Geometric Origins of Spin-1/2 and Relativistic Kinematics

Defect Migration in the

Vacuum Lattice:

Hop Parity, Site Symmetry, and Soliton Kinematics

Raghu Kulkarni

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

August 2026

Abstract

We study how a localized defect moves through the

vacuum lattice of the Selection-

Stitch Model. One coordinate is selected as time. The spatial slice is then exactly

, the

face-centered cubic lattice, and a particle is a defect that migrates between its tetrahedral

voids.

The voids form a simple cubic lattice of spacing

a/2

. Its two-coloring by coordinate sum

modulo

is the positive/negative void distinction. Each void has six neighbors, all of the

opposite color, so every hop ips the void class

A hop has spatial length

, while the spatial bonds have length

√

, so only the time-

mixed bonds match it. Assume further that a worldline advances by one bond per update.

A hop then carries exactly one time step, and no update leaves a defect in place. A defect

at rest oscillates between two adjacent voids. This resembles Dirac zitterbewegung, but we

stop at the resemblance.

A hop induces a canonical bijection on the bounding atoms: shared atoms persist and

released ones correspond to engaged ones by a translation of the atom lattice. If the winding

labels that x the charge follow that bijection, baryon species is preserved. We mark the

transport rule as an assumption, since the defect center moves by half the atom displacement

and no defect bond is translated rigidly. An anchor-conserving rule is separately excluded,

because it would conne the defect to one octant.

Two gradings must be kept apart. The void class ips at every step; the matter/antimatter

grading must not, since baryon number is conserved. We prove the second cannot come from

the void, because each class admits exactly one bond-direction set. The worldline supplies

it instead, through its time orientation.

The rotational symmetry of a void is

∼

, whose double cover has faithful irreducible

representations only in dimension

. Among half-integer spins, only

j =

stays irreducible

on restriction. This selects rather than determines. Finally, a localized solution satisfying

the von Laue condition obeys

E = γE

rest

; carrying this to the trapped defect requires

conditions we state but do not establish.

1 Introduction

The quantum mechanical description of matter is empirically exact. Its geometric origin is not

settled. The wave function

is treated as a fundamental object, and its physical meaning is

still debated [1]. The de Broglie relation

λ = h/p

is postulated rather than derived. Spin-

introduced as an abstract representation of SU(2), formally disconnected from spatial geometry.

The Selection-Stitch Model (SSM) proposes a geometric origin for these things. This paper

works inside that framework, so we set it out before proceeding.

raghu@idrive.com

1.1 The Selection-Stitch Model

The SSM models the physical vacuum as a discrete close-packed lattice rather than a continuum.

Three claims dene the framework, and each has been developed separately.

The vacuum crystallizes.

The early universe passes from a geometrically frustrated tetra-

hedral phase, in which each node has

K = 4

neighbors, to an ordered close-packed phase with

K = 12

[2]. The ordered phase is selected because it saturates the sphere-packing bound.

Regular tetrahedra cannot tile space, so the intermediate phase is unstable and the transition

proceeds to the densest packing.

Matter is incomplete crystallization.

If a node from the earlier phase fails to integrate

during the transition, it remains trapped at the center of a tetrahedral void, bonded to the four

surrounding lattice sites. This creates a local

K = 4

pocket inside the

K = 12

bulk. That

trapped node is a baryon. Four results follow from this single premise [2]. The tetrahedral

bond angle gives fractional electric charges of

−1/3

and

+2/3

. The three skew-edge pairs of the

bounding tetrahedron give three color charges. The metric obstruction to extracting the node

gives linear connement. And the structural counts give the proton-to-electron mass ratio

= (K + 1)K

− c

skew

K = 13 · 144 − 3 · 12 = 1836.

(1)

Mass is verication cost.

The vacuum is modeled as a stabilizer code, and a particle is a

defect that the stabilizer measurements detect. On the spatial lattice this is a

[[192, 130, 3]]

weight-12 CSS code [4]. Detecting a defect costs classical bits, and by Landauer's principle

each bit costs energy. Because the vacuum has no separate hardware substrate, that cost is the

particle's entire mass. Mass ratios are then ratios of integer verication costs,

= C

with the temperature canceling [3].

1.2 Where this paper stands

All three claims above are static. They describe what a particle is, not how it moves. The mass

formulas, Eq. (1) among them, evaluate a defect frozen in place, and nothing in them refers to a

trajectory. This paper asks the kinematic question instead: how does a trapped defect migrate

through the lattice, and what does the lattice permit?

The question turns out to be sensitive to the number of lattice dimensions in a way the static

calculations are not, which is why the setting below is four-dimensional.

We separate three kinds of statement, and mark each one in the text where it occurs.

Kind Examples Status

Lattice geometry Props. 17, Cor. 1 Veried; Appendix A

Framework premise Defect

particle; mass

verication cost Taken from [2, 3, 4]

Interpretation Worldline orientation; particle identication Stated as such

The distinction matters for how much of the paper depends on the framework. The geometric

results are statements about the lattice and hold whether or not the SSM reading of them is

correct. What the framework supplies is the identication of a defect with a particle, which is

what makes those statements physically interesting.

We do not defend the framework here. The reader who does not accept the identication

can still read Sections 2 through 7 as results about defect migration on

1.3 Setting: the lattice and its foliation

We work on the four-dimensional root lattice



x ∈ Z

≡ 0 (mod 2)



(2)

the densest lattice packing in four dimensions [6]. Each point has

nearest neighbors at

distance

√

, obtained by setting any two coordinates to

±1

Foliation.

One coordinate, written

, is selected as time. The remaining three are spatial.

We take this selection as given; a mechanism for it lies outside this paper. The selection splits

the

bonds into two sets of twelve. The twelve with

-component zero involve only spatial

coordinates and are the nearest-neighbor bonds of



x ∈ Z

≡ 0 (mod 2)



(3)

which is the face-centered cubic (FCC) lattice. The twelve with

-component

±1

connect

adjacent time slices. We verify in Appendix A that the slice has exactly twelve neighbors at

√

as FCC requires.

Throughout, the conventional cubic cell of the spatial slice has side

a = 2

, so the nearest-

neighbor distance is

√

2 =

√

. Lengths are quoted in these units.

A correction to the earlier framework.

The migration analysis forces a revision of the

static matter/antimatter identication in Ref. [2]: inversion-related void classes are propagation

sublattices, not particle and antiparticle states. Section 6 states the correction and Section 7

supplies the replacement.

Interactive visualization.

