An Electroweak-Shaped Structure on the FCC Vacuum

An Electroweak-Shaped Structure on the FCC Vacuum:

Static Centrosymmetry Forbids Chirality, but a Defect Worldline

Relaxes the Obstruction to PT and Violates Parity

Raghu Kulkarni

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

raghu@idrive.com

June 2026

Abstract

In the Selection-Stitch Model the FCC vacuum has already been shown to carry color: the

tetrahedral cage of a trapped defect yields a qutrit and selects SU(3) on the triangular faces of the

nearest-neighbor cuboctahedron. We ask whether the same shell carries an electroweak-shaped

group-theoretic structure, and report a layered result. The six square faces of the cuboctahedron

carry an su(2): three antipodal axis-pairs along the ⟨100⟩ directions that close as so(3) = su(2).

We are explicit that this is a group-theoretic correspondence, not yet a demonstration that these

spatial rotation axes act as internal gauge generators. With the diagonal bond-phase U(1) the

rank equals that of SU(2) ×U(1)—a necessary structural condition, not a strong identiﬁcation.

Read as interlayer objects—the square faces are the inter-sheet faces created by the out-of-plane

Lift in the crystallization picture—the structure supplies a two-dimensional up/down doublet

and an order-two grading that we treat as a candidate chirality (a model grading, not yet the

Lorentz/spinor chirality of the Standard Model), so a chiral SU(2)

is geometrically available

as a target. Whether it is realized is then governed by symmetry, and we trace the obstruction

through three stages. Statically, the chirality is forbidden: FCC is centrosymmetric (space

group F m

3m), and we prove that at any bulk node the up-Lift and down-Lift environments

are exact inversion partners, so their handedness cancels under parity (P ). On a chronological

defect worldline threading the stack, however, the up-connection lies in the future and the down-

connection in the past; spatial inversion no longer pairs them, and the cancellation relaxes from a

P relation to a P T relation. A chiral coupling cannot survive a P cancellation but is compatible

with a P T (and hence CP T ) relation—the weak interaction itself violates P while respecting

CPT —so this relaxation removes the prohibition rather than merely weakening it. That a lattice

resists chiral fermions is the content of the Nielsen–Ninomiya no-go theorem [4, 5, 6], and the

centrosymmetry obstruction found here is a concrete geometric realization of it; the worldline

relaxation is, in the same spirit as the Ginsparg–Wilson construction [7], a route by which the

strict prohibition is loosened rather than evaded outright. Beyond removing the prohibition,

the oriented Lift supplies a temporal generator that is odd under P and even under P T (the

direction of the doublet): the worldline genuinely violates parity, which the static geometry

forbade. What remains unforced is the ﬁnal step—the left-handed projection P

that turns

parity violation into a chiral gauge coupling; the oriented generator alone produces a vector

su(2), acting equally on both chiralities. The result is therefore a three-tier boundary: static

chirality is forbidden by centrosymmetry, a defect worldline relaxes the obstruction to P T and

the oriented Lift violates parity, and only the chiral projection remains unforced. Hypercharge

and anomaly cancellation are untouched. The boundary locates precisely where the geometric

derivation of the electroweak sector stands: the gauge structure and genuine parity violation are

reached on defect worldlines; the chiral projection that completes SU(2)

is not yet compelled.

1 Introduction

The Selection-Stitch Model (SSM) treats the vacuum as a face-centered-cubic (FCC) entanglement

network carrying a stabilizer code, with matter as defects in that network [1, 2]. Its strong-sector

development establishes color geometrically: a quark is a node trapped in a tetrahedral void, the

four-vertex cage gives a color qutrit through the three perfect matchings of K

, and SU(3) is selected

on the triangular faces of the nearest-neighbor shell [3]. The nearest-neighbor shell of an FCC site

is a cuboctahedron, whose fourteen faces divide into eight triangles and six squares. Color occupies

the triangles. This paper asks whether the squares carry the electroweak sector, and reports the

answer with its boundary made explicit.

