neighbor edges, Z-checks on vertices, X-checks on octahedral voids—yields a valid

trapped node

(a) defect bond complex

(b)

= 3

(electric/color),

= 1

(magnetic)

defect bonds tetrahedron edges

cycles (

)

surface

Figure 2: The trapped-node defect. (a) The bond complex. (b) Its two homology sectors: three one-

cycles, H

= 3, coinciding with the skew-edge pairs and the three color charges (electric), and one

enclosing surface, H

= 1 (magnetic). Both nonzero force θ = −1.

CSS code (every Z–X pair commutes, bulk weight 12 in both sectors), so the identiﬁcation of H

, H

with the electric and magnetic charges is forced, not posited. Both nonzero gives, by (2),

θ = (−1)

= −1. (4)

Proposition 2. In the FCC/D4 CSS code, the trapped-node defect supports an odd electric parity sector,

represented by the H

cycles of its bond complex, and an odd magnetic parity sector, represented by

the enclosing H

surface. By Lemma 1, the charge-carrying string and the enclosing Gauss membrane

have a single mod-2 linking, so the defect lies in the fermionic e × m sector with θ = −1, in the charge–

ﬂux linking sense appropriate to the three- and four-dimensional code. No Pauli operator is assigned

by hand, and the rank dim H

= 3 simultaneously gives the color multiplicity, so color charge and

fermionic statistics are two readings of the same homology.

6.3 The minimal logical qubit and the emergent group

The size of the minimal logical qubit that completes the defect depends on the lattice, and the dependence

is instructive. On the three-dimensional FCC code the single tetrahedral void hosts no complete logical

qubit—its six local edges carry three X-type and three Z-type operators commuting with the stabilizers

but with no anticommuting pair, so no qubit closes locally—and the minimal complete qubit ﬁrst appears

on ﬁve edges, its logical Z internal and its logical X reaching to one neighbor. This ﬁve-edge value is a

feature of the cuboctahedral void’s local structure, not a signal of a larger algebra.

On the four-dimensional D4 code—the lattice the electroweak structure of Section 4 actually re-

quires, since D4 is the root lattice of so(8)—the count is different and simpler. The deep-hole void

hosts its logical qubit, and the minimal complete qubit closes on four edges. The four-edge D4 closure

is obtained by the same GF(2) stabilizer reduction used for the FCC ﬁve-edge result, on the D4 code

built from its 24-neighbor root system with 16-cell deep-hole X-checks; the accompanying script con-

structs the code, veriﬁes CSS validity, and reports the four-edge anticommuting pair and its support. By

the Clifford correspondence an object on N edges carries Cl(2N), whose Fock space is the spinor of

Spin(2N); four edges give Spin(8), whose spinors are 8

⊕8

. The minimal D4 defect therefore carries

exactly the Spin(8) spinor content that the bulk lattice supports—no larger group is forced, and none

needs to be broken. This is the representation the triality program of Refs. [4, 5, 6, 8] uses for a fermion

generation: the lattice organically populates the so(8) spinors rather than requiring a grand-uniﬁed group

above them.

We note two honest limits. First, the correspondence gives 8

⊕ 8

, both spinor chiralities on equal

footing; which is the left-handed weakly-coupled fermion, and why only that one couples, is the chiral

selection question left open in Section 7, and it is not resolved by the branching. Second, the D4 code

is high-rate: the logical structure does not saturate at the minimal qubit but grows with region size, so

“Spin(8)” is the group of the minimal logical qubit, not a unique feature of the geometry.

7 Fermionic versus chiral order: the obstruction and the open question

The result of Section 6 is that the defect is a fermion. This is weaker than, and must be distinguished

from, the claim that the construction realizes chiral electroweak structure, and the distinction is where

the relevant no-go theorems act.

A commuting-projector (stabilizer) code in two dimensions cannot realize chiral topological order

with nonzero chiral central charge: the c

−

extracted from the anyon data via the Gauss sum e

2πic

−

|A|

−1/2

θ(a) is obstructed for such models [17, 23]. Fermionic topological order—an excitation

with θ = −1—is a distinct and weaker property, and it is realized by stabilizer codes; recent three-

dimensional stabilizer constructions explicitly realize fermionic and chiral topological orders [22]. Our

θ = −1 establishes the fermionic property. It does not by itself establish c

−

= 0, and the identiﬁcation

with chiral electroweak structure requires that additional step.

Three features place the construction outside the hypotheses of the two-dimensional obstruction,

and we state them as the reasons the question is open rather than closed. First, the code is three/four-

dimensional, not two-dimensional, and chiral fermionic stabilizer codes are known to exist in three

dimensions [22]; the known examples show the obstruction is not automatically fatal outside the two-

dimensional commuting-projector hypotheses, but they do not prove that the present code is chiral.

Second, the matter defect’s logical operator is nonlocal—the minimal logical operator of Section 6 (ﬁve

edges on FCC, four on D4)—and no-go results for chiral states from local circuits explicitly exempt

constructions that are nonlocal in a spatial direction [24]. Third, the object we exhibit is a defect carrying

fermionic self-statistics, not a claimed chiral phase with a computed edge central charge. Whether the

construction realizes genuinely chiral (c

−

= 0) order remains open, and settling it—by computing the

anyon data and the Gauss sum for the FCC/D4 code, or by exhibiting the chiral edge mode—is the

natural next problem. We note that the framework must ultimately confront the Nielsen–Ninomiya

theorem [25] in its appropriate code-theoretic form; the present work does not claim to have done so.

8 Discussion

The defensible content of this paper is contained and checkable. The FCC no-go for a local inter-

nal SU (2)

is a representation-invariant exclusion. The main result is that the FCC/D4 matter defect

is a fermion, forced by the homology H

= 3, H

= 1 of its bond complex on the veriﬁed CSS

code, with the striking feature that the electric homology H

= 3 is the color multiplicity—so within

this framework color and fermionic statistics are one topological datum. The supporting representation

theory—so(8) triality, the 8

chiral fermion, the Spin(10) generation—is not original to this work; it

is the program of Refs. [4, 5, 6, 7, 8, 9], to which we add a stabilizer-code realization of the fermionic

statistics.

We have been explicit about the boundary of the claim. Fermionic statistics (θ = −1) is established;

chiral order (c

−

= 0), which the electroweak identiﬁcation requires, is not, and faces a genuine ob-

struction for commuting-projector codes that the construction plausibly but not provably evades. The

homological result stands on its own regardless of that outcome. Its main contingency is the identiﬁ-

cation of the published FCC/D4 stabilizer generators with the topological CSS code used here, veriﬁed

against every stated property and closable against the exact generators by a reader with the source con-

struction.

Code availability

All computations—the Frobenius–Schur indicators, the so(8) branchings, the Finkelstein–Rubinstein

isotropy check, the topological-spin evaluator validated on the toric code, the defect-complex homology,

the single-void and minimal-string computations, and the FCC code CSS-validity check—are repro-

ducible with the scripts at https://github.com/raghu91302/ssmtheory/blob/main/

electroweak-scripts.zip. Each script names the section it reproduces.

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