One Chiral Bit: The Electroweak Structure of the
Selection–Stitch Vacuum from e
6
Raghu Kulkarni
SSMTheory Group, IDrive Inc., Calabasas, CA 91302, USA
raghu@idrive.com ORCID 0009-0008-8364-7155
September 2026
Abstract
In the Selection–Stitch Model the vacuum is a face-centered-cubic lattice carrying a stabilizer
code. Its bonds and matter close into e
6
, where the lattice’s own charge conjugation and parity
select a unique electric-charge operator. Its symmetries do not select handedness. We take
handedness as one binary input, the choice of weak factor, and derive its consequences by root-
system methods, checked by a verification script. An appendix asks how the lattice could supply
the input. The input fixes the weak group up to a relabeling of states and gives hypercharge as
Y = Q − T
3
. The 27 becomes one Standard Model generation, with its right-handed singlets and
the vector-like states of trinification, and all anomalies cancel. The lattice’s flip symmetry forces
equal couplings for the two non-color factors, which gives sin
2
θ
W
=
3
8
at the breaking scale. The
boson masses cannot be local verification costs. In the appendix, one-way layer growth traps
every tetrahedral node from the growing surface in the void class pointing along the growth, and
this class is conserved along D
4
worldlines, given the propagation postulate of earlier work. If our
observable universe grew one way, handedness follows from its growth direction. The next test is
a breaking scale near 10
13
GeV.
Keywords: E
6
trinification; chirality; weak mixing angle; left–right symmetry; hypercharge;
anomaly cancellation; D
4
root lattice; face-centered-cubic lattice; Selection–Stitch Model
1 Introduction
In the Selection–Stitch Model the vacuum is an FCC lattice. It grows by two moves: a stitch, which
builds flat sheets, and a rare lift, which stacks them [1]. Appendix A.2 summarizes this growth
picture. The lattice carries a [[192, 130, 3]] CSS stabilizer code [2]. Matter is a node trapped in a
tetrahedral void [1]. A trapped defect moves along six edge channels [4]. In Ref. [4], each hop changes
the class of its void. That result assumes a fixed spatial slice. Appendix A shows that along physical
worldlines the class is conserved. The lattice’s four-dimensional home is the D
4
root lattice. Its bond
algebra so(8) and the matter weights close into f
4
under a triality twist. Doubling the matter weights
by the void-orientation label gives e
6
. Ref. [5] presents both steps. In e
6
the lattice’s own charge
conjugation and parity select a unique electric-charge operator. This operator gives the Standard
Model charges, including the integer unit [5].
That program leaves handedness open. The orientation flip fixes color and exchanges the other two
su(3) factors of the trinification decomposition. The lattice treats these two factors alike. So nothing
in its structure says which one contains the weak interaction. Handedness is a hard problem on any
lattice. The Nielsen–Ninomiya theorem forbids net chirality for local Hermitian lattice actions [6].
1
Ginsparg–Wilson fermions [7, 8] and domain-wall fermions [9] realize chiral fermions on a lattice.
They impose the handedness. They do not explain it. In this framework, a verification cost that
treats the creation and destruction of bonds alike gives reciprocal hops at any fixed frontier. So it
cannot prefer a handedness either.
We therefore take handedness as an input. We keep it small: one bit, stated once, at the level of
e
6
. Appendix A shows that directed growth of the lattice selects one void class for trapped matter.
The class is conserved along physical worldlines, given the propagation postulate of Ref. [4]. So the
selection supplies the bit, under the stated conditions. The Standard Model likewise takes its chiral
field content as an input. This paper derives what the bit unlocks. That includes the weak group,
the hypercharge, and a complete generation with its right-handed singlets. It also includes anomaly
cancellation, the mixing angle at the breaking scale, and the breaking sector. Once the structure
and the bit are given, these results are those of e
6
trinification [10, 11, 12, 13]. We label them as
inherited. The framework contributes four things. It supplies the e
6
structure. It fixes the charge
operator through the lattice’s C and P . It links the input to the lattice’s orientation label. Its flip
symmetry sets the boundary condition that gives sin
2
θ
W
=
3
8
.
Scripts check every claim (see Data availability). Each claim carries a status tag. Derived means
it follows here from the stated inputs. Inherited means it is a standard result of e
6
trinification,
reproduced here. Imported means it comes from a cited paper of the series. Assumed marks an input
not derived here, chiefly the chiral bit. Open means it is not settled here. The appendix uses its own
labels, stated there.
2 The inherited structure
The root system. We use coordinates R
4
bond
⊕ R
2
class
. The 24 bond roots ±e
i
± e
j
of D
4
are joined
by the 24 short matter weights of 8
v
⊕ 8
s
⊕ 8
c
. Each matter weight is lifted twice, as (s, ±v
class
(s)).
The three glue-class vectors are v
v
= (1, 0), v
s
= (−
1
2
,
√
3
2
) and v
c
= (−
1
2
, −
√
3
2
). They sit at 120
◦
[5].