An animation of defect migration accompanies this paper. It runs

in any WebGL browser, with no installation, at

https://raghu91302.github.io/ssmtheo

ry/d4_hopping.html

. Four modes are available. Three successive hops along one axis, with

the shared, released and engaged bounding atoms marked and the tetrahedron edges colored so

that the ve broken, ve formed and one preserved can be counted. A worldline at rest. Free

drift. And a static view of the escape channels, showing the six open edge channels and the four

closed face channels of Proposition 2, with the count of atoms inside the metric wall at each. The

second and third modes assume Postulate 1, and the interface says so. Corner and face-center

atoms are shaded dierently and one conventional cubic cell is outlined, so the face-centered

pattern is visible, with an optional cuboctahedral

K = 12

shell. The projection is orthographic,

so that equal lattice vectors subtend equal lengths on screen; perspective is available as a toggle.

Readers may nd it useful alongside Sections 2 and 3, where the hop and its parity are dened.

Organization.

Section 2 xes the geometry of the tetrahedral-void sublattice on which defects

migrate. Section 3 locates the hop in the bond set, states the propagation rule, and draws its

consequences. Section 4 examines what the site symmetry xes about spin, and Section 5

treats the resulting oscillation. Sections 6 and 7 separate the two orientation gradings and prove

that one of them cannot be geometric. Sections 8 and 9 treat annihilation and baryon transport.

Section 10 states the conditions under which a localized solution of a Lorentz-invariant continuum

theory obeys

E = γE

rest

, and what would be needed to apply them to the trapped defect.

2 The Tetrahedral-Void Sublattice

2.1 Void enumeration

The conventional cubic cell of the spatial slice contains four lattice sites and eight tetrahedral

interstitial voids [7, 8]. We call a lattice site an atom throughout, reserving

node

for the trapped

defect of Section 1.1. The voids split by orientation:



Positive (P) voids

(a/4)(1, 1, 1)

(a/4)(3, 3, 1)

(a/4)(3, 1, 3)

and

(a/4)(1, 3, 3)

. The

four bounding atoms point along the

[111]

family.



Negative (N) voids

(a/4)(3, 3, 3)

(a/4)(1, 1, 3)

(a/4)(1, 3, 1)

and

(a/4)(3, 1, 1)

. The

four bounding atoms point along the

[

family.

What the labels mean.

The two classes are related by spatial inversion, which is improper

and is not a rotation. That relation is the geometric content of the P/N labels, and Proposition

7 states it precisely.

2.2 The voids form a simple cubic lattice

The void positions.

Writing the eight void centers in units of

a/4

gives

(1, 1, 1), (3, 3, 1), (3, 1, 3), (1, 3, 3), (3, 3, 3), (1, 1, 3), (1, 3, 1), (3, 1, 1),

which is every triple of odd entries drawn from

{1, 3}

. Extending periodically, the voids occupy

all points

(a/4)(

odd

)

Proposition 1

(Void lattice)

The tetrahedral voids of the FCC slice form a simple cubic lattice

of spacing

a/2

. The P voids are those with coordinate sum

≡ 3 (mod 4)

in units of

a/4

, and

the N voids those with sum

≡ 1

. The two classes are the checkerboard two-coloring of that cubic

lattice.

2.3 A closed form for the void class

Denition.

Take a void center

and its four bounding atoms

. The direction vectors

(v −

c)/(a/4)

are triples of

±1

. Dene

σ = d

(4)

for any one of the four; the value is the same for all four. Direct enumeration gives:

Void class Bond direction set

(−1, −1, −1)

(−1, 1, 1)

(1, −1, 1)

(1, 1, −1) −1

(1, 1, 1)

(1, −1, −1)

(−1, 1, −1)

(−1, −1, 1) +1

Inversion.

Spatial inversion maps each direction

−d

. This maps the P set onto the N set

exactly. So

is odd under inversion.

2.4 Migration by edge-adjacent hopping

Migration and connement.

Reference [2] shows that the trapped node cannot be extracted

from the void network: the surrounding lattice resists radial displacement with a potential

linear in the displacement. That is a statement about removal into the bulk. A hop is a

dierent process. The defect leaves one void only by entering an adjacent one, and remains four-

coordinated throughout, so connement bounds migration by a barrier rather than forbidding

it.

The move.

A defect occupying a void with bounding atoms

{A, B, C, D}

migrates to an

adjacent void sharing one tetrahedral edge, that is, two bounding atoms:

{A, B, C, D} −→ {A

′

, D

′

, B, C},

(5)

where

{B, C}

is the shared edge. Figure 1 shows the geometry. Five bonds among

{A, B, C, D}

are broken, ve among

′

, D

′

, B, C}

are created, and the bond



is preserved. Appendix A

conrms this for all

(void, neighbor) pairs of the cell.

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

(σ = −1)

(σ = +1)

trapped K = 4 node shared edge {B, C} (kept) released {A, D}

engaged {A

, D

}

Figure 1:

One hop. The defect's void

(red, solid) and its image

(blue, dashed) share two bounding

atoms

and

, which form one tetrahedral edge. The other two are exchanged,

{A, D} → {A

′

, D

′

}

The trapped

K = 4

node sits at the centroid of each tetrahedron. The two tetrahedra are related by

inversion, which is the geometric content of the class ip.

Enumeration.

Direct enumeration over the eight voids, including periodic images, gives the

following.

1. Each void has exactly six neighbors, at

±a/2

along each spatial axis.

2. Every neighbor is of the opposite class. There are no same-class adjacencies.

3. All hops have uniform distance

hop

= a/2

4. There are

hop channels per conventional cell.

5. Every hop shares exactly two bounding atoms and rearranges bonds as

2.5 Why migration goes by edges

The choice of channel is not free. It follows from the exclusion condition of Ref. [2], and we

record the argument because it also xes the coordination number.

The exclusion condition.

A tetrahedral void is bounded by four atoms forming a regular

tetrahedron of edge

. Reference [2] rejects any conguration in which a node lies within

√

all three vertices of a bounding triangle at once: it would then be at unit-bond-length range of all

three simultaneously, which is over-constrained. The length

√

is the in-plane circumradius

of such a triangle and is called the metric wall. Note that the condition involves three atoms.

Proximity to one or two is not excluded.

Face exits are closed.

To leave through a face, the defect must cross the plane of three

bounding atoms. The centroid of a triangle of edge

lies at exactly

√

from each of its

vertices, so a defect at that point is at the wall with respect to all three at once. All four faces of

the tetrahedron are therefore closed. Each face is shared with an octahedral void, so migration

to an octahedral site is excluded.

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

(a) Voids of the spatial slice

P void (σ = −1)

N void (σ = +1)

one hop, a/2

simple cubic, spacing a/2; 6 neighbors in 3D, 4 shown;

each hop flips σ

(b) A slice of the void lattice

Figure 2:

The void sublattice. (a) The conventional cubic cell of the spatial slice, with four P voids (red,

up) and four N voids (blue, down). Dark spheres are lattice atoms and green segments are hop channels.

