Geometric Origins of Spin-1/2 and Relativistic Kinematics:
Defect Migration in the FCC Vacuum Lattice
Raghu Kulkarni
*
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
Draft manuscript May 2026
Abstract
We identify a non-trivial geometric structure in the Face-Centered Cubic (FCC, K=12)
vacuum lattice of the Selection-Stitch Model: tetrahedral void defects can migrate to ad-
jacent void sites only by edge-adjacent hopping (sharing two bounding atoms), and the
resulting adjacency graph on the eight tetrahedral voids per unit cell is exactly bipartite:
every hop necessarily ips a positive-orientation void (P) to a negative-orientation void
(N), or vice versa. We verify this computationally: each tet void has exactly three edge-
adjacent neighbors at uniform hop distance
a/2
, all of opposite orientation, with zero same-
orientation edge-adjacencies anywhere in the unit cell. Three structural correspondences
follow between this bipartite adjacency and features of relativistic quantum mechanics: the
720
◦
rotational periodicity of spinors maps onto the geometric inversion required to tra-
verse P
→
N
→
P; Dirac's zitterbewegung maps onto the mandatory P
↔
N alternation during
defect propagation; and matterantimatter annihilation maps onto the meeting of opposite-
orientation defects at adjacent voids, with shared-edge cancellation restoring local K=12
saturation. Finally, we derive
E = M c
2
lat
for any topologically stable defect in the lattice's
Lorentz-invariant continuum limit by direct integration of the energy-momentum tensor on a
Lorentz-contracted soliton prole, obtaining
E(v) = γE
rest
across the full relativistic range.
The result is species-agnostic and applies uniformly to K=4 trapped tet defects (baryonic
matter) and K=6 trapped oct defects (dark matter). Combined with the MassEnergy
Information equivalence, this yields a single unied mass formula
M
x
= C
x
·kT ln 2/c
2
lat
for
every defect species in the framework, identifying inertial mass with the literal Landauer
thermodynamic cost of maintaining the vacuum's topological defect structure.
1 Introduction
The quantum mechanical description of matter (wave-particle duality, the uncertainty principle,
spin-
1
2
statistics, the Dirac equation) is empirically exact but lacks a settled geometric origin. The
wave function
ψ
is treated as a fundamental object whose physical meaning remains debated [1].
The de Broglie relation
λ = h/p
is postulated, not derived. Spin-
1
2
is introduced as an abstract
representation of SU(2), formally disconnected from spatial geometry.
The Selection-Stitch Model (SSM) identies the physical vacuum with a discrete tensor
network on the FCC lattice [2], the densest sphere packing in three dimensions [6]. In the SSM,
baryonic matter is identied with K=4 tetrahedral remnants trapped in the K=12 FCC bulk
during incomplete crystallization; the proton's mass-to-electron ratio
m
p
/m
e
= (K + 1)K
2
−
c
skew
K = 13 · 144 − 3 · 12 = 1836
follows directly from this identication [2]. The underlying
quantum error-correcting code is a
[[192, 130, 3]]
weight-12 CSS code on the same lattice [5].
*
raghu@idrive.com
1
In this paper we focus on a non-obvious geometric structure of the tetrahedral-void sublattice
and its consequences for relativistic quantum mechanics. We identify a bipartite adjacency on
tet voids under edge-adjacent hopping, verify it computationally, and examine three structural
correspondences with relativistic quantum phenomena: spin-
1
2
rotational periodicity, Dirac zit-
terbewegung, and matterantimatter annihilation. The bipartite adjacency provides a geometric
template that connects directly to the standard tight-binding description of a hopping defect,
and to the spin-
1
2
representation theory of the rotation group. We outline natural extensions of
the framework in Section 8.
1
2 Tetrahedral Void Adjacency
2.1 Void enumeration
The FCC conventional cubic cell of side
a
contains 4 lattice sites (one at the origin, three at face
centers) and 8 tetrahedral interstitial voids [7, 8]. The voids partition by orientation:
Positive (P) tetrahedra
, centered at
(a/4)(1, 1, 1)
and its three permutations. The four
bounding atoms point along the
[111]
family.
Negative (N) tetrahedra
, centered at
(a/4)(3, 3, 3)
and its three permutations. The
four bounding atoms point along the
[
¯
1
¯
1
¯
1]
family.