The 72 vectors are closed under reflection and span rank 6. We use the simple roots of Ref. [5]:
α
1
= (0, 1, −1, 0; 0, 0), α
2
= (0, 0, 1, −1; 0, 0),
α
3
= (
1
2
, −
1
2
, −
1
2
,
1
2
; −
1
2
,
√
3
2
), α
4
= (
1
2
, −
1
2
, −
1
2
,
1
2
;
1
2
, −
√
3
2
),
α
5
= (−
1
2
,
1
2
,
1
2
,
1
2
; −
1
2
, −
√
3
2
), α
6
= (−
1
2
,
1
2
,
1
2
,
1
2
;
1
2
,
√
3
2
).
(1)
The Cartan matrix has determinant 3, and α
2
is the trivalent node. The highest root is θ =
2α
1
+ 3α
2
+ 2α
3
+ 2α
4
+ α
5
+ α
6
. The system is E
6
. [imported; verified]
Three su(3) factors and the flip. Set α
0
= −θ. Deleting the mark-3 node α
2
from the affine
diagram leaves three mutually orthogonal A
2
subsystems (Figure 1):
su(3)
c
= ⟨α
0
, α
1
⟩, su(3)
A
= ⟨α
3
, α
5
⟩, su(3)
B
= ⟨α
4
, α
6
⟩. (2)
The orientation flip reverses the class-plane coordinates and fixes the bond roots. It fixes su(3)
c
and
exchanges su(3)
A
with su(3)
B
. The fixed factor is color [5]. All three factors have the same root
length. So their embedding indices in e
6
are equal. Color is built only from bond roots of D
4
. The
other two factors are built only from matter roots. [imported; verified; the bond and matter content
is derived]
The charge operator. Let {ω
i
} be the coweights dual to the simple roots. The charge operator
selected by the lattice’s geometric C and P is [5]
Q = ±
2
3
ω
2
− ω
3
− ω
4
+ ω
5
+ ω
6
. (3)
2
α
2
α
1
α
0
α
3
α
5
α
4
α
6
color: flip-fixed
su(3)
A
su(3)
B
orientation flip
delete the mark-3 node α
2
:
e
6
⊃ su(3)
c
⊕ su(3)
A
⊕ su(3)
B
the chiral bit: weak su(2) ⊂ su(3)
A
Figure 1: The affine E
6
diagram in the simple roots of Ref. [5]. Removing the trivalent node leaves
three A
2
arms. The orientation flip fixes the color arm and exchanges the other two. The flip-
symmetric structure of the lattice does not prefer either of them (Appendix A). Choosing the one
that contains the weak interaction is the one input of this paper.
We take the minus sign. This sign gives the quark doublet positive hypercharge (Section 4). Q
commutes with color and is invariant under the flip. In each of su(3)
A
and su(3)
B
, two of the three
root pairs carry charge ±1, and one is neutral. The 27 is the Weyl orbit of ω
5
, at the end of the
su(3)
A
arm. The orbit of ω
6
, at the end of the su(3)
B
arm, is the 27. On the 27, Q takes the values
±
2
3
and ±
1
3
on the colored states. On the color singlets it takes −1 twice, 0 five times, and +1 twice.
[imported; verified]
3 The chiral input
The two long-arm ends of the diagram carry the 27 and the 27. The orientation flip exchanges them,
together with the two non-color factors. Spatial inversion acts on the gauge labels by this flip. It
also exchanges left- and right-handed fields. The one input of this paper is the choice between the
two worlds that inversion relates.
Input (one bit). The weak su(2) lies in su(3)
A
, not in su(3)
B
. The left-handed Weyl fields of a
generation are in the 27. [assumed]
Appendix A shows how directed growth could supply this bit. Its last step, conservation of the void
class along worldlines, rests on the propagation postulate of Ref. [4]. The main text keeps the bit as
an input, so that its results do not depend on the conditions of the appendix.
Only the pairing of factor and representation matters. Relabeling by the flip maps (su(3)
A
, 27) to
(su(3)
B
, 27). This is the same world, described in other labels. The other pairing is (su(3)
B
, 27),
which equals (su(3)
A
, 27). It is the mirror world. Its quark doublet has Y = −
1
6
, the quantum
numbers of left-handed antiquarks. If the representation is held fixed, the bit is the choice of factor.
If the factor is held fixed, the bit is the choice between the 27 and the 27. [derived; verified]
In the lattice this bit is tied to the void orientation. Spatial inversion exchanges the two worlds, and
it also exchanges the two void orientations [5]. So the input has a clear place in the lattice, even
though the lattice does not choose it. Appendix A shows that the orientation is a conserved label
3
along a worldline, given the propagation postulate of Ref. [4].
Proposition 1 (The weak group) Assume the input. Take a weak su(2) inside su(3)
A
such that
Y = Q − T
3
commutes with it and with color. Then it is generated by a charged root pair, with
Q · β = ±1. There are two such pairs, ±α
3
and ±α
5
. Both give a Y that commutes with su(2) and
with color. Both give the same spectrum of (color, T
3
, Q, Y ) on the 27. So the input fixes the weak
group up to a relabeling of states.