(b) A plane of the void lattice. The voids form a simple cubic lattice of spacing

a/2

, two-colored as a

checkerboard, so every hop changes color.

Edge exits are open.

To leave across an edge, the defect passes two bounding atoms. At the

midpoint of a tetrahedron edge it lies

L/2

from each of those two, and the remaining atoms are

further away. Direct enumeration along the whole path nds at most two atoms within

√

at any point, never three, so the exclusion never applies. The six edges of the tetrahedron are

therefore the six open channels.

Proposition 2

(Edge channels and coordination)

Of the ten ways out of a tetrahedral void, the

four through faces are excluded by the metric wall and the six across edges are permitted. Each

edge is shared with exactly one neighboring tetrahedral void, and the six shared atom pairs are

precisely the six edges of the bounding tetrahedron. The coordination number six of Section 2 is

a consequence, not an assumption.

Both statements are veried in Appendix A.

Proposition 3

(Bipartite migration)

Every hop ips

This follows from Proposition 1: a hop changes one coordinate by

in units of

a/4

, so the

coordinate sum changes by

and its residue modulo

alternates between

and

. Figure 2

shows the structure.

3 Hop Parity

We now come to the feature that distinguishes migration in four dimensions from migration in

three.

3.1 The hop displacement in the bond set

The displacement.

A hop displaces the defect by

a/2

along one spatial axis. In lattice units

that displacement has length

. We rst record where such a displacement sits in the bond set.

The twelve spatial bonds have spatial length

√

. The twelve time-mixed bonds have spatial

length

. So the hop matches the spatial part of a time-mixed bond, and of no spatial bond. The

same statement in coordinates: with no time component the hop is

(0, 1, 0, 0)

, whose coordinate

sum is odd, while

(±1, 1, 0, 0)

has even sum.

4-vector sum

in D

(0, 1, 0, 0)

1 no hop, no tick

(1, 1, 0, 0)

2 yes

hop, +1 tick

(−1, 1, 0, 0)

0 yes

hop, −1 tick

(1, 0, 0, 0)

1 no tick, no hop

(2, 0, 0, 0)

2 yes two ticks, no hop

a hop is a D

vector only with a tick;

a tick is a D

vector only with a hop

(a) Which steps are D

vectors

a/2

position

time (ticks)

net motion: none

amplitude

a/2

(b) A worldline at rest, under Postulate 1

Figure 3:

The parity obstruction. (a) Admissible steps. A hop with no time step and a time step with

no hop both have odd coordinate sum and are not

translations. A hop together with one tick has

even sum and is admissible. (b) A worldline at rest. Since the defect cannot advance one tick while

staying at one void, it alternates between two adjacent voids, returning to its starting position every two

ticks and alternating

as it goes.

Proposition 4

(Bond realisation of a hop)

Among the nearest-neighbor bonds of

, the spatial

displacement of a hop is realized only by the time-mixed bonds. No spatial bond has spatial length

What this does not say.

Proposition 4 is a statement about vectors, not about dynamics.

The defect occupies an interstitial void rather than a lattice point, and the void sublattice has

spacing

a/2

while the atom lattice has nearest-neighbor distance

√

. The translation group of

the void sublattice is therefore strictly larger than that of the atom lattice, and a hop is already

not a lattice translation. Nothing so far forbids a defect from moving at xed time, or from

persisting at one void for one update. A rule for how the defect propagates is needed, and we

state one.

3.2 The propagation rule

Postulate 1

(Bond propagation)

The worldline of a defect advances by one nearest-neighbor

bond of

per update.

What the rule is.

Postulate 1 species a discrete-time local update. Evolution proceeds in

steps, and each step moves the defect along one bond. This is the structure of a discrete-time

quantum walk, or of a quantum cellular automaton [13, 14], and not of continuous-time evolution

under a lattice Hamiltonian.

The distinction matters. Continuous evolution

−iHt

under a nearest-neighbor

does not

conne amplitude to one bond per unit time. Higher powers of

generate amplitude at all

distances. What holds instead is a Lieb-Robinson bound [11], giving an eective light cone with

exponentially small leakage outside it rather than a strict one. The modern statement and its

reach are reviewed in [12]. A strict one-bond step is a property of a discrete update rule and

must be imposed as one.

Status of the rule.

We do not derive Postulate 1 from the bond Hamiltonian, and we do

not claim it follows from one. It is an assumption about the dynamics, and every result in this

section depends on it. Section 11 lists its derivation as an open problem, together with the

construction of the update operator itself.

3.3 Consequences of the rule

From vectors to worldlines.

Given Postulate 1, the parity facts become statements about

worldlines.

Proposition 5

(A hop carries a tick)

Under Postulate 1, a spatial hop is accompanied by exactly

one time step.

This follows from Proposition 4: a step is a bond, the hop has spatial length

, and only the

time-mixed bonds have that spatial length.

Corollary 1

(Rest requires alternation)

Under Postulate 1, a defect cannot advance one tick

while remaining at a single void. All twelve one-tick bonds have exactly one nonzero spatial

component, so every update displaces the defect. A worldline with zero net spatial displacement

over two ticks consists of two one-tick steps in opposite spatial directions, for example

(1, 1, 0, 0)

followed by

(1, −1, 0, 0)

, with net displacement

(2, 0, 0, 0)

Status.

Proposition 4 is unconditional and is veried in Appendix A. Proposition 5 and Corol-

lary 1 hold given Postulate 1 and not otherwise.

What this means physically.

Figure 3 shows both statements. A particle at rest is not

stationary on the lattice. It oscillates between two adjacent voids with amplitude

a/2

and

period two ticks. By Proposition 3 it alternates

as it does so:

P → N → P → N → ···

(6)

4 Spin-1/2 from the Site Symmetry

The question.

A defect occupies a tetrahedral void, so its internal states carry a representation

of the symmetry group of that void. This section shows that the group in question admits spin-

, and that

j =

is the only half-integer representation of

SU(2)

that remains irreducible on

restriction to it. Higher half-integer representations are admitted, but they decompose.

The site group.

The proper rotations that map a regular tetrahedron to itself form the

tetrahedral group

∼

, of order

. Spin transforms under rotations, so

is the relevant

group. Improper operations map a void to one of the opposite class, and are treated in Section

Its double cover.

Spin states carry projective rather than linear representations of the rotation

group. A projective representation of

is a linear representation of its preimage under the

covering map

SU(2) → SO(3)

. That preimage is the binary tetrahedral group

, of order

, realized concretely as the

unit Hurwitz quaternions. Direct computation gives seven

conjugacy classes, hence seven irreducible representations, of dimensions

1, 1, 1, 2, 2, 2, 3, 1 + 1 + 1 + 4 + 4 + 4 + 9 = 24.