A regular tetrahedron and its inversion through its body center are related by a
180
◦
orien-
tation ip; this ip is the geometric content of the P/N labels.
2.2 Hopping by edge-adjacent void pairs
A defect occupying tet void
T
with bounding atoms
{A, B, C, D}
can migrate to an adjacent
tet void
T
′
that shares one tetrahedral edge with
T
, i.e. exactly two bounding atoms. Two of
T
's four bounding atoms remain (the shared edge), and two are replaced by the two bounding
atoms of
T
′
that are not shared:
{A, B, C, D} −→ {A
′
, D
′
, B, C}
(1)
where
{B, C}
is the shared edge and
{A, D} → {A
′
, D
′
}
are the swapped pairs. We refer to this
as
edge-adjacent
hopping.
During such a hop the bond rearrangement is:
5 bonds among
{A, B, C, D}
involving
A
or
D
are broken (the 3 bonds
A
-
{B, C, D}
plus
the 2 bonds
D
-
{B, C}
, since the bond
A
-
D
is counted once);
5 bonds among
{A
′
, D
′
, B, C}
involving
A
′
or
D
′
are created;
the single bond
B
-
C
is preserved.
There is also a parallel adjacency relation in which two voids share a single bounding vertex
(and lie at distance
a
√
2/2
apart). In the FCC unit cell this vertex-adjacent relation always
connects voids of the same orientation (P-P or N-N) and is not edge-adjacent. We focus on
edge-adjacency throughout, since this is the channel through which a defect's local coordination
structure is most coherently preserved.
1
An interactive 3D visualization that walks through the seven-step geometric construction of this paper, from
the K=12 FCC vacuum lattice through the bipartite tet-void adjacency, the spin-
1
2
correspondence, and shared-
edge annihilation, is available at
https://raghu91302.github.io/ssmtheory/ssm_qm_hopping_visualization.
html
.
2
A
D
B
C
A'
D'
Edge-adjacent hop: T
P
(red) and T
N
(blue) share edge BC; atoms swap
©
A; D
ª
!
©
A
0
; D
0
ª
Shared edge BC
Trapped K=4 node
Shared atoms (kept)
Released atoms (A; D)
Engaged atoms (A
0
; D
0
)
Hop distance jT
P
¡ T
N
j = a=2. Bond rearrangement: 5 broken, 5 created, 1 (BC) preserved.
Figure 1: Edge-adjacent hopping: a defect's tet void
T
P
(left, red, P-orientation) and its post-
hop image
T
N
(right, blue, N-orientation) share two bounding atoms B, C (one tetrahedral edge),
while the other two bounding atoms are exchanged:
{A, D} → {A
′
, D
′
}
. The trapped K=4 node
(yellow star) sits at the centroid of each tetrahedron. The hop distance is
|T
P
− T
N
| = a/2
,
identical for all 12 such adjacencies in the FCC unit cell, and every edge-adjacent hop ips
orientation P
→
N (or N
→
P).
2.3 Computational verication
Direct enumeration over the 8 tet voids of the FCC unit cell (reproduced as runnable code in
Appendix A) yields:
1. Each tet void has exactly
3
edge-adjacent neighbors.
2. All adjacencies are of
opposite orientation
: every P-void's three neighbors are N-voids,
and vice versa. There are
zero
same-orientation adjacencies in the unit cell.
3. All hops have
uniform distance
d
hop
= a/2
.
4. The adjacency graph is
bipartite
, with the two parts being the four P-voids and the four
N-voids.
These facts are veriable in under one second by the code in Appendix A and follow from
FCC crystallography alone. Item (2), the bipartite structure, is the central geometric input for
the rest of this paper.
Every edge-adjacent hop in the FCC tet-void sublattice ips orientation P
↔
N. (2)
3
0.0
0.5
1.0
x / a
0.0
0.5
1.0
y / a
0.0
0.5
1.0
z / a
(a) FCC unit cell
P-orientation tet void N-orientation tet void
P
1
P
2
P
3
P
4
N
1
N
2
N
3
N
4
P-orientation N-orientation
12 edges, all P↔N
Each void: 3 neighbors
0 same-orientation edges
(b) Bipartite adjacency graph
Figure 2: The FCC tet-void sublattice and its bipartite adjacency.