The neutral pair is excluded. For it, Y would not commute with the weak group. Commutation
requires (Q −
1
2
β) · β = 0. For the positive root this means Q · β = 1. The two charged choices are
related by a relabeling of states with identical Standard Model quantum numbers. This ambiguity
is familiar in trinification. We take β = α
3
, for which Q · α
3
= +1. We set T
3
=
1
2
β as a coweight
and Y = Q − T
3
. [derived; verified]
4 One generation
Table 1 lists the 27 as left-handed Weyl fields under su(3)
c
⊕ su(2)
L
⊕ u(1)
Y
. It uses the charge
operator of Section 2 and the T
3
and Y of Section 3.
Table 1: The 27 of e
6
with the chiral input. The counts include color. Every value is computed from
the weights. The identification column is the standard one.
color T
3
Q Y states identification role
3 +
1
2
+
2
3
+
1
6
3
Q
L
= (u
L
, d
L
) quark doublet
3 −
1
2
−
1
3
+
1
6
3
¯
3 0 −
2
3
−
2
3
3 u
c
conjugate of u
R
¯
3 0 +
1
3
+
1
3
3 d
c
conjugate of d
R
1 ±
1
2
0, −1 −
1
2
2 L = (ν
L
, e
L
) lepton doublet
1 0 +1 +1 1 e
c
conjugate of e
R
1 0 0 0 1 ν
c
conjugate of ν
R
3 0 −
1
3
−
1
3
3 D vector-like
¯
3 0 +
1
3
+
1
3
3 D
c
vector-like
1 ±
1
2
+1, 0 +
1
2
2 H
u
Higgs-type
1 ±
1
2
0, −1 −
1
2
2 H
d
Higgs-type
1 0 0 0 1 N singlet
Theorem 1 (One generation) Assume the input and set Y = Q − T
3
. Then the 27 contains
exactly one Standard Model generation: Q
L
at Y = +
1
6
, u
c
at −
2
3
, d
c
at +
1
3
, L at −
1
2
, and e
c
at +1.
It also contains a right-handed neutrino ν
c
and the vector-like states D + D
c
, H
u
+ H
d
and N of e
6
trinification. The right-handed fields appear as the conjugates u
c
, d
c
, e
c
and ν
c
, with the Standard
Model hypercharges.
Three identifications in the table are conventions, not computations. One is which Y = +
1
3
antitriplet
is d
c
and which is D
c
. Another is which Y = −
1
2
doublet is L and which is H
d
. The third is which
neutral singlet is ν
c
and which is N . The two states in each pair have identical gauge quantum
4
numbers. Telling them apart needs the Yukawa couplings and the breaking pattern. We do not
construct those here. [derived (theorem); verified; the content is inherited from trinification]
5 Consistency
Anomalies. We treat the 27 as left-handed Weyl fields. Then
Tr Y = Tr Y
3
= Tr Y T
2
su(2)
= Tr Y T
2
su(3)
= 0. (4)
So the gravitational, cubic and mixed hypercharge anomalies vanish [14, 15]. The pure color anomaly
cancels between three triplets and three antitriplets. There are six weak doublets, an even number,
so there is no global SU (2) anomaly [16]. All gauge and gravitational anomalies therefore vanish. e
6
itself has no anomalies, so this result is guaranteed. It checks the charge operator and the input. It
is not independent evidence. [derived; verified; inherited]
Normalization. Over the 27, Tr T
2
3
= 3, Tr Y
2
= 5 and Tr Q
2
= 8. So the hypercharge has the
unified normalization Tr Y
2
/Tr T
2
3
=
5
3
. [derived; verified; inherited]
The mixing angle. The hypercharge has no color part. It splits between the two non-color factors.
With g
2
= g
A
, its coupling obeys
1
g
2
Y
=
1/3
g
2
A
+
4/3
g
2
B
. (5)
This is the standard trinification relation. [derived; verified; inherited]
Proposition 2 (The weak boundary condition) Where the orientation flip is unbroken, g
A
=
g
B
. Then
sin
2
θ
W
=
Tr T
2
3
Tr Q
2
=
3
8
, (6)
for any value of the color coupling g
c
. The gauge fields and complete 27 fermion multiplets preserve
the equality under running. So it persists down to the scale where SU(3)
A
× SU(3)
B
breaks, provided
the scalar and threshold sector also respects the flip up to that scale.