(7)

Which are spinorial.

The quotient

2T/{±1}

∼

, whose irreducible representations

have dimensions

1, 1, 1, 3

. Those four are exactly the representations of

on which the central

element

−1

acts trivially, so they are tensorial. The remaining three of Eq. (7), those of dimension

, are the faithful ones, on which

−1

acts as minus the identity. This gives the bound that does

the work:

no faithful irreducible representation of

has dimension greater than

(8)

The selection.

Restrict the spin-

representation of

SU(2)

, of dimension

2j + 1

, to

and

compute

⟨χ

, χ

⟩

over the group:

dimension

⟨χ

, χ

⟩

irreducible under

1/2

2 1 yes

3/2

4 2 no

5/2

6 3 no

7/2

8 6 no

9/2

10 9 no

Proposition 6

(Compatibility and selection)

The tetrahedral site symmetry admits faithful

two-dimensional projective representations, and these are compatible with spin

. Among the ir-

reducible representations of

SU(2)

j =

is the only half-integer representation whose restriction

remains irreducible; every higher one decomposes, by the bound

(8)

The computation is in Appendix A.

What this does not establish.

Proposition 6 is a selection criterion and not a determina-

tion of spin. A nite site group cannot in general recover the continuum parent representation,

because a two-dimensional irrep of

may arise as one component of the crystal-eld decom-

position of a higher-spin representation. Direct computation conrms this. The faithful irrep

carried by

j =

also occurs inside

j =

, with multiplicity one, and inside

j =

, with multi-

plicity two. The representation

j =

decomposes into the other two faithful irreps. Observing

a two-dimensional faithful representation therefore does not identify the parent as

j =

What would be needed.

To conclude that a defect carries spin

, two further things must

hold and neither is derived here. The whole physical state must transform irreducibly under

, rather than as one component of a decomposition. And it must extend to an irreducible

representation of the emergent continuous rotation group with no additional components. Both

are listed in Section 11.

The projective assumption.

Separately, the argument assumes the state is spinorial rather

than tensorial. A tensorial state would carry one of the representations of dimension

1, 1, 1, 3

and would have integer spin. Which case is realized is not derived.

What the void class does not do.

The alternation of

along a worldline is a parity grading,

not a rotation. It distinguishes the two void orientations, which are related by inversion, and

inversion is improper. A

grading of this kind cannot by itself select a spin value, because the

central element of

SU(2)

acts as

−1

on every half-integer representation alike. The selection in

Proposition 6 comes from the site group and its cover, not from

. The role of

is taken up in

Sections 6 and 7.

5 A Two-Sublattice Oscillation

A free Dirac electron exhibits zitterbewegung, a high-frequency oscillation between positive- and

negative-energy states [10, 9]. The eect also arises in discrete models. A one-dimensional Dirac

quantum cellular automaton reproduces it directly [16], and lattice automata of this kind go back

to Bialynicki-Birula [15]. We compare the present lattice oscillation with it, and state carefully

how far the comparison reaches. The two states are eigenstates of the Dirac Hamiltonian with

opposite-sign energy eigenvalues, and their interference produces a trembling motion superposed

on the mean trajectory.

The lattice picture.

Corollary 1 produces a supercially similar picture, given Postulate 1.

A defect at rest does not sit still. It oscillates between two adjacent voids with amplitude

a/2

, returning to its starting position every two ticks, and its mean position is stationary. The

oscillation is not superposed on the rest state; it is what the rest state is made of, because no

admissible step leaves the defect where it is.

Why this is not yet zitterbewegung.

The resemblance is structural only, and we do not

claim more. Establishing that the oscillation is zitterbewegung would require several things. A

lattice Dirac operator with positive- and negative-energy branches. Interference between those

branches. A position or velocity operator. The frequency

= 2mc

/ℏ

, with its characteristic

dependence on mass. None of these is derived here. They are available for related discrete

models, where the continuum limit of a quantum walk is known to give the Dirac equation in

one and in higher dimensions [17, 18, 19]. The claim would also be circular as things stand,

since Postulate 1 was itself motivated by the Dirac velocity spectrum.

What is settled and what is not.

The existence of the oscillation, its amplitude

a/2

, and its

two-tick period follow from Postulate 1, not from the geometry alone. The physical frequency

does not. Relating a tick to a physical time requires the lattice time unit, which this paper does

not x, and the rule gives the same two-tick period for every defect whereas

depends on

mass. Section 11 states this gap precisely.

Which grading is involved.

Zitterbewegung is interference between two energy eigenstates

of a single particle. It does not convert a particle into its antiparticle, and baryon number

and charge are unchanged throughout. The grading that alternates is therefore

, not the

matter/antimatter grading of the next section.

6 Two Gradings

The lattice oers a natural

label, the void class

. It is tempting to identify it with the

matter/antimatter distinction, since the P/N ip is an inversion. That identication cannot be

made, and saying why requires correcting an earlier claim.

Relation to the earlier static identication.

Reference [2] associated the inversion-related

tetrahedral congurations with matter and antimatter. The migration analysis shows that this

identication cannot be maintained dynamically. Adjacent voids are inversion-related, and every

allowed hop exchanges their classes; identifying that exchange with charge conjugation would

make an ordinary propagating baryon alternate between baryon and antibaryon. We therefore

revise the earlier interpretation. The P/N distinction is a host-void parity

, while the parti-

cle/antiparticle distinction is carried by the hop-invariant worldline grading

of Section 7. Both

particles and antiparticles consequently traverse both void classes. This is a correction to [2],

not a consequence of it.

The parity role requires alternation.

records which of the two void orientations the

defect occupies. By Proposition 3 it alternates at every step, and by Corollary 1 it alternates

even at rest. Any grading identied with

inherits that alternation.

The matter role forbids alternation.

alternates at every step, at rest or in motion. If

the matter/antimatter label alternated with it, a baryon would oscillate between baryon and

antibaryon along its own worldline, and baryon number would not be conserved. Baryon number

is believed to have been conserved since baryogenesis [5].

The cost of merging them.

A single label carrying both roles also misdescribes matter.