(a)
The FCC conventional
cubic cell contains 4 P-orientation tet voids (red triangles, pointing up) and 4 N-orientation tet
voids (blue triangles, pointing down). FCC lattice atoms are shown as small dark spheres at the
cube corners and face centers. Green edges connect all 12 edge-adjacent pairs; every such pair
has opposite orientation.
(b)
The same adjacency graph drawn as a bipartite graph between
the four P-voids and four N-voids. The graph is regular of degree 3: each void has exactly three
opposite-orientation neighbors, and there are no same-orientation edges.
3 Tight-Binding on the Bipartite Sublattice
Given the bipartite adjacency (2), the natural description of a single defect hopping coherently
through the lattice is the standard tight-binding Hamiltonian [8]:
iℏ
∂ψ
n
∂t
= E
0
ψ
n
− t
X
m∈N(n)
ψ
m
,
(3)
where
ψ
n
is the amplitude for the defect to occupy site
n
,
E
0
is the on-site energy,
t
is the
hopping amplitude, and
N(n)
is the set of three opposite-orientation neighbors of
n
. In the
long-wavelength continuum limit this reduces to the Schrödinger equation,
iℏ
∂ψ
∂t
= −
ℏ
2
2m
eff
∇
2
ψ + V ψ,
(4)
with eective mass
m
eff
determined by the curvature of the tight-binding band, and the potential
V
encoding site-energy modulation from external elds or other defects. The de Broglie relation
λ = h/p
likewise emerges as a standard property of the tight-binding band dispersion, with the
lattice constant
a
setting the natural length scale.
The bipartite adjacency on the tet-void sublattice is therefore compatible with the standard
quantum-mechanical description of a hopping defect: the geometry supports the Schrödinger
equation as its long-wavelength continuum limit and reproduces the de Broglie dispersion. The
substantive open question, whether the Schrödinger equation can be derived from a more fun-
damental classical lattice dynamics with
ℏ
and the imaginary unit emerging rather than being
inputs, is a natural target for further work.
4
4 Spin-1/2 from Mandatory Orientation Alternation
The strongest structural correspondence the bipartite adjacency oers is a geometric route to
the
720
◦
rotational periodicity of spin-
1
2
wave functions.
4.1 Per-hop orientation ip
By Eq. (2), every edge-adjacent hop ips defect orientation
P ↔ N
. A coherent propagating
defect therefore alternates orientation at every step of its trajectory:
P → N → P → N → ···
(5)
This is a
geometric
fact about the lattice: there is no adjacency channel that preserves orienta-
tion, so the alternation cannot be circumvented by any local hopping mechanism.
4.2 Geometric inversion and the spinor phase
A regular tetrahedron and its inversion through the body center dier by a parity ip; geomet-
rically, rotating a tetrahedron by
360
◦
about a body axis is equivalent to inverting it (mapping
P
→
N or N
→
P), not to bringing it back to its starting orientation. A second
360
◦
rotation
(total
720
◦
) returns the tetrahedron to its original orientation.
If we identify the wave function's transformation under spatial rotation with the geometric
orientation of the underlying tet-void defect, then:
R(360
◦
)ψ = −ψ
(one P
↔
N ip, phase
e
iπ
= −1)
(6)
R(720
◦
)ψ = +ψ
(two ips, phase
(−1)
2
= +1)
(7)
This is the dening property of spin-
1
2
particles [9].
Note on the identication.
The geometric content of the lattice is that one P
↔
N ip
is the orientation eect of one
360
◦
rotation. The further identication of this ip with the
spinor phase factor
e
iπ
= −1
uses the standard representation theory of the rotation group on
the wave function: the bipartite adjacency naturally produces a
Z
2
phase per hop, and the
spinor representation is the unique faithful representation in which a
Z
2
orientation ip acts
as
ψ → −ψ
. The geometric template for spin-
1
2
is provided by the lattice; the representation-
theoretic identication makes the structure operative.
5
+Ã
0
±
¡Ã
360
±
(one inversion)
+Ã
720
±
(two inversions)
R(360
±
): P ! N flip, Ã ! e
i¼
à = ¡Ã
R(360
±
): N ! P flip, Ã ! (¡1)
2
à = +Ã
White-rim marker: tracer vertex showing the geometric inversion. Gold star: trapped K=4 node at the centroid.
Figure 3: Geometric origin of the
720
◦
rotational periodicity of spin-
1
2
wave functions.
Left:
P-orientation tetrahedron at
0
◦
, with phase
+ψ
.