Spatial inversion is a symmetry of the lattice and the code (Appendix A). It acts on the gauge labels
by the flip, which exchanges su(3)
A
and su(3)
B
. So the two couplings must be equal wherever this
symmetry holds. The condition matters. A 20% difference between g
A
and g
B
moves sin
2
θ
W
to 0.29
or 0.44. The flip cannot also tie g
c
to the weak couplings. Color is built only from bond roots, and
the other two factors only from matter roots (Section 2). So no lattice symmetry maps color onto a
weak factor. The value
3
8
is the familiar unified value [17]. What is new is its source. In trinification
the condition g
A
= g
B
is usually imposed as a discrete left–right parity. Here the lattice’s own
flip supplies it. The proviso matters. Suppose scalar thresholds broke the flip above the breaking
scale. Then they could split g
A
from g
B
, even if the two couplings start equal in the ultraviolet. So
the breaking sector must respect the flip down to that scale. This is a concrete condition on the
scalar sector, which is open here. The growth direction of Appendix A also breaks the flip, but only
through where matter is trapped, which is an initial condition. We assume that it does not enter the
gauge dynamics, so it does not split g
A
from g
B
. [derived; verified; the scalar condition is open; the
separation of trapping from the gauge dynamics is assumed]
6 The breaking sector
Two stages. The 27 has what both stages of breaking need. Color stays unbroken. SU(3)
A
×SU(3)
B
breaks to SU (2)
L
×U (1)
Y
through expectation values of the two neutral singlets N and ν
c
. Both carry
5
Q = T
3
= Y = 0. Electroweak breaking then needs a doublet with a neutral member. The photon
stays massless only if it annihilates the condensate. So that member must have Q = T
3
+ Y = 0,
which forces Y = ±
1
2
. The doublets of the 27 with a neutral member are exactly those with Y = ±
1
2
:
H
u
, H
d
and L. [derived; verified; the breaking pattern is inherited]
Statistics. In non-supersymmetric trinification the breaking fields are scalars in separate 27 multi-
plets. This framework has not yet fixed the exchange statistics of its defects. So it is open whether
the Higgs-type states of Table 1 are fermions or scalars, or whether they condense through a scalar
bilinear. [open]
The boson masses are not local counts. Suppose the W and Z masses were local verification
costs of the code, in the sense of Ref. [3]: C = E
s
× C
s
. The natural unit is a square plaquette of the
photon channels. On the code it has E
s
= 4 edges and C
s
= 9 detecting checks (four vertex and five
octahedral), so C = 36. The largest local electromagnetic-sector count is one plaquette per square
face of the coordination shell, which gives 216. The measured ratios are m
W
/m
e
= 1.573 × 10
5
and
m
Z
/m
e
= 1.785 × 10
5
[19]. They exceed these counts by factors of 700 to 5 000. So the boson masses
must come from the breaking scale. They cannot come from local verification cost. [derived; verified]
7 Next step: running from the breaking scale
Let M
T
be the scale where SU(3)
A
× SU(3)
B
breaks to the electroweak group. By Proposition 2,
sin
2
θ
W
=
3
8
at M
T
, provided the scalar and threshold sector respects the flip down to M
T
. The
measured value at the Z mass is sin
2
θ
W
≈ 0.231. Reaching it requires running from M
T
down
to the electroweak scale. At M
T
the condition g
A
= g
B
means α
1
= α
2
in unified normalization.
Color does not need to meet them. With Standard Model content alone, one-loop running makes
α
1
and α
2
meet at about 1.0 × 10
13
GeV. So the framework needs M
T
near that scale, shifted by
thresholds. The thresholds are the vector-like states D, D
c
, H
u
and H
d
, and the extra gauge bosons
of SU(3)
A
× SU(3)
B
. Each changes the running above its mass. The singlets ν
c
and N do not. The
lattice does not yet fix M
T
or these masses. The next computation is the running with these states
as thresholds. It asks whether the framework’s own scale for M
T
lands where α
1
and α
2
meet. This
is the first place where the framework could make a quantitative electroweak prediction. [open; the
Standard Model crossing scale is derived]
8 Status of the claims
Table 2 gives the status of every claim. The main assumption of the main text is the chiral bit.
Everything marked inherited is standard e
6
trinification, reproduced from the lattice’s own structure
and charge operator.
9 Outlook
The main text takes the bit as an input. Appendix A shows how the growth direction of the lattice
could supply it. No flip-invariant dynamics can explicitly lift the degeneracy between the two worlds,
and the site symmetry of a void forbids a spin–hop helicity coupling. The growth direction is the
only fixed background that tells the two void classes apart. In layer-by-layer growth in one direction,
a node can be trapped only in the class that points along it. So directed growth selects one class
for trapped matter, under two conditions. Growth runs one way across our patch, which holds if
our observable universe lies far from the seed, the point where growth began and spread in both
6
Table 2: Status of every claim. Derived: follows here from the stated inputs and is checked by the
script. Inherited: a standard result of e
6
trinification, reproduced. Imported: from a cited paper of
the series. Assumed: an input not derived here. Conditional: derived in Appendix A from stated
conditions. Open: not settled here.