The result is layered, and we state its tiers at the outset. On the positive side, the six square

faces carry an su(2), the rank equals that of SU(2) × U(1) once the diagonal U(1) is included

(a necessary condition, not a strong identiﬁcation), and—read in the layered language natural to

the crystallization picture—the structure provides a two-dimensional doublet and an order-two

grading we treat as a candidate chirality, so a chiral SU(2)

is available as a target. Whether it

is realized is decided by symmetry, in three stages. Statically the chirality is forbidden: FCC is

centrosymmetric, and the up- and down-handedness a chiral assignment would need cancel under

parity at every bulk node. On a chronological defect worldline the obstruction relaxes from parity

(P ) to P T , because the up-connection lies in the future and the down-connection in the past, so

spatial inversion no longer pairs them; since a chiral coupling is compatible with P T /CP T , this

removes the prohibition. The oriented Lift then supplies a P -odd, P T -even temporal generator, so

the worldline genuinely violates parity. What is not reached is the ﬁnal projection: the oriented

generator produces a vector su(2), and nothing yet forces the left-handed projection P

that would

make the coupling chiral. The boundary is therefore precise: static chirality forbidden, worldline

obstruction relaxed to P T , parity violated, chiral projection open.

We regard the boundary as the contribution. A framework that derives part of the Standard Model

is most useful when it says precisely where the derivation stands and what would be required to

continue. Section 2 establishes the su(2) structure; Section 3 gives the interlayer reading and the

chirality operator; Section 4 proves the static centrosymmetry obstruction; Section 5 shows its

relaxation to P T and the parity violation on a defect worldline; Section 6 records what is reached,

what is open, and what is untouched.

2 An SU(2) × U(1)-shaped structure on the square faces

The twelve nearest neighbors of an FCC site lie at the vertices of a cuboctahedron, the ⟨110⟩/

√

directions. Its surface has eight triangular and six square faces (Figure 1). The triangular faces are

the close-packed {111}-plane triangles that carry color [3]; we examine the squares.

A direct enumeration of the cuboctahedron (12 vertices, 24 edges) returns 8 triangular and 6 square

faces. The six squares point along the ±x, ±y, ±z axes and pair antipodally into three axes. The

three axes carry the rotation generators J

, J

, which close as

, J

] = J

, [J

, J

] = J

, [J

, J

] = J

, (1)

the algebra so(3)

∼

su(2). The square faces therefore carry an su(2), of rank one, in the group-

theoretic sense that their three axes furnish generators with the su(2) commutation relations; we

do not yet claim these spatial rotation axes act dynamically as internal gauge generators. Together

with the diagonal bond-phase U(1) already present in the model (the overall phase modulo the

6 square faces →

SU(2)-shaped sector (⟨100⟩)

8 triangular faces →

SU(3)-shaped sector (⟨111⟩)

Figure 1: The FCC nearest-neighbor shell is a cuboctahedron (12 vertices, 24 edges). Its fourteen

faces divide into eight triangles, the close-packed ⟨111⟩ planes that carry color SU(3) (green), and

six squares along the ⟨100⟩ axes (blue), which are the subject of this paper. Front faces are shaded;

the square and triangular faces interleave over the surface.

color-traceless directions), the total rank is two, equal to that of SU(2) × U(1). Rank equality

is a necessary condition for such an identiﬁcation, not strong evidence for it. With that caveat,

the cuboctahedron separates the candidate gauge structure by face type: color SU(3) on the eight

triangles, an SU(2) × U(1)-shaped structure on the six squares.

This is a statement about the gauge group structure carried by the geometry, not yet about the

electroweak interaction, whose deﬁning feature—chirality—is the subject of the remaining sections.