Center:
after a
360
◦
rotation, the tetrahedron
has inverted to N-orientation (one P
↔
N ip), and the wave function picks up a phase factor
e
iπ
= −1
.
Right:
after a second
360
◦
rotation (total
720
◦
), the tetrahedron has inverted again
back to P-orientation, and the wave function returns to
+ψ
. The yellow star marks the trapped
K=4 node at the centroid; the geometric inversion maps centroid to centroid, so the trapped
node sits at the xed point of the rotation throughout.
4.3 Discreteness of the lattice and absence of intermediate spins
The FCC unit cell contains exactly two tetrahedral orientations (P and N). A particle whose spin
character was tied to defect orientation would therefore admit only a
Z
2
-valued internal structure,
matching spin-
1
2
but excluding fractional values intermediate between 0 and
1
2
. Lorentz-invariant
quantum eld theory similarly excludes such intermediate values, so this is not a new prediction;
it is a consistency between the lattice geometry's discreteness and the established representation
theory of the Lorentz group.
5 Zitterbewegung as P
↔
N Alternation
A free Dirac electron exhibits zitterbewegung: a high-frequency oscillation between positive and
negative energy states at frequency
ω
Z
= 2m
e
c
2
/ℏ
, rst identied by Schrödinger [10] in his
analysis of the Dirac equation [9]. The two states involved in the oscillation are eigenstates of
the Dirac Hamiltonian with opposite-sign energy eigenvalues; their interference produces the
trembling motion.
The bipartite adjacency (2) suggests a structural mapping: the two energy eigenstates of the
Dirac oscillation correspond to the two orientation classes (P and N) of the tet-void sublattice,
and the zitterbewegung frequency corresponds to the rate at which a propagating defect alter-
nates orientation. Each hop is one P
↔
N transition; a sequence of hops at hopping rate
ν
hop
produces alternation at the same rate.
From qualitative to quantitative.
The qualitative correspondence (two-state alternation
during free propagation, mapped onto two orientation classes of the underlying lattice) is a
structural feature of the bipartite adjacency. The quantitative correspondence, in which the
rest-frame hopping rate would equal
2mc
2
/ℏ
and reproduce the Dirac zitterbewegung frequency
exactly, would follow from a derivation of the rest-frame hopping rate from the bond Hamiltonian
of the FCC lattice. Such a derivation is a natural extension of the SSM framework [2], which
6
already identies the defect's rest mass with its static topological verication cost via Landauer's
principle and the MassEnergyInformation equivalence [3].
6 Annihilation as Shared-Edge Cancellation
A second structural correspondence concerns matterantimatter annihilation. In the SSM, bary-
onic matter is identied with K=4 defects in the K=12 bulk, and antimatter with the inverted-
orientation counterpart [2]. We can identify matter with P-orientation defects and antimatter
with N-orientation defects without loss of generality.
When a P-defect and an N-defect occupy adjacent voids that share an edge, the local K=4
decit 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 disappear from the lattice. The
total stored energy, equal to the sum of the two defects' disrupted-bond contributions, is released
into outgoing lattice excitations.
Energy partition.
The released energy
E
annihilation
= 2 N
disrupted
× E
bond
partitions among
outgoing lattice excitations. Identifying which lattice modes carry which fraction of this energy,
and how this relates to the empirical
p¯p
annihilation spectrum, is a natural extension of the
model. The geometric content here is the mechanism: opposite-orientation defects at adjacent
(shared-edge) voids mutually cancel and release their stored verication energy as lattice waves.
The bipartite adjacency (2) ensures that any two defects close enough to share an edge are
automatically of opposite orientation, so spatial proximity is sucient for annihilation.
7 Derivation of
E = mc
2
for Solitonic Defects
The relation
E = mc
2
holds for any topologically stable defect in the lattice's Lorentz-invariant
continuum limit, by direct integration of the energy-momentum tensor. The K=4 tetrahedral
defect of the matter paper [2] is one such topological defect; the calculation below applies to
it once the specic solitonic prole is specied, and reproduces the standard relativistic energy
formula
E(v) = γE
rest
, from which
E = mc
2
follows by identifying
M = E
rest
/c
2
lat
.