statement status
E
6
from the D
4
lift; simple roots; highest root imported; verified
three su(3) factors; the flip fixes color and exchanges A and B imported; verified
charge operator Q from the lattice’s C and P imported; verified
equal embedding indices of the three factors derived
the flip exchanges 27 and 27 derived
weak su(2) ⊂ su(3)
A
, left-handed fields in the 27 assumed (one bit)
the other pairing is the mirror world, with Y (Q
L
) = −
1
6
derived
weak su(2) fixed up to relabeling by [Y, su(2)] = 0 derived
27 = one SM generation + ν
c
+ D, D
c
, H
u
, H
d
, N, SM hypercharges derived; inherited
d
c
versus D
c
, L versus H
d
, ν
c
versus N open (needs Yukawas)
all gauge and gravitational anomalies vanish derived; inherited
Tr Y
2
/Tr T
2
3
=
5
3
; 1/g
2
Y
= (1/3)/g
2
A
+ (4/3)/g
2
B
derived; inherited
the flip forces g
A
= g
B
, so sin
2
θ
W
=
3
8
for any g
c
derived
color cannot be tied to the weak factors (bond versus matter roots) derived
gauge fields and complete 27 multiplets run g
A
and g
B
alike derived
scalar and threshold sector respects the flip down to M
T
open (condition)
the growth direction does not enter the gauge dynamics assumed
two neutral singlets for the first breaking stage derived; inherited
electroweak condensate needs Y = ±
1
2
; H
u
, H
d
, L qualify derived
statistics of the Higgs-type states open
boson masses are not local verification costs derived
Standard Model α
1
and α
2
meet near 10
13
GeV derived
running from M
T
with vector-like thresholds open
directed growth selects one void class for trapped matter conditional (App. A)
the selected class persists, supplying the bit conditional (App. A)
7
directions (Appendix A.2). Matter is trapped from the growing surface. The selection supplies the
bit if a worldline keeps its void class. Only the D
4
reading of the time slices fits the D
4
bonds, and
under it the class is conserved, given the propagation postulate of Ref. [4]. A second test is a growth
simulation that builds the crystal layer by layer in one direction, with realistic defects.
Apart from the input, the next quantitative step is the running of Section 7.
10 Conclusion
The Selection–Stitch lattice supplies e
6
and a unique electric-charge operator. Its symmetries do
not supply handedness. We take handedness as one bit: the weak interaction lies in one of the two
flip-exchanged factors, and the left-handed fields are in the 27. This bit fixes the weak group up to
relabeling. It gives the hypercharge as Y = Q−T
3
. It turns the 27 into one complete Standard Model
generation, with its right-handed singlets and the vector-like states of trinification. The anomalies
cancel. The hypercharge has the unified normalization. The lattice’s flip forces equal couplings for
the two non-color factors, and that gives a mixing angle of
3
8
at the breaking scale. The breaking
fields are present, and the boson masses must come from the breaking scale. The algebra after
the input is standard. The framework’s contribution is the source of the structure, the charge, the
input, and the condition behind
3
8
. Appendix A shows that directed growth selects one void class for
trapped matter. The class is conserved along physical worldlines, given the propagation postulate of
Ref. [4], so the selection supplies the bit under the stated conditions. Its next test is whether its own
breaking scale lands near 10
13
GeV, where α
1
and α
2
meet.
A The growth direction and the chiral bit
This appendix asks what the framework can say about where the input of Section 3 comes from.
It first summarizes the growth picture it uses. It then proves one constraint, rules out the simplest
candidate, and identifies the fixed background from which a selection could come. Finally, it shows
that the void class is conserved along physical worldlines, and that this background selects one void
class for trapped matter. Each statement carries one of six labels. Established means proved here and
checked by the script. Measured means found in the growth simulation. Conditional means derived
here from conditions stated with it. Assumed marks an assumption that is not derived. Imported
means taken from a cited paper. Open means not settled here.
A.1 What must be derived
The input chooses between two worlds that spatial inversion exchanges. With the left-handed fields
in the 27, the weak su(2) lies either in su(3)
A
or in su(3)
B
(Section 3). The same inversion also
exchanges the two void orientations. A mechanism must pick one world, either by preferring it or by
choosing it spontaneously.
A.2 The growth picture
The rest of this appendix uses the growth picture of Ref. [1], which we summarize here. The vacuum
is built from nodes joined by bonds of one fixed length L. Growth starts from a single bond, and
two moves add nodes (Figure 2).
A stitch places a new node at the apex of an equilateral triangle on an existing edge, bonded to the
two ends of that edge. Repeated stitches grow a flat, triangulated sheet. Each interior node of a
8
(a)
stitch: two bonds,
stays in the sheet
(b)
p
2/3 L
lift: three bonds,
starts the next sheet
Figure 2: The two growth moves of Ref. [1]. Gray nodes exist; the teal node is added, and dashed
lines are its new bonds, all of length L. (a) A stitch bonds a new node to both ends of an edge,
in the plane of the sheet. (b) A lift bonds a new node to the three corners of a triangle, at height
p
2/3 L above its center. Lifts are rare. Stitches build flat sheets with K = 6, and lifts stack them.
sheet has six neighbors (K = 6). A lift places a new node above the center of an existing triangle, at
height
p
2/3 L, bonded to its three corners. A lift leaves the plane and starts the next sheet. Lifts
are rare: the simulation of Ref. [1] takes a lift probability of e
−3
≈ 5% per step. In that simulation,
two nodes may not come closer than 0.95L, and nodes within 1.05L of each other are bonded.
