From F
4
to E
6
: The Worldline Unfolds the Vacuum—Chirality,
Lepton Families, and Trinification
*
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, and matter is a node trapped in a tetrahedral void; a predecessor construction closes the
bond algebra of vacuum and matter into the exceptional algebra f
4
, with color a factor of the
fixed subalgebra of a triality twist and electric charge quantized in thirds. This paper is an
algebraic result with a physical interpretation, and the two are kept separate throughout. The
two tetrahedral void orientations are exact inversion partners in the static crystal, so orientation
is not a label there; but on a chronological defect worldline, the pairing relaxes from P to P T ,
and doubling every matter weight by the now-physical orientation label—with the three glue
classes at 120
◦
in a new plane—assembles exactly the 72-root system of E
6
, verified by reflection
closure, rank, and Cartan determinant, the charge-based competitor failing. The orientation flip
is an automorphism folding e
6
back to f
4
; trinification follows from the root system, and the 27’s
color-singlet block is a lepton-family-structured sector whose fold is the predecessor’s octet plus
one singlet. The same geometry fixes the electric charge: the Cartan charges that commute with
color, are odd under the lattice’s charge conjugation and even under its parity form a single ray,
and on it the 27 carries the Standard Model charges, integer unit included, selected uniquely
among eighteen candidate charge operators. No gauge dynamics is constructed; a consolidated
table maps each claim to its type, status, source, and falsifier, and the open frontier is the selection
of the chiral projection.
Keywords: E
6
; F
4
; trinification; chirality; lepton-family structure; triality; D
4
lattice; parity viola-
tion; Nielsen–Ninomiya; Selection–Stitch Model
1 Introduction
The Selection–Stitch Model (SSM) treats the vacuum as a face-centered-cubic (FCC) lattice grown
by two kinematic operators—a planar stitch and a rare out-of-plane lift—whose simulation saturates
the K = 12 bulk with growth confined to a surface [1]. The lattice carries a [[192, 130, 3]] CSS
stabilizer code [2]. Matter is a node trapped in a tetrahedral void: from that single defect, the
model derives the charge thirds from the tetrahedron’s bond angle, three colors from its skew-edge
pairs, and the proton-to-electron mass ratio [1]; the light-particle spectrum follows from the defects’
verification costs in the code [3]. The lattice’s four-dimensional home is the D
4
root lattice, and its
*
Author’s version. Accepted for publication in Symmetry (2026), manuscript symmetry-4567866. This version
includes a correction to Section 8 that was submitted to the journal on 23 September 2026.
1
bond algebra is so(8); the fixed subalgebra of a twisted triality automorphism is an su(3), the only
non-abelian candidate the classification allows once G
2
is excluded by dimension count, and color is
a factor of the fixed subalgebra this twist acquires in f
4
computations, and it distinguishes verified,
derived, and open claims explicitly; in this paper, that distinction is carried by the consolidated table
of Section 11. The present paper is the next step in that chain.
Relation to prior unification programs. E
6
as a grand unification algebra dates to Ref. [5];
its trinification subgroup SU(3)
3
, with the 27 as the fermion multiplet, dates to Refs. [6, 7]; the
representation-theoretic toolkit is standard [8]. Octonionic and exceptional-Jordan routes to adjacent
algebras are pursued in Ref. [9]. Those programs postulate the algebra and fit matter into it. The
present series runs in the opposite direction: the lattice and its defect are fixed first, by a published
simulation and a published code, and the algebra is whatever their bond structure closes into. The
result reproduces the trinification kinematics without adopting it, and the comparison is drawn where
the two directions meet.
The companion paper, Ref. [4], derived, on the D
4
home of the face-centered-cubic (FCC) vac-
uum [2], a charge quantized in thirds, the identity of charge conjugation with spatial inversion, and
the closure of vacuum and matter into f
4
, with an emergent second su(3) whose 26-branching put
a family-structured block of colored matter at the down-quark charge. It also left three marks that
this paper takes as its starting questions. Appendix A reconstructs this predecessor material—the
D
4
→F
4
construction, the void/matter identification, and the code—in full, so the present paper can
be read without the cited works. Why is charge conjugation the same operation as inversion? Why
do the uncolored matter states form a horizontal octet instead of three lepton families? And where
is the integer charge unit that up-type quarks require?