7.1 The bond Hamiltonian and its Lorentz-invariant continuum limit
A nearest-neighbor harmonic FCC bond Hamiltonian augmented by a nonlinear binding poten-
tial takes the schematic form
H
bond
=
X
i
1
2m
s
p
2
i
+
X
⟨i,j⟩
1
2
κ (|r
i
− r
j
| − L)
2
+
X
i
V
bind
(u
i
),
(8)
where
m
s
is the inertial mass at each lattice site,
κ
is the NN bond spring constant,
L
is the
equilibrium NN distance, and
V
bind
is a nonlinear potential that admits topologically stable
defect congurations of the displacement eld
u
i
. The binding
V
bind
is what makes a localized
K=4 decit energetically favorable against being absorbed into the K=12 bulk.
In the long-wavelength continuum limit, Eq. (8) produces an eective Lagrangian density
L =
m
s
2a
3
(∂
t
u)
2
−
κ a
−1
2
(∇u)
2
− a
−3
V
bind
(u),
(9)
which, after rescaling, is Lorentz-invariant with signal speed
c
2
lat
=
κ a
2
m
s
.
(10)
7
Lorentz invariance of Eq. (9) is exactly the structure tensor identity
S
µν
= 4 δ
µν
established in
the matter paper [2]: that identity is the statement that the harmonic part of
H
bond
has an
isotropic continuum limit of the form (9).
7.2 Static defect energy
A static topological defect
u
static
(x)
is a solution of
δL/δu = 0
with
∂
t
u = 0
. Its rest energy is
the integral of the static Hamiltonian density,
E
rest
=
Z
d
3
x
κ a
−1
2
(∇u
static
)
2
+ a
−3
V
bind
(u
static
)
.
(11)
This integral is nite and positive for any topologically stable defect; its specic value depends
on
V
bind
and the defect type.
7.3 Moving defect energy by direct integration
A defect moving with velocity
v
is described by the Lorentz-boosted prole
u(x, t) = u
static
Λ
−1
v
(x − vt)
, γ =
1
q
1 − v
2
/c
2
lat
,
(12)
where the prole is contracted by
γ
along the direction of motion. For motion along
ˆ
x
, this gives
∂
t
u = −γv u
′
static
and
∂
x
u = γ u
′
static
(with
u
′
static
= ∂
s
u
static
evaluated at
s = γ(x − vt)
), so the
moving Hamiltonian density is
H
moving
= γ
2
(u
′
static
)
2
m
s
v
2
2a
3
+
κ a
−1
2
+ a
−3
V
bind
(u
static
).
(13)
Using
c
2
lat
= κa
2
/m
s
to rewrite
m
s
v
2
/a
3
= (κa
−1
) β
2
with
β = v/c
lat
, and integrating over
x
at
xed
t
(Jacobian
dx = ds/γ
along the boost direction,
dy dz
unchanged),
E(v) =
Z
d
3
x H
moving
=
1
γ
Z
d
3
s
γ
2
(1 + β
2
)
κ a
−1
2
(u
′
static
)
2
+ a
−3
V
bind
(u
static
)
.
(14)
For any topological soliton of Bogomolnyi type, the gradient and potential integrals satisfy
equipartition:
R
κ a
−1
2
(u
′
static
)
2
d
3
s =
R
a
−3
V
bind
d
3
s = E
rest
/2
. Substituting,
E(v) =
1
γ
γ
2
(1 + β
2
)
E
rest
2
+
E
rest
2
=
E
rest
2γ
γ
2
(1 + β
2
) + 1
.
(15)
The bracket simplies using
γ
2
= 1/(1 − β
2
)
:
γ
2
(1 + β
2
) + 1 =
1 + β
2
1 − β
2
+ 1 =
(1 + β
2
) + (1 − β
2
)
1 − β
2
=
2
1 − β
2
= 2γ
2
,
(16)
so
E(v) =
E
rest
2γ
· 2γ
2
= γ E
rest
.
(17)
The kinetic and gradient
β
2
contributions cancel against the prole-contraction Jacobian, leav-
ing the standard relativistic energy
E(v) = γ E
rest
. For non-Bogomolnyi defects the gradient and
potential integrals individually depend on the prole, but the Lorentz invariance of the under-
lying Lagrangian guarantees the same nal result
E(v) = γ E
rest
by general energymomentum
tensor covariance.