3 The interlayer reading and a chirality seed

The square faces admit a second, physically motivated reading. In the SSM crystallization pic-

ture [1] the lattice grows from a seed triangle by two operators: the in-plane Stitch, which builds

the {111} hexagonal sheets, and the rare out-of-plane Lift, which projects a new node orthogonally

from a triangular face at the tetrahedral height h =

2/3 L. The Lift is the intersection of three

unit spheres, a zero-parameter solution consisting of two points—above and below the face—one of

which is selected by orientation. The square faces of the cuboctahedron are precisely the interlayer

faces: they appear where one sheet stacks on the next, and the 6+3+3 = 12 coordination of a bulk

node decomposes into six in-plane neighbors and three in each adjacent sheet, reached by Lifts.

Read this way, the relevant object is not three ⟨100⟩ axes in three dimensions but a two-dimensional

doublet: the upper-sheet and lower-sheet directions. On this doublet,

= σ

, γ

= ⊮, eig(γ

) = {+1, −1}, (2)

is a valid order-two grading of the doublet, which we adopt as a candidate chirality operator, with

(⊮ + γ

) projecting onto the upper sheet. We are careful here: this is an order-two Z

grading, the algebraic prerequisite for a chirality, not yet the Lorentz/spinor chirality of a Standard

Model fermion, which is tied to the Dirac representation and is not established by a two-state

grading alone. What the grading provides is the structure a chiral assignment requires, and it is

exactly what the three-axis adjoint reading did not provide; promoting it to physical chirality is

part of what remains open.

There is also a physical chirality seed. The two sheets are not built by the same process: the

upper sheet is reached by the oriented Lift (the rare event that selects a sense), while the lower

sheet pre-exists and is closed by proximity bonding. The up and down directions are therefore not

interchangeable; the oriented Lift breaks the symmetry between them. A chiral SU(2)

—an su(2)

acting only on the upper-sheet (left) projection—is therefore geometrically available, and carries a

seed from the crystallization dynamics rather than being imposed by hand.

Availability is not necessity. Whether the geometry forces the chiral assignment, or merely permits

it, is decided in the next section, and the answer is set by a symmetry of the lattice.

4 The static obstruction: parity cancellation from centrosymme-

try

A chiral SU(2)

requires the up/down asymmetry to be uniform: every node must see the same

handedness, so that the left projection is globally well-deﬁned. We show this fails statically, and

fails for a structural reason—FCC centrosymmetry—which makes the static obstruction a parity

(P ) relation. That it is a P relation, and not a PT relation, is what the next section exploits.

The FCC lattice is centrosymmetric: its space group F m

3m contains inversion, and every lattice

node is a center of inversion. Consider a bulk node in a single ABC-stacked domain—layer B at

the origin, with layer C above and layer A below. The three upper-sheet (Lift) partners are the

nearest C nodes; the three lower-sheet partners are the nearest A nodes. With the close-packing

sublattice oﬀsets o

= 0, o

+ a

), o

+ a

), the oﬀsets satisfy o

+ o

= 2o

, so A

and C are inversion images through B. A direct computation conﬁrms that the down-Lift partners

are exactly the inversion of the up-Lift partners through the node,

{down partners} = {2B − u : u ∈ {up partners}}, (3)

to machine precision.

Proposition 1 (Bulk chirality cancellation). In a single-domain FCC crystal, the up-Lift and

down-Lift environments of any bulk node are related by inversion through that node. Inversion

reverses handedness; therefore the up-Lift and down-Lift carry opposite chirality, and the local

chirality cancels at every bulk node.

The consequence for the electroweak question, statically, is decisive. A single nucleation seed sets

a single global stacking sense (ABC rather than ACB), and that sense is a genuine handedness of

the crystal as a whole—ABC and ACB are mirror images. But by Proposition 1, that global sense

does not imprint a uniform per-node chirality: at each interior node the inversion symmetry pairs

the up-handedness against the down-handedness and cancels it. The crystallization handedness is

real but global, living at the boundary of the domain, not in the bulk where the interlayer su(2)

acts.

The essential feature for what follows is that this cancellation is a parity relation. The pairing

operator is spatial inversion through the node—a pure P operation, with no time component—

and it is this that forbids a static chiral assignment, because a P -cancellation is exactly what a

chiral (parity-violating) coupling cannot survive. The obstruction is not a feature of the model’s

internal bookkeeping; it is the classical centrosymmetry of FCC [9], the same property that makes

elemental FCC crystals optically inactive, and it can be checked independently of the model. The

next section shows that the relation is P only as long as the Lifts are treated statically; on a

chronological worldline it becomes P T , with which a chiral coupling is compatible.