Bipartiteness guarantees that adjacent voids lie in opposite classes. If opposite class meant

n+1

n+2 n+3

n+4

step along the worldline

−1

one step

void parity

matter /

antimatter

(a) The two gradings along a worldline

P N P N P

P voids

all four identical

(-1, -1, -1)

(-1, +1, +1)

(+1, -1, +1)

(+1, +1, -1)

N voids

all four identical

(+1, +1, +1)

(+1, -1, -1)

(-1, +1, -1)

(-1, -1, +1)

d ↦ − d

one bond-direction set per class

⇒ the void carries exactly one bit

(b) The void cannot supply ε

Figure 4:

The two gradings. (a) Along a worldline the void class

alternates at every step, while

stays

xed, so that baryon number is conserved. No single quantity can occupy both rows. (b) Proposition

7. Each class admits exactly one bond-direction set, and inversion maps one onto the other, so the void

carries exactly one bit of orientation information.

matter meeting antimatter, then any two baryons at adjacent voids would annihilate on contact,

and bound matter would not exist.

Denition 1

(The two gradings)

(void class).

Given by Eq.

(4)

. It is a property of

the site. It ips at every step and is odd under spatial inversion. It is a parity grading and

carries no spin information.

(worldline orientation).

A property of the defect's worldline, not of its host void. It

must be invariant under hopping. It carries the matter/antimatter distinction.

The corrected assignment is summarized below.

Quantity Meaning Changes on a hop?

host-void parity, P/N yes

matter/antimatter worldline label no

spatial inversion exchanges the void classes yes

charge conjugation exchanges particle and antiparticle not on an ordinary hop

7 The Void Cannot Supply

The natural rst attempt is to build

from the geometry of the host void. This cannot work.

Proposition 7

(One bit only)

The bond direction set of a tetrahedral void is determined by

alone. There is exactly one such set for

σ = −1

and exactly one for

σ = +1

Verication.

Figure 4(b) shows the two sets. Collecting the bond direction set of each of the

eight voids as an unordered set of

±1

triples, all four P voids give the identical set and all four

N voids give the identical set. Inversion maps one onto the other exactly (Appendix A).

□

Consequence.

The host void carries exactly one bit of orientation information, and that bit

. Any quantity built from the void geometry alone is a function of

, so it either alternates

at every step or is constant. Neither is what

requires. The spatial slice therefore exhausts its

available structure at one bit.

7.1 The worldline supplies what the void cannot

The construction.

In four dimensions a defect is a worldline rather than a point. The world-

line carries structure the void does not. By Proposition 5, under Postulate 1, every step advances

±1

. A worldline whose steps all have

∆x

= +1

is future-directed, and one whose steps

all have

∆x

= −1

is past-directed. Write

ε = +1

and

ε = −1

for the two cases.

It has the required properties.

A hop changes the spatial coordinate and leaves the sign

∆x

alone, so

is unchanged by hopping. Reversing the worldline exchanges it. And it is

independent of

, which is a property of position rather than of the worldline.

Identication.

We identify

ε = −1

with antimatter, following the standard reading of an

antiparticle as a particle propagating backward in time. This is an interpretive step and not

a lattice theorem, and we mark it as such. What the lattice supplies is a hop-invariant

worldlines, which is the structure Denition 1 requires and which Proposition 7 rules out at the

level of the void.

Why a purely spatial construction fails.

A grading built from spatial data alone has no

worldline direction to appeal to, and Proposition 7 then leaves nothing for it to be. The carrier

exists because the defect is an extended object in time, and it requires no assumption about

the defect beyond the foliation itself.

8 Annihilation

Setting.

Baryonic matter is the trapped defect of Section 1.1, a

K = 4

pocket in the

K = 12

bulk. Antimatter is its counterpart with

ε = −1

The mechanism.

When two defects with opposite

occupy adjacent voids that share an edge,

the local

K = 4

decit can be resolved by cancellation. The two trapped nodes drift toward the

shared boundary, all bounding bonds return to

K = 12

saturation, and the defects leave the

lattice. The stored energy, equal to the sum of the two defects' disrupted-bond contributions,

is released into outgoing lattice excitations. In worldline terms a future-directed and a past-

directed segment meet and join, which is the lattice form of a pair-annihilation vertex.

The criterion.

Annihilation requires opposite

at adjacent voids. It does not require opposite

. The consequences are as they should be.



Two protons at adjacent voids have the same

and do not annihilate. Bound matter is

stable and nuclei exist.



A proton and an antiproton at adjacent voids have opposite

and annihilate by the

mechanism above.



A propagating baryon alternates

at every step and keeps

throughout, so baryon number

is conserved during propagation.

Energy partition.

The released energy

annihilation

= 2N

disrupted

× E

bond

partitions among

outgoing lattice excitations. Identifying which modes carry which fraction, and relating this to

the empirical

p¯p

annihilation spectrum, is an open extension.

9 Baryon Transport

The migration picture makes contact with a recent measurement. It is worth stating both what

the contact is and what it is not.

The measurement.

The STAR Collaboration has reported measurements of baryon number

transport in nuclear collisions [5]. Two observables are involved: the ratio of net-baryon number

to the net-charge dierence between isobar collision systems, and the rapidity dependence of the

net-proton yield in photonuclear collisions. Both disfavor the picture in which valence quarks

carry the baryon number, and both are consistent with an alternative in which a

-shaped

gluonic conguration traces it. The relevant feature is not the shape but that baryon number

can travel even when the original valence quarks are displaced or separated from the structure

that carried them.

The correspondence.

Equation (5) is the relevant structure. A hop keeps the shared edge and

replaces two of the four bounding atoms with atoms drawn from the destination void. Section

1.1 decomposes the trapped defect into one anchor bond and three valence bonds [2]. Read

together with Eq. (5), the node carrying baryon identity migrates while two of its four bonds

are reconstituted from the lattice at each step. This is the structural content of the junction

picture, and by Proposition 2 it follows from the crystallography rather than being assumed.

The two quantities separate.

What makes the correspondence sharper is that the framework

treats baryon number and electric charge dierently under migration, and the dierence is forced.

Baryon number is carried by the trapped node itself, which survives every hop, together with

the hop-invariant grading

of Section 7. Electric charge is carried by the winding numbers on

the three valence bonds [2], and enumeration over all

(anchor, hop) pairs shows that one or

two of those three bonds are replaced at every hop, never none. Baryon number therefore travels

with the carrier, while charge is re-established from the local embedding as the defect moves.

That the two transport dierently is the qualitative direction of the isobar measurement.

What the correction of Section 6 secures.

Baryon number conservation under migration

is not automatic in this framework. Had the void class carried the matter/antimatter label, a

propagating baryon would alternate between baryon and antibaryon and the comparison with

[5] could not be made at all. The separation of

from

is what makes the conserved quantity

available.

Section 10 gives conditional continuum soliton kinematics, whose application to the migrating

defect remains open.

9.1 Baryon species under migration

The observable in [5] is the net-proton yield, so it distinguishes baryon species. In [2] the species

is xed by which of the four bonds serves as the anchor and by the winding numbers

∈ {0, 1}

on the other three, through

Q = −1 + W

with

W =

. A hop replaces two of the four

bonds, so the question is whether

survives.