8
7.4 Identication of the inertial mass
Expanding Eq. (17) for small
β
,
E(v) = E
rest
+
1
2
E
rest
c
2
lat
v
2
+ O(v
4
),
(18)
identies the inertial mass
M =
E
rest
c
2
lat
(19)
as the coecient of
1
2
v
2
in the kinetic-energy expansion. Equivalently, the relativistic energy
E(v) = γMc
2
lat
at
v = 0
gives
E
rest
= Mc
2
lat
.
7.5 Numerical verication
We verify Eq. (17) numerically for the canonical topological soliton, the
1+1
D sine-Gordon
kink, for which the calculation can be done in closed form. Setting
κ = m
s
= a = 1
and
V
bind
(u) = (V
0
/a)(1 − cos u)
with
V
0
= 1
gives
c
lat
= 1
, soliton width
ξ = 1
, and analytical rest
energy
E
rest
= 8
√
κV
0
= 8
. Numerical integration of the moving-soliton Hamiltonian density at
velocities
v = β c
lat
yields:
β E(v)/E
rest
(numerical)
γ
(Lorentz) agreement
0.1 1.005038 1.005038 exact to 6 d.p.
0.3 1.048285 1.048285 exact to 6 d.p.
0.5 1.154701 1.154701 exact to 6 d.p.
0.7 1.400280 1.400280 exact to 6 d.p.
0.9 2.294157 2.294157 exact to 6 d.p.
0.95 3.202563 3.202563 exact to 6 d.p.
The relation
E(v) = γE
rest
holds exactly across the full relativistic range, conrming that
E = Mc
2
lat
is a property of the lattice's Lorentz-invariant continuum limit applied to topological
solitons, not an externally imposed equivalence.
7.6 Application to the K=4 tet defect
The K=4 trapped tet defect of the matter paper [2] is a 3D topological defect of the FCC
structure. The argument of Sections 7.17.3 applies to it directly: its rest energy is the inte-
gral (11) over the FCC displacement eld, and Eq. (19) identies the inertial mass from the
Lorentz-invariant continuum limit. The MassEnergyInformation (MEI) framework's identi-
cation
E
rest
= C
x
·kT ln 2
[3] provides the value of
E
rest
for each defect species (electron, muon,
pion, proton, neutron, K=6 dark matter defect [4]) in terms of the topological verication cost
C
x
. Combining this with Eq. (19),
M
x
=
C
x
· kT ln 2
c
2
lat
.
(20)
The lattice signal speed
c
lat
, identied with the empirical speed of light through the matter
paper's emergent Lorentz invariance, closes the chain:
E
rest
= Mc
2
, with both sides computed
independently from the same underlying bond Hamiltonian.
Physical interpretation.
Equation (20) identies inertial mass with the Landauer thermody-
namic cost of maintaining the vacuum's topological defect structure. The numerator
C
x
·kT ln 2
is the minimum free energy required, by Landauer's principle, to verify (or equivalently, to erase
upon annihilation) the
C
x
bits of topological information that distinguish a particle of species
9
x
from the saturated K=12 vacuum. The denominator
c
2
lat
converts that energetic cost into
inertia through the soliton-kinematics relation
E = Mc
2
lat
established in Sections 7.17.3. Mass
is therefore neither postulated nor a free parameter in this framework; it is a derived thermo-
dynamic quantity equal to the bookkeeping cost the vacuum pays to keep a topological defect
distinct from its surroundings.
8 Natural Extensions
The bipartite adjacency identied here, taken together with the structural correspondences of
Sections 46 and the
E = mc
2
derivation of Section 7, provides a geometric template on which
a microscopic derivation of relativistic quantum mechanics in the SSM framework can be built.
Four extensions remain open:
1.
Hopping amplitude
t
from the FCC bond Hamiltonian.
The tight-binding ampli-
tude
t
in Eq. (3) governs the inertial response of the defect through the band-curvature
relation
m
eff
∝ 1/ta
2
. A derivation of
t
from the energy cost of edge-adjacent vertex ex-
change, computed against the bond Hamiltonian of the K=12 bulk, would tie the eective
mass directly to the underlying crystallography.
2.
The rest-frame hopping rate.
The matter paper [2] relates the defect's rest mass
to its static topological cost via Landauer's principle and the MassEnergyInformation
equivalence [3]. Bridging this static cost to a dynamical hopping rate
ν
hop
= 2mc
2
/ℏ
would
quantitatively reproduce Dirac zitterbewegung from the P
↔
N alternation.
3.
The lattice signal speed
c
.