Where lifts pile up, nodes form tetrahedra, and each node has four neighbors (K = 4). Regular
tetrahedra cannot fill space, so this K = 4 phase is a frustrated foam and does not last. Where
sheets stack and interlock, the lattice settles into the face-centered-cubic arrangement. Each node
then has twelve neighbors (K = 12), and the sheets stack in the order ABC along one axis, the
stacking axis. The simulation of Ref. [1] reproduces this: its bulk reaches K = 12, with ABC
stacking and the ideal spacing between sheets. The simulation code is published with that paper.
We use four more terms. The crystal grows outward from where it began, which we call the seed.
A lift can go to either side of a sheet, so growth proceeds in both directions from the seed. The
growing boundary of the crystal is the front, and the direction in which it advances locally is the
growth direction. Behind the front the lattice has crystallized. Ahead of it lies what is left of the
K = 4 foam, which we call the frozen foam. A node of that foam that does not join the crystal
can be caught in a tetrahedral void of the lattice, bonded to the four atoms around it. We call it a
remnant node. Ref. [1] identifies such a trapped node with matter. [imported]
A.3 The variables
The framework supplies five discrete ingredients at a defect. The first is the void class σ = ±1. The
second is the worldline time orientation ε = ±1 [4]. The third is the spin-
1
2
doublet selected by the
double cover of the site group [4]. The fourth is the direction
ˆ
v of a hop. The fifth is the growth
direction
ˆ
g of the front (Section A.2). [imported]
A.4 A symmetry constraint
Proposition 3 (Degeneracy) Spatial inversion maps the D
4
lattice to itself at every time slice. It
maps the code’s vertex and octahedral checks to themselves, and it exchanges the two void classes.
Let F be the operator that implements it. F maps the states of one world onto those of the other. Let
H be any dynamics built only from the lattice, the code and the void geometry. Then F HF
−1
= H.
We call dynamics with this property flip-invariant, since F acts on the gauge labels by the flip. It
follows that F H
A
F
−1
= H
B
for the restrictions of H to the two worlds, and
Spec(H
A
) = Spec(H
B
). (7)
9
So no such dynamics can explicitly lift the degeneracy or prefer one world. This holds whether H
is Hermitian or not, since a similarity transformation preserves the spectrum. It also holds with or
without a verification front that treats the two classes alike.
The proposition does not rule out a choice between the two degenerate worlds. A flip-invariant
H can have two degenerate ground states, and the system can settle into one of them. Selection
therefore requires one of three things: a flip-odd term in the dynamics, a flip-odd order parameter
chosen spontaneously, or a flip-asymmetric initial condition. [established; the action of inversion on
the gauge labels by the flip is imported from Ref. [5]]
A.5 Spin–hop couplings at a void
The natural candidate couples the spin to the direction of motion, H
χ
= λ σ ·
ˆ
v. This coupling is odd
under parity, because
ˆ
v is polar and σ is axial. Its sign would select a helicity. The site symmetry
decides whether it is allowed.
Proposition 4 (Allowed couplings) The site group of a tetrahedral void is T
d
. It has order 24
and 12 improper elements, and it does not contain inversion. Under it
ˆ
v transforms as T
2
and σ as
T
1
. The product T
1
⊗ T
2
contains no invariant. Hence:
1. No spin–velocity bilinear is allowed. In particular, σ ·
ˆ
v is odd under every improper site
element. It is invariant only under the chiral subgroup T .
2. No coupling quadratic in
ˆ
v is allowed either. The lowest allowed spin–hop coupling is cubic,
and it is unique at that order:
H
D
= γ
σ
x
v
x
(v
2
y
− v
2
z
) + σ
y
v
y
(v
2
z
− v
2
x
) + σ
z
v
z
(v
2
x
− v
2
y
)
. (8)
It is T
d
-invariant and odd under inversion. It is the analogue of the Dresselhaus coupling of
zincblende crystals [18].
3. H
D
carries no helicity. The trace Tr [(σ ·
ˆ
v)H
D
] vanishes identically in
ˆ
v.
So a helicity selector of the form σ ·
ˆ
v needs a local environment with no mirrors. The void (T
d
)
has mirrors. So does the trapped defect with its anchor bond (C
3v
). The coupling that the site does
allow splits spin states, but it does not prefer a helicity. [established]
A.6 The action of inversion
Inversion carries a void of one class onto a void of the other, and it reverses H
D
. So the coefficient
has opposite signs on the two classes: γ
−
= −γ
+
. The same inversion exchanges the two worlds of
Section 3. So the sign of the spin–hop coupling that a defect feels and the choice of world change
together. The two labels are tied. By Proposition 3 they are tied, but flip-invariant dynamics does
not prefer either value. [established; the action of inversion on the gauge labels is imported]
A.7 The void class along a D
4
worldline
Proposition 5 (Class along a worldline) In D
4
the slice at time t consists of the spatial points
whose coordinate sum has the parity of t. So consecutive slices are the two FCC cosets. Three facts
follow. A fixed void changes class at every tick. A spatial hop within a frozen slice changes class,
which is the rule of Ref. [4]. A time-mixed hop preserves the class. Such a hop is a spatial step
together with one tick, and the propagation rule of Ref. [4] makes every hop of this kind.