The anchor cannot simply be conserved.

The obvious rule is to require that the anchor lie

in the shared edge, so that it is never released. Direct enumeration shows this fails. For a given

anchor, the three hops that preserve it are exactly the three axis projections of that anchor's own

bond direction: for the anchor along

(−1, −1, −1)

they are

(−1, 0, 0)

(0, −1, 0)

and

(0, 0, −1)

No axis is reversible. A defect obeying that rule could drift into one octant and never return,

which is not a possible history for a particle. The anchor must therefore be reassigned, and a

transport rule is needed.

Four distinct objects.

The argument below turns on keeping these apart. A

bounding-lattice

edge

joins two bounding atoms and is an edge of the tetrahedron. A

defect bond

joins the trapped

node to one bounding atom; there are four. The

anchor

and

valence

labels are an assignment

to those four defect bonds. A

winding class

∈ {0, 1}

is carried by each valence bond. A

hop is dened by which bounding atoms it keeps, so it acts rst on the atoms and only then on

everything attached to them.

The lattice supplies a bijection on atoms.

For every hop, from every void, the two released

atoms are carried onto the two engaged atoms by the single displacement

, where

is the hop

vector. That displacement has length

and even coordinate sum, so it is a translation of the

atom lattice. Together with the two shared atoms, which persist, this gives a canonical bijection

between the four initial bounding atoms and the four nal ones.

It is not an isometry on defect bonds.

The defect center moves by

while a released atom

moves by

, so the defect bond

d = v − c

becomes

′

= (v + 2h) − (c + h) = d + h,

(9)

not

. Enumeration shows the same for the shared atoms, whose bonds change by

−h

: no defect

bond is carried to itself, and every bond direction has its component along the hop axis reversed.

All four bonds retain their length

√

3/4

, but the frame rotates. A winding class attached to a

defect bond therefore has no automatic reason to follow the atom bijection, and we do not claim

that it does.

Proposition 8

(Canonical species-transport map)

Every hop induces a canonical bijection

between the initial and nal bounding-atom data: shared atoms persist, and each released atom

corresponds to an engaged atom through the Bravais translation

. If the winding labels are

transported equivariantly under this bijection, then

, the charge

Q = −1 + W

, and the baryon

species are preserved, and a released anchor corresponds to the anchor of the destination.

The bijection is veried in Appendix A, together with the octant-trapping property that rules

out anchor conservation.

Status of the transport rule.

Equivariant transport is at present a dynamical assumption,

not a theorem. It would become one only if the denition of

in [2] forced it, and that denition

refers to traversal of the bond endpoint under the bulk translation group, whereas Eq. (9) shows

the bond itself is not translated. Whether free translation alone xes the winding labels is stated

in Section 11.

What this would settle, if granted.

Under the rule, charge is conserved locally rather than

only globally, and the defect's bond content is relabeled rather than redrawn. Baryon number

and charge would still travel by dierent routes, which is the structural point of Section 9: the

node moves by

a/2

while the atom data are transported by

Limits.

We state these plainly. The physical hopping rate is not derived. Postulate 1 xes

the period at two updates but not the duration of one, so there is no transport probability and

no prediction for either observable. The rapidity slope requires a production and scattering

framework this model does not have. The gap between the lattice scale and the scale of the

measurement is not bridged.

We cite the result for one purpose. It is the observation that requires baryon number to be

conserved under migration, and that requirement is what forces the separation of

from

Section 6.

10 Relativistic Kinematics of a Continuum Soliton

10.1 Bond Hamiltonian and continuum limit

We start from a nearest-neighbor harmonic bond Hamiltonian with a nonlinear binding potential:

bond

⟨i,j⟩

(|r

− r

| − L)

bind

(10)

with

the inertial mass per site,

the bond spring constant,

the equilibrium bond length,

and

the displacement eld. In the long-wavelength limit this gives

L =

(∂

−

κa

−1

(∇u)

− a

−3

bind

(u).

(11)

After rescaling this is Lorentz-invariant, with signal speed

lat

κa

(12)

Isotropy of the bond set.

For the

unit bond vectors

, the moments

(k) =

(k · n

)

are

= 6|k|

, M

= 3



|k|



(13)

both exactly direction-independent, with anisotropy rst appearing at order six.

Scope.

Equation (11) is a long-wavelength statement, valid for

ka ≪ 1

. The defects of Sections

27 are lattice-scale objects at

ka ∼ 1

and are not covered by it.

10.2 The kinematic statement

A standard result.

The following is a theorem of continuum eld theory, quoted here for

reference. It is conditional, and we state its hypotheses explicitly because they carry the weight.

Proposition 9

(Relativistic transformation of a localized solution)

Let a Lorentz-invariant

continuum theory possess a static, localized, nite-energy solution with stress tensor

µν

. Static

equilibrium and localization give the von Laue condition

x = 0,

(14)

which follows from

∂

= 0

on multiplying by

and integrating. Under

(14)

the quantity

0µ

transforms as a four-vector, so for the boosted solution

E(v) = γE

rest

, M =

rest

lat

(15)

Why the hypotheses are not automatic.

The proposition presupposes that such a solution

exists. For a canonical scalar theory of the form (11) in three spatial dimensions, it generally

does not. Take the rescaling

(x) = u(λx)

. The gradient and potential contributions then scale

−1

and

−3

. The energy therefore has no stationary point at nite

, and the conguration

is unstable against collapse. This is the Derrick scaling obstruction [20]. Evading it requires

additional structure: further elds, higher-derivative terms, gauge couplings, or a conserved

charge [21]. None has been supplied here.

10.3 An illustrative example in one spatial dimension

Purpose of the example.

To show that a continuum theory of this type

can

carry relativistic

soliton kinematics, we evaluate the sine-Gordon kink, which lives in one spatial dimension and is

not subject to the obstruction above. Setting

κ = m

= a = 1

and

bind

(u) = (V

/a)(1 − cos u)

with

= 1

gives

lat

= 1

and

rest

= 8

. Integrating the boosted prole

u(x, t) = u

static

(γ(x −

vt))

gives:

β E(v)/E

rest

(Lorentz)

0.10 1.005038 1.005038

0.30 1.048285 1.048285

0.50 1.154701 1.154701

0.70 1.400280 1.400280

0.90 2.294157 2.294157

0.95 3.202563 3.202563

How to read it.

Agreement is exact to six decimal places. This is an illustration in

1 + 1

dimensions. It is not a calculation about the trapped defect, which is neither one-dimensional

nor a sine-Gordon kink, and it should not be read as numerical support for the three-dimensional

case.