The matter paper [2] establishes emergent Lorentz invari-
ance through the structure tensor identity
S
µν
= 4 δ
µν
and the centrosymmetry-cancelled
odd tensor
T
µνλ
= 0
, giving an exactly isotropic linear dispersion
ω = c
lat
|k|
at long wave-
lengths. Pinning
c
lat
to a microscopic FCC bond-vibration scale would make the empirical
c
a derived quantity rather than a parameter inherited from the dispersion's slope.
4.
The full lattice Dirac equation.
The two-component P/N alternation provides a ge-
ometric realization of the Dirac spinor's two-component structure. Writing down the full
lattice Dirac equation, including the realization of the
γ
-matrix algebra through FCC crys-
tallographic operations, would tie the spin-
1
2
structural correspondence of Section 4 to a
complete relativistic dynamics on the lattice.
The structural facts of Sections 26 and the
E = mc
2
derivation of Section 7 are independent
of extensions 14: bipartite adjacency is a geometric fact about FCC, the spin-
1
2
structural corre-
spondence rests on the tetrahedron's inversion-rotation relationship, the annihilation mechanism
is a local geometric picture, and
E = mc
2
follows from the lattice's Lorentz-invariant continuum
limit applied to topological solitons. These provide the rm ground on which the extensions
above can be developed.
10
9 Summary of Correspondences
Quantum feature Lattice realization Basis
Single-particle wave function
ψ
n
Defect amplitude on tet-void site
n
Tight-binding
Schrödinger equation Continuum limit of Eq. (3) Tight-binding
de Broglie
λ = h/p
Tight-binding band dispersion Tight-binding
Spin-
1
2
rotation:
R(720
◦
) = +1
Two P
↔
N ips Geometric
R(360
◦
) = −1
One P
↔
N ip + spinor rep. Geometric + rep. theory
Zitterbewegung
ω
Z
= 2mc
2
/ℏ
P
↔
N alternation rate Geometric (qualitative)
Annihilation
P +
¯
P
Shared-edge cancellation to K=12 Geometric mechanism
E = mc
2
M = E
rest
/c
2
lat
from soliton kinematics Sec. 7
Lorentz invariance Emergent from
S
µν
= 4δ
µν
[2] Cited from matter paper
Speed of light
c
Lattice signal speed
c
lat
Extension (Sec. 8)
Full Dirac equation Two-component lattice Dirac Extension (Sec. 8)
Table 1: Structural correspondences between FCC tetrahedral-void defect migration and features
of relativistic quantum mechanics. The Basis column indicates whether the entry follows from
geometry of the FCC lattice (this paper), from the standard tight-binding Hamiltonian on the
bipartite adjacency graph, from the
E = mc
2
derivation of Section 7, from the matter paper [2],
or from one of the natural extensions outlined in Section 8.
10 Conclusion
The FCC tetrahedral-void sublattice has a non-trivial bipartite structure under edge-adjacent
hopping: every hop necessarily ips orientation P
↔
N, with each void having exactly three
opposite-orientation neighbors at uniform distance
a/2
. This is a veriable geometric fact about
the FCC lattice (Appendix A).
Three structural correspondences follow directly from this bipartite structure: the
720
◦
ro-
tational periodicity of spin-
1
2
wave functions maps onto the two-inversion structure of the P
↔
N
ip; Dirac zitterbewegung maps onto P
↔
N alternation during defect propagation; and matter
antimatter annihilation maps onto the meeting of opposite-orientation defects at adjacent voids,
with shared-edge cancellation restoring local K=12 saturation.
Section 7 derives
E = Mc
2
lat
for any topologically stable defect in the lattice's Lorentz-
invariant continuum limit, by direct integration of the energy-momentum tensor on a Lorentz-
contracted soliton prole. The result
E(v) = γE
rest
holds across the full relativistic range
(veried to six decimal places at
β = 0.1, 0.3, 0.5, 0.7, 0.9, 0.95
for the canonical sine-Gordon
test case) and is species-agnostic: the same formula applies to the K=4 trapped tet defects
identied with baryonic matter and to the K=6 trapped oct defects identied with the dark-
matter sector [4]. Combined with the MassEnergyInformation equivalence [3]'s identication
E
rest
= C
x
· kT ln 2
, this gives a unied mass formula
M
x
= C
x
· kT ln 2/c
2
lat
for every defect
species, identifying inertial mass with the literal Landauer thermodynamic cost the vacuum pays
to maintain its topological defect structure.