10
The two readings give different answers. In a frozen slice the class alternates along a worldline, as
Ref. [4] concludes. In the D
4
slices it is conserved. The next result decides between them.
Proposition 6 (The reading of the time slices) Suppose every time slice were the same FCC
lattice. Then each of the twelve time-mixed D
4
bonds, starting from an atom, would end on an
octahedral site, not on an atom. Every D
4
root joins two vacuum nodes only if consecutive slices are
the two FCC cosets.
The series treats all 24 roots of D
4
as bond directions of the vacuum [5]. The migration rule of Ref. [4]
also depends on them: a hop is a time-mixed D
4
bond, so every hop carries one tick. Both require
the D
4
reading, and the frozen slice does not describe physical worldlines. Under the D
4
reading
the class is conserved at every hop. A defect at rest oscillates between two adjacent voids [4], and
it keeps one class at every tick. The only dynamical input is the propagation rule of Ref. [4]: each
update advances the worldline by one D
4
bond. That rule is a postulate there. So class conservation
rests on that postulate, not on a choice of reading. It holds for the six edge channels, which are
the migration channels of Ref. [4]. A move along a purely spatial bond would flip the class, but
no migration channel is of that kind. [established; conservation is conditional on the propagation
postulate of Ref. [4]]
By the previous subsection, the sign of the spin–hop coupling is then conserved too. So each worldline
carries a fixed class, and parity reverses it. This revises one conclusion of Ref. [4], which argued that
the class cannot carry a conserved label because it flips at every hop. The class still cannot be
the matter–antimatter label: voids that are neighbors in one slice have opposite classes, so bound
matter would annihilate [4]. Instead it is a conserved label that parity reverses, the natural carrier of
handedness. The time orientation ε of the worldline remains the matter–antimatter label [4]. Charge
conjugation combined with parity then reverses both labels, just as CP relates left-handed matter
to right-handed antimatter in the Standard Model. [conditional]
A.8 Directed growth selects one class
Proposition 7 (Growth distinguishes the classes) Take the growth direction
ˆ
g = [111]. The
subgroup of the FCC point group that preserves
ˆ
g is C
3v
, of order 6. It maps each void class to itself.
Every element of the point group that exchanges the classes reverses
ˆ
g.
So in a crystal that grows along
ˆ
g, the two classes are not equivalent. One class points its apex along
the growth direction. The other points it against. This agrees with Proposition 3, because
ˆ
g is itself
the flip-odd ingredient. [established]
Trapping at a growing surface. In the growth picture of Section A.2, a remnant node of the
frozen foam is trapped when the growing crystal covers it. Suppose growth proceeds layer by layer
in one direction. Below we assume this holds across our observable universe, a patch far from where
growth began. Before the next layer arrives, a remnant node on the surface can rest only where the
surface holds it (Figure 3).
Proposition 8 (Surface trapping) Take a growing (111) surface, and keep a remnant node at
least 0.612L from every atom, its closest approach in a void. The node has three bonds in each of
the two kinds of three-fold hollow, at a depth of a quarter of the layer spacing. Atop a single atom it
has only one bond. When the next layer arrives, a node in the hollow that the layer covers ends in a
tetrahedral void whose apex points along
ˆ
g. A node in the hollow that the layer leaves open ends in
an octahedral void. This holds for both stacking choices of the next layer. A tetrahedral void whose
apex points against
ˆ
g is centered above a single atom of the grown layer, where no node can be held.
11
So every node trapped in a tetrahedral void points along the growth direction. [established]
×
(a)
top view of the surface layer
hollow covered by the next layer: tetrahedral trap, apex along
ˆ
g
hollow left open: becomes an octahedral void
×
atop an atom: one bond, a node cannot be held
an atom of the next layer
grown layer
next layer
held in a hollow,
capped: apex along
ˆ
g
×
apex against
ˆ
g:
center above one atom,
nothing can wait there
ˆ
g
(b)
side view (schematic)
Figure 3: Trapping at a surface that grows along
ˆ
g. (a) Top view of the surface layer. A remnant
node is held in either kind of three-fold hollow, but not atop an atom. The next layer covers one
kind of hollow and leaves the other open. (b) Side view, schematic. A node held in a covered hollow
is capped by the arriving atom and ends in a tetrahedral void whose apex points along
ˆ
g. A void
whose apex points against
ˆ
g is centered above one atom. No node can wait there before the layer
above closes it.
Interactive simulation. An interactive simulation shows this process in a growing patch. It starts
from three FCC sheets and grows them in one direction through the frozen K = 4 foam. As the front
reaches each band of foam, its remnant nodes settle into surface sites. The next sheet then caps the
nodes in covered hollows. The simulation keeps a running count of voids formed and nodes trapped.
It is available at raghu91302.github.io/ssmtheory/ssm one sided growth.html.