10.4 Application to the trapped defect

What would be required.

To obtain

E(v) = γE

rest

for the trapped defect, three things are

needed and none is established here. An eective continuum action valid at the defect's scale,

which Eq. (11) is not, since the defect sits at

ka ∼ 1

. A static localized nite-energy solution

of that action, which the Derrick argument shows a pure scalar sector will not supply in three

dimensions. And verication of the von Laue condition (14) for it. The alternative route is to

solve for the moving defect conguration directly on the discrete lattice, without passing through

a continuum limit at all.

The relation to mass, if those conditions are met.

Suppose they are established. Section

1.1 gives

rest

= C

· kT ln 2

for each defect species [3]. Here

is the defect's topological

verication cost. Combining this with Eq. (15),

· kT ln 2

lat

(16)

This would identify inertial mass with the Landauer cost of maintaining the vacuum's topological

defect structure. We record it as a conditional consequence, not a result.

Scope across defect types.

Proposition 9 nowhere uses the defect type. What restricts its

application is the set of hypotheses above, not the identity of the defect.

11 Open Problems

Deriving the propagation rule.

Postulate 1 is assumed. A derivation from the bond

Hamiltonian would remove the only dynamical assumption in the paper and make Propo-

sition 5 and Corollary 1 unconditional. The geometry alone does not supply it: the defect

is interstitial, and its moves need not be lattice translations.

From selection to determination of spin.

Proposition 6 is a compatibility criterion.

Upgrading it to a determination needs three things: that the state is spinorial rather than

tensorial; that the whole physical state transforms irreducibly under

rather than as one

component of a decomposition; and that it extends to an irreducible representation of the

emergent rotation group with no further components.

A lattice Dirac operator.

Postulate 1 denes a discrete-time update on a two-colored

lattice. Constructing that update operator explicitly, and obtaining its spectrum, would

show whether positive- and negative-energy branches and a mass-dependent frequency

emerge. Only then could the oscillation of Section 5 be identied with zitterbewegung.

The construction is well posed. Quantum walks and cellular automata whose continuum

limit is the Dirac equation are known in one and in higher dimensions [15, 17, 18], and the

two-color structure of the void sublattice is the natural coin degree of freedom.

Kinematics of the trapped defect.

The relation

E(v) = γE

rest

holds for a localized

solution of a Lorentz-invariant continuum theory satisfying the hypotheses of Proposition

9. Extending it to the trapped defect requires either exhibiting that defect as a broad

smooth soliton, or solving for its moving conguration directly on the discrete lattice.

The physical tick.

The rule xes the rest oscillation at two updates. Converting that

to a frequency requires the lattice time unit. Fixing it from the bond Hamiltonian would

turn the zitterbewegung correspondence from a structural statement into a quantitative

one.

Mass dependence of the trembling frequency.

The Dirac frequency

= 2mc

/ℏ

depends on mass, while the rule gives every defect the same two-tick period. Either the tick

is defect-dependent, or the observed frequency is a coarse-grained quantity whose relation

to the microscopic period involves the defect's internal structure. Deciding between these

is the sharpest open question raised here.

Sublattice degeneracy and the gap.

P and N voids are related by spatial inversion

(Proposition 7), and the lattice is centrosymmetric, so the two classes are physically equiv-

alent sites and cannot carry an on-site energy dierence. A two-branch spectrum with a

gap therefore cannot come from the void class. The candidate that remains is the worldline

orientation of Section 7, which is also where

lives. Whether the Dirac two-component

structure is that doublet is open.

Does free translation x the winding labels?

Proposition 8 supplies a canonical

bijection on bounding atoms and shows that species is preserved if the winding labels

follow it equivariantly. It does not establish that they must. The obstruction is explicit:

the defect center moves by

and a released atom by

, so no defect bond is translated

rigidly and the local frame rotates. Deciding whether the denition of

in [2] forces

equivariance under free translation is the immediate question. Only after that does the

secondary one arise, of whether interactions other than free translation can change

Foliation.

The selection of

is taken as given. A mechanism, and a demonstration that

the resulting dynamics is Lorentzian rather than merely isotropic, are outside this paper.

10.

The full lattice Dirac equation.

Realizing the

-matrix algebra through lattice oper-

ations would tie the spin and zitterbewegung results to a complete relativistic dynamics.

12 Summary

Feature Lattice realization Basis

Void sublattice Simple cubic, spacing

a/2

Prop. 1

Migration alternates class Checkerboard two-coloring Prop. 3

Hop is a time-mixed bond Spatial length

√

Prop. 4

Hop carries a tick Postulate 1 Prop. 5

Rest state trembles No zero-displacement update Cor. 1

Spin

1/2

compatible Faithful

irreps have dimension 2 Prop. 6

Unique irreducible half-integer

Higher

decompose under

Prop. 6

Spin determined  Open

Two-sublattice oscillation Period two updates, amplitude

a/2

Cor. 1

Zitterbewegung proper  Open

Trembling frequency  Open

Matter / antimatter Worldline orientation

Sec. 7

Annihilation Opposite

at adjacent voids Sec. 8

Quark exchange in transport Two of four bonds redrawn per hop Sec. 9

Canonical map on bounding atoms Shared persist, released shift by

Prop. 8

Species preserved Requires equivariant winding transport Open

E = γE

rest

, continuum soliton Boosted-prole integration Eq. (15)

Same for the trapped defect  Open

Bond-set isotropy

direction-independent Eq. (13)

13 Conclusion

The tetrahedral voids of the spatial slice form a simple cubic lattice of spacing

a/2

, two-colored

as a checkerboard, with six neighbors per void. Migration between adjacent voids therefore

alternates the void class at every step. The displacement involved has spatial length

, which

among the nearest-neighbor bonds of

is realized only by the time-mixed ones.

That last fact becomes dynamical once one assumes that a worldline advances by one bond

per update. A hop then carries exactly one time step, and no update leaves a defect where it

is. A defect at rest oscillates between two adjacent voids with amplitude

a/2

and period two

updates. The assumption species a discrete-time local update, not continuous evolution under

a lattice Hamiltonian, and we do not derive it. The oscillation resembles zitterbewegung. We

stop there, because identifying the two would require a lattice Dirac operator, energy branches

and a mass-dependent frequency, none of which is obtained.

On spin, the rotational symmetry of a tetrahedral void is

∼

, and the faithful irreducible

representations of its double cover all have dimension two, which is compatible with spin one

half. Among the half-integer spins only

j =

survives restriction to that group irreducibly. This

selects rather than determines: the same two-dimensional irrep also occurs inside higher-spin

decompositions, so a nite site group cannot x the continuum parent. The void class plays no

part either way, being a parity grading.