5 Relaxation to P T and parity violation on a defect worldline

The static obstruction treats the up-Lift and down-Lift as simultaneous spatial structures. A

physical defect, however, threads the stack in time: it is at layer A in the past, B now, and C in

the future. The down-connection (to A) lies in the past and the up-connection (to C) in the future,

so the two are separated not only in space but in the time coordinate.

5.1 The pairing relaxes from P to P T

Assign each Lift step its time coordinate: the future-up step at t = +δt, the past-down step at

t = −δt. Spatial inversion P through the node acts on the spatial coordinates and leaves time

untouched. It still maps the up-step’s spatial part onto the down-step’s spatial part—but it leaves

it at t = +δt, whereas the down-step sits at t = −δt. As four-vectors, P (up) and down no longer

coincide. The operator that does pair them is the full spacetime inversion P T , which ﬂips the time

coordinate as well, sending +δt 7→ −δt. A direct computation conﬁrms that P T through the node

maps the future-up worldline step onto the past-down step, while P alone does not.

The cancellation that forbade static chirality is therefore a P relation that relaxes to a P T relation

on a chronological worldline. This distinction is decisive: a parity (P ) cancellation forbids a

chiral coupling, but a relation that pairs the partners only under the combined P T operation does

not, because a chiral theory may violate P while respecting CP T (the weak interaction is the

physical example). A handedness whose cancellation requires PT rather than P alone is therefore

not forbidden; the prohibition is removed, not merely weakened. This is the sense in which the

geometric obstruction here is milder than the general lattice no-go [4, 6]: the worldline carries a

complex, time-ordered structure that the static lattice action [8] lacks.

5.2 The oriented Lift violates parity

The relaxation removes the prohibition; the oriented Lift then supplies actual parity violation. On

the up/down doublet, parity is represented by P = σ

(it exchanges the two sheets). Classifying

the doublet generators by their behavior under P and under P T , the unique generator that is odd

under P and even under P T is σ

P σ

P = −σ

, (P T ) σ

(P T ) = +σ

. (4)

The oriented Lift, which selects the sense (above versus below), contributes precisely this σ

direction—the imaginary, sense-carrying oﬀ-diagonal of the interlayer hop. The temporal gen-

erator on a defect worldline therefore contains a P-odd, P T -even piece: the worldline genuinely

violates parity, which the static centrosymmetric geometry forbade. This is a positive result, not

a negative one—parity violation is a necessary ingredient of the electroweak interaction, and it is

supplied here by the oriented out-of-plane growth.

5.3 What remains unforced: the chiral projection

Parity violation in the generator is necessary but not suﬃcient for a chiral gauge coupling. A chiral

SU(2)

requires the gauge generators to act through the left projector, P

with P

(⊮ + γ

on the left sheet only. The σ

generator supplied by the oriented Lift is P -odd, but it generates

su(2) rotation on both sheets equally: evolving the doublet under exp(−i δt σ

) moves the left-sheet

and right-sheet populations by the same amount. It is a parity-violating but vector su(2), not a

chiral one. Nothing established here forces the P

projection that would restrict the coupling to

the left sheet. Whether a further structure—the anchor, the stabilizer protection, or the coupling

to the color sector—supplies that projection is open, and we do not claim it.

The worldline therefore carries the obstruction two tiers further than the static geometry: from

P -forbidden to P T -permitted, and from parity-conserving to parity-violating. Only the chiral

projection remains, sharply isolated as the open frontier.

6 Scope and status

The result is best stated in three classes.

Reached. The six square faces of the FCC cuboctahedron carry an su(2) (three ⟨100⟩ axis-pairs

closing as so(3)), distinct from the color SU(3) on the eight triangular faces; with the diagonal U(1)

the rank matches SU(2) ×U(1). Read as interlayer objects the square faces give a two-dimensional

doublet with a valid order-two chirality operator, so a chiral SU(2)

is geometrically available.