The count of voids. The number of voids tells the same story from the other side. In the growth
simulation of Ref. [1], described in Section A.2, with its rules unchanged, the two classes exist in
nearly equal numbers. Deep in the bulk, the class that points along the local growth direction exceeds
the other by only 3.8% ± 0.7%. So the selection does not come from the number of voids. It comes
from which voids can capture a node. That simulation grows in both directions with random lift
directions, so it does not model a front that moves one way. [measured]
Our patch and the seed. In the growth picture of Section A.2, the crystal grows from a seed, and
growth spreads from it in both directions. We assume that our observable universe is a small patch
of a much larger universe, far from that seed. Across such a patch, growth ran in a single direction
ˆ
g. This assumption is not derived here. [assumed]
Two consequences follow. First, Proposition 8 applies across the whole patch, so the patch has one
handedness. Second, on the far side of the seed, growth ran the opposite way, so that side has the
opposite handedness. The universe as a whole then consists of two domains, which meet at the seed.
This agrees with Proposition 3. The dynamics prefers neither world, and globally both occur. Each
side is fixed by its own growth direction, which acts as an initial condition. The assumption also
predicts that no boundary between the two handedness domains crosses our patch. That agrees with
the uniform handedness of the weak interaction that we observe. [conditional]
12
From selection to the bit. Suppose three things hold. Growth runs in one direction across our
patch. Matter consists of nodes trapped from the growing surface. A worldline keeps its void class,
which follows from Propositions 5 and 6 given the propagation postulate of Ref. [4]. Then all matter
sits in the class that points along the growth direction. So the handedness of the weak interaction
is fixed by the direction in which the vacuum crystallized. The link between the void class and the
choice of world uses the action of inversion on the gauge labels, imported from Ref. [5]. [conditional]
Under these conditions the physical content of the bit follows: all matter has one handedness, the
same across our patch. Which of the two worlds we live in is set by the sign of the growth direction,
an initial condition. That is the expected form of the answer. The label is a convention, since “left”
is defined by the weak interaction itself. Of the three conditions, the first is the patch assumption
above. The second belongs to the growth picture of Section A.2. The third carries the weight. The
first two only select a class at the moment of trapping. The third keeps it: by Proposition 6, the
class is conserved along physical worldlines, given the propagation postulate of Ref. [4]. The fate of
nodes trapped in octahedral voids is also open.
A.9 Summary
Table 3 collects the statements of this appendix.
Table 3: Status of the statements of Appendix A.
statement level
the flip exchanges su(3)
A
and su(3)
B
, and 27 and 27 established
spatial inversion exchanges the two worlds established
flip-invariant dynamics cannot lift the degeneracy of the two worlds established
site group T
d
; no spin–velocity bilinear; σ ·
ˆ
v forbidden established
no coupling below cubic order; H
D
unique; no helicity established
H
D
changes sign with the void class; tied to the choice of world established
class flips per tick and per frozen-slice hop; preserved per D
4
hop established
growth direction makes the two classes inequivalent established
spin-
1
2
from the site group; inversion acts on gauge labels by the flip imported
only the D
4
reading fits the D
4
bonds established
class conserved along physical worldlines conditional (postulate)
a remnant node is held only in hollows, not atop an atom established
the covered hollow traps along
ˆ
g; the other is octahedral established
void counts nearly balanced: along-growth excess 3.8% ± 0.7% in the bulk measured
fate of nodes trapped in octahedral voids open
our observable universe is a patch far from the seed, grown one way assumed
the far side of the seed carries the opposite handedness conditional
one-way growth and surface trapping select one class for trapped matter conditional
the selected class persists along worldlines, supplying the bit conditional
The appendix shows that directed growth selects one void class for trapped matter, under two
conditions. Flip-invariant dynamics cannot prefer one world, and no helicity coupling exists at a
void. The growth direction is the only fixed flip-odd background of the framework, and it acts as an
initial condition. In layer-by-layer growth in one direction, a node can be trapped only in the class
that points along it. The class is then conserved along physical D
4
worldlines, given the propagation
postulate of Ref. [4]. So the selection supplies the bit, under the stated conditions.
13
Declarations
Competing interests. The author is employed by IDrive Inc. The author declares no competing
financial interests or personal relationships that could have appeared to influence the work reported
in this paper.
Funding. No external funding was received.
Data availability
One Python script checks every algebraic and geometric claim of the paper. It uses NumPy only.
It builds the E
6
root system from the D
4
lift, the three factors, the flip, the charge operator and
the 27. It then imposes the chiral input. It checks the generation content, the anomalies, the
normalization and mixing angle, the equal running of the two non-color couplings, the Standard
Model crossing scale, and the breaking constraints. It also checks the plaquette count, the reciprocity
of symmetric verification costs, and the site-symmetry, slice, growth-direction and surface-trapping
statements of Appendix A. The script is available at github.com/raghu91302/ssmtheory/raw/main/
one chiral bit scripts.zip. The growth simulation of Appendix A, with its scripts and data, is available
at github.com/raghu91302/ssmtheory/raw/main/one chiral bit growth.zip. It uses NumPy, SciPy
and NetworkX. The interactive simulation of Appendix A is at raghu91302.github.io/ssmtheory/
ssm one sided growth.html.
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