The bipartite adjacency, the structural correspondences, and the
E = M c
2
lat
derivation
together provide a geometric template on which a microscopic derivation of relativistic quantum
dynamics in the SSM framework can be built. Section 8 outlines four remaining extensions:
derivation of the hopping amplitude from the FCC bond Hamiltonian, of the rest-frame hopping
rate, of the lattice signal speed
c
as a microscopic quantity, and of the full lattice Dirac equation.
The structural template identied here motivates that program.
11
Data Availability
The geometric facts asserted in Section 2 (bipartite adjacency, three neighbors per void, uniform
hop distance
a/2
, zero same-orientation hops) are veried by the Python script in Appendix A,
which is self-contained and runs in under one second on a standard interpreter. 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 inuence the work reported in this paper.
A Computational verication of FCC tet-void adjacency
The following self-contained Python script enumerates the 8 tetrahedral voids of the FCC con-
ventional cubic cell, computes the edge-adjacency graph by checking shared bounding atoms,
and veries the four claims of Section 2. It requires only NumPy and runs in under one second.
import numpy as np
a = 1.0 # lattice constant
# FCC basis atoms in conventional cubic cell
basis = np.array([
[0, 0, 0 ],
[0.5, 0.5, 0 ],
[0.5, 0, 0.5],
[0, 0.5, 0.5]
]) * a
# 4 positive tetrahedral voids (centers along [111] family)
tet_pos = np.array([
[0.25, 0.25, 0.25],
[0.75, 0.75, 0.25],
[0.75, 0.25, 0.75],
[0.25, 0.75, 0.75]
]) * a
# 4 negative tetrahedral voids (centers along [-1,-1,-1] family)
tet_neg = np.array([
[0.75, 0.75, 0.75],
[0.25, 0.25, 0.75],
[0.25, 0.75, 0.25],
[0.75, 0.25, 0.25]
]) * a
all_tets = np.vstack([tet_pos, tet_neg])
types = ['P']*4 + ['N']*4
def get_bounding_atoms(center):
"""Return the set of FCC atoms at distance a*sqrt(3)/4 from center."""
verts = []
for dx in range(-1, 2):
for dy in range(-1, 2):
for dz in range(-1, 2):
for b in basis:
pos = np.array([dx, dy, dz]) * a + b
if abs(np.linalg.norm(pos - center) - a*np.sqrt(3)/4) < 0.05:
verts.append(tuple(np.round(pos, 6)))
12
return set(verts)
# Build adjacency: voids share edge iff they share 2 bounding atoms
adjacency = {}
for i in range(8):
vi = get_bounding_atoms(all_tets[i])
neighbors = []
for j in range(8):
if i == j:
continue
vj = get_bounding_atoms(all_tets[j])
if len(vi & vj) >= 2:
d = np.linalg.norm(all_tets[i] - all_tets[j])
neighbors.append((j, types[j], d))
adjacency[i] = neighbors
# Verification
n_per_void = [len(adjacency[i]) for i in range(8)]
same_hops = sum(1 for i in adjacency
for n in adjacency[i] if n[1] == types[i]) // 2
opp_hops = sum(1 for i in adjacency
for n in adjacency[i] if n[1] != types[i]) // 2
hop_dists = sorted({round(n[2]/a, 6)
for i in adjacency for n in adjacency[i]})
is_bipartite = all(n[1] != types[i]
for i in adjacency for n in adjacency[i])
print(f"Neighbors per void: {n_per_void} (expected: all 3)")
print(f"Same-orientation hops: {same_hops} (expected: 0)")
print(f"Opposite-orientation hops: {opp_hops} (expected: 12)")
print(f"Unique hop distances (a): {hop_dists} (expected: [0.5])")
print(f"Bipartite (P<->N only): {is_bipartite}")
Expected output:
Neighbors per void: [3, 3, 3, 3, 3, 3, 3, 3] (expected: all 3)
Same-orientation hops: 0 (expected: 0)
Opposite-orientation hops: 12 (expected: 12)
Unique hop distances (a): [0.5] (expected: [0.5])
Bipartite (P<->N only): True
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Decoherence, the measurement problem, and interpretations of quantum mechan-
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Matter as Incomplete Crystallization: Quark Charges, Color Connement, and the
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13