The matter/antimatter distinction requires a grading that does not alternate. It cannot be

the void class, because each class admits exactly one bond-direction set and the void therefore

carries exactly one bit. The worldline supplies what the void cannot: its time orientation is

invariant under hopping and is exchanged by reversal. Annihilation is then the meeting of

oppositely oriented worldlines at adjacent voids, and matter is stable against contact. This

corrects the earlier static identication of inversion-related voids with matter and antimatter

[2]. Those classes are propagation sublattices, and both particles and antiparticles traverse both

of them.

A hop also induces a canonical bijection on the bounding atoms, shared atoms persisting

and released ones shifting by a lattice translation. Baryon species is preserved if the winding

labels follow that bijection, which we state as an assumption rather than derive: the defect

center moves by half the atom displacement, so no defect bond is carried rigidly. Conserving

the anchor instead is excluded, because it would conne the defect to a single octant.

What remains open is set out in Section 11. The propagation rule is assumed rather than

derived. The oscillation has no rate, since the duration of an update is not xed, and it carries

the same period for every defect while the Dirac frequency depends on mass. The spin result

is a criterion rather than a determination. And the soliton kinematics hold only for a localized

solution meeting the stated hypotheses, not for the trapped defect itself. Each is a denite

calculation rather than a matter of principle.

Data Availability

Two les accompany this paper and are also supplied as ancillary les. The verication script is

https://github.com/raghu91302/ssmtheory/blob/main/d4_hopping_verify.py

and the

visualization source at

https://github.com/raghu91302/ssmtheory/blob/main/d4_hopping

.html

. Appendix A lists what the script checks. No external data are used.

Declaration of Competing Interest

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

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

A Computational Verication

Every geometric claim in this paper is veried by a single self-contained script,

d4_hopping_verify.py

available at

https://github.com/raghu91302/ssmtheory/blob/main/d4_hopping_verify.py

and supplied as an ancillary le. It requires only NumPy and SymPy and runs in under a

minute. It performs twelve checks in order:

1. the spatial slice of

is FCC;

2. the void enumeration and Proposition 1;

3. the adjacency data of Section 2, namely the coordination number, the uniform hop dis-

tance, the shared-atom count and the bond rearrangement;

4. the closed form (4) for

, and Proposition 3;

5. Proposition 7;

6. the hop parity of Proposition 4;

7. Corollary 1;

8. the moments (13);

9. the binary tetrahedral group, Proposition 6, and the multiplicity of its two-dimensional

irrep inside higher-spin restrictions;

10. the metric-wall channel count of Proposition 2: four faces closed, six edges open;

11. the anchor and valence-bond bookkeeping under migration used in Section 9;

12. the octant-trapping property of an anchor-conserving rule, and the bond bijection of Propo-

sition 8.

Each check prints

FAIL

against the value the text claims. All thirty-eight pass.

References

[1] M. Schlosshauer,

Decoherence, the measurement problem, and interpretations of quantum

mechanics

, Rev. Mod. Phys.

, 1267 (2005).

doi:10.1103/RevModPhys.76.1267

[2] R. Kulkarni,

Matter as Incomplete Crystallization: Quark Charges, Color Connement, and

the Proton Mass from a Single Extra Node in the Vacuum Lattice

, Phys. Open

, 100423

(2026).

doi:10.1016/j.physo.2026.100423

[3] R. Kulkarni,

The MassEnergyInformation Equivalence: A bottom-up identication of

the particle spectrum via FCC lattice error correction

, Phys. Open

, 100414 (2026).

doi:10.1016/j.physo.2026.100414

[4] R. Kulkarni,

A 67%-rate CSS code on the FCC lattice:

[[192, 130, 3]]

from weight-12 stabi-

lizers

(2026).

arXiv:2603.20294

[5] STAR Collaboration,

Tracking the baryon number with nuclear collisions

, Science (2026).

doi:10.1126/science.ads5962

;

arXiv:2408.15441

[6] J. H. Conway and N. J. A. Sloane,

Sphere Packings, Lattices and Groups

, Springer, 3rd ed.

(1999).

doi:10.1007/978-1-4757-6568-7

[7] H. S. M. Coxeter,

Regular Polytopes

, Dover (1973).

[8] N. W. Ashcroft and N. D. Mermin,

Solid State Physics

, Saunders College (1976).

[9] P. A. M. Dirac,

The quantum theory of the electron

, Proc. R. Soc. A

117

, 610 (1928).

doi:10.1098/rspa.1928.0023

[10] E. Schrödinger,

Über die kräftefreie Bewegung in der relativistischen Quantenmechanik

, Sitz.

Preuss. Akad. Wiss.

, 418 (1930).

[11] E. H. Lieb and D. W. Robinson,

The nite group velocity of quantum spin systems

, Commun.

Math. Phys.

, 251 (1972).

doi:10.1007/BF01645779

[12] C.-F. A. Chen, A. Lucas and C. Yin,

Speed limits and locality in many-body quantum

dynamics

, Rep. Prog. Phys.

, 116001 (2023).

doi:10.1088/1361-6633/acfaae

[13] T. Farrelly,

A review of Quantum Cellular Automata

, Quantum

, 368 (2020).

doi:10.22331/q-2020-11-30-368

[14] P. Arrighi,

An overview of quantum cellular automata

, Natural Computing

, 885 (2019).

doi:10.1007/s11047-019-09762-6

[15] I. Bialynicki-Birula,

Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular

automata

, Phys. Rev. D

, 6920 (1994).

doi:10.1103/PhysRevD.49.6920

[16] A. Bisio, G. M. D'Ariano and A. Tosini,

Dirac quantum cellular automaton in one di-

mension: Zitterbewegung and scattering from potential

, Phys. Rev. A

, 032301 (2013).

doi:10.1103/PhysRevA.88.032301

[17] P. Arrighi, V. Nesme and M. Forets,

The Dirac equation as a quantum walk:

higher dimensions, observational convergence

, J. Phys. A

, 465302 (2014).

doi:10.1088/1751-8113/47/46/465302

[18] G. M. D'Ariano and P. Perinotti,

Quantum cellular automata and free quantum eld theory

Front. Phys.

, 120301 (2017).

doi:10.1007/s11467-016-0616-z

[19] S. Miyashita et al.,

Digital quantum simulator for the time-dependent Dirac equation using

discrete-time quantum walks

(2023).

arXiv:2305.19568

[20] G. H. Derrick,

Comments on nonlinear wave equations as models for elementary particles

J. Math. Phys.

, 1252 (1964).

doi:10.1063/1.1704233

[21] N. Manton and P. Sutclie,

Topological Solitons

, Cambridge University Press (2004).

doi:10.1017/CBO9780511617034