On a chronological defect worldline the static parity obstruction relaxes to P T , and the oriented

Lift supplies a P -odd, P T -even temporal generator, so the worldline genuinely violates parity—

a necessary ingredient of the electroweak interaction, here delivered by the oriented out-of-plane

growth.

Forbidden statically, open on the worldline. Statically the chirality is forbidden: at every bulk node

the up-Lift and down-Lift environments are exact inversion partners (Proposition 1), a consequence

of FCC centrosymmetry, so the handedness cancels under parity. This prohibition is lifted on the

worldline, where the cancellation is P T rather than P , a relation with which a chiral coupling

is compatible. What is not reached, at either level, is the ﬁnal step: the left projection P

that

turns the parity-violating but vector su(2) into a chiral gauge coupling. The oriented generator

alone rotates both chiralities equally; nothing established here forces the P

projection. Whether

a further structure supplies it is the isolated open frontier.

Not touched. The electroweak sector requires more than a chiral SU(2) × U(1): the chiral hy-

percharge assignments and the cancellation of gauge anomalies across a generation. Neither is

addressed here. Establishing them is necessary before any claim that the geometry reproduces the

electroweak interaction, and they are independent of, and additional to, the chirality question.

The boundary this paper draws is therefore the following. Color is selected on the triangular faces

of the FCC shell; an electroweak-shaped SU(2) ×U(1) structure exists on the square faces; a static

chiral assignment is forbidden by centrosymmetry, but a defect worldline relaxes the obstruction to

P T and the oriented Lift violates parity; and the chiral projection that would complete SU(2)

, to-

gether with hypercharge and anomalies, remains open. The geometric derivation of the electroweak

gauge sector, in this framework, reaches the gauge structure and genuine parity violation on defect

worldlines, and stops one identiﬁed step short of a chiral coupling.

Data availability

The scripts reproducing the su(2) structure, the inversion (centrosymmetry) computation, the

P → P T relaxation on a defect worldline, and the P -odd/P T -even classiﬁcation of the temporal

generator are available at https://github.com/raghu91302/ssmtheory/raw/main/ew_scripts.

zip. They require NumPy (the generator classiﬁcation also SciPy). No other data were generated

or analyzed in this study.

References

[1] R. Kulkarni, Matter as incomplete crystallization: quark charges, color conﬁnement, and the

proton mass from a single extra node in the vacuum lattice, Phys. Open 27 (2026) 100423.

doi:10.1016/j.physo.2026.100423.

[2] R. Kulkarni, A face-centered-cubic stabilizer code for the vacuum lattice, arXiv:2603.20294

[quant-ph] (2026). arXiv:2603.20294.

[3] R. Kulkarni, A self-correcting discrete vacuum: baryon conservation and color, from an emer-

gent qutrit to dynamical su(3), on one FCC code (2026). doi:10.5281/zenodo.18503168.

[4] H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice. (I) Proof by homotopy theory,

Nucl. Phys. B 185 (1981) 20 [Erratum: Nucl. Phys. B 195 (1982) 541]. doi:10.1016/0550-

3213(81)90361-8.

[5] H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice. (II) Intuitive topological

proof, Nucl. Phys. B 193 (1981) 173. doi:10.1016/0550-3213(81)90524-1.

[6] H. B. Nielsen and M. Ninomiya, A no-go theorem for regularizing chiral fermions, Phys. Lett.

B 105 (1981) 219. doi:10.1016/0370-2693(81)91026-1.

[7] P. H. Ginsparg and K. G. Wilson, A remnant of chiral symmetry on the lattice, Phys. Rev. D

25 (1982) 2649. doi:10.1103/PhysRevD.25.2649.

[8] K. G. Wilson, Conﬁnement of quarks, Phys. Rev. D 10 (1974) 2445.

doi:10.1103/PhysRevD.10.2445.

[9] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York,

1976.