The lattice has two kinds of tetrahedral voids, the static crystal cannot tell them apart, and a
defect worldline can; the algebra of the orientation-blind crystal is f
4
, the algebra of the orientation-
aware worldline is e
6
, and the three marks are what folding e
6
down to f
4
looks like from below.
Protocol and provenance. As throughout the series, the structure was computed first and
compared with known physics afterward; comparisons are labeled as comparisons. No new substrate
is introduced: the lattice, its code, and matter as defects [1, 3] are unchanged, and the doubling datum
used below is not a new field but a label the worldline is shown to possess. Every construction is
verified by the scripts linked in the Data Availability statement. Standard background on E
6
, its
27, and trinification-type decompositions is in Refs. [8, 10, 11], and finite-order automorphisms in
Ref. [12]; the positioning against prior unification programs is given above.
Scope: algebra, not gauge dynamics. Every theorem in this paper is a statement about
root systems, their automorphisms, and their representations: reflection closure, rank, Cartan data,
branching, and invariant subalgebras. No gauge field is constructed, no action or Hamiltonian for
an E
6
gauge dynamics is proposed, and no claim is made that the algebra is dynamically realized;
connecting these generators to physical operators and interactions is the framework’s registered
continuation, not a result of this paper. Physical words (“lepton,” “chirality,” “worldline”) name
interpretations of algebraic objects and are flagged as such where they occur.
2 Methods: Construction and Verification
The vacuum roots. The D
4
root system is taken in its standard coordinates: the 24 vectors
±e
i
±e
j
, 1 ≤ i < j ≤ 4, each of norm 2. These are the bond directions of the four-dimensional home
of the FCC vacuum, and they are used exactly, with no scaling freedom.
The matter weights and the class plane. The predecessor’s 24 matter weights are doubled by
2
the void-orientation label into 48. Each orientation pair is separated in a fifth and sixth coordinate:
the up copy receives a vector u and the down copy a vector d, with
u · u = d · d = 1 (completing each norm-1 weight to norm 2) , u · d = −
1
2
, (1)
so the two copies sit at 120
◦
in the new plane. Section 5 proves this Gram matrix is not a choice: unit
completion, the zero of the pair sum against the vacuum, and the discriminant form of D
∗
4
/D
4
—glue
classes [0], [v], [s], [c] with discriminant norms 0, 1, 1, 1, the three nontrivial classes equinormal—force
it.
A candidate set R passes only if the following are true: (i) for every α, β ∈ R, the reflection
s
α
(β) = β − 2
(α,β)
(α,α)
α lies in R (closure); (ii) the integer span of R has rank 6; (iii) simple roots
extracted from a generic linear functional have Cartan matrix A
ij
= 2(α
i
, α
j
)/(α
j
, α
j
) equal, up to
relabeling, to the standard E
6
matrix, of determinant 3; (iv) the resulting diagram is the E
6
diagram
and no other rank-6 diagram. The natural competitor—doubling along the glue class [v] without the
orientation label—is run through the identical battery and fails (i): specific reflections leave the set.
The failure is reported as a result.
The fold and the charge, as computed. The fold is the involution that exchanges the
two orientation copies and fixes the vacuum roots; its fixed subsystem is computed by direct orbit
enumeration and matched against F
4
by the same battery. The charge is computed directly: the
Cartan generators of the assembled e
6
that commute with color, are odd under the lattice’s geometric
C and even under its geometric P form a single ray (Section 8), and the charges of the 27 on that
ray are compared with the Standard Model. Uniqueness is tested by scanning all Standard Model
charge operators built from the two non-color su(3) factors.
Reproducibility. Every numbered claim is reproduced by one script of 26 labeled checks
(NumPy 2.4 on Python 3.12 only, about one minute), in the order this paper states them: the
f
4
closure; the unfold; the competitor’s failure; rank, Cartan determinant, and diagram; the fold;
trinification; the families; the discriminant-derived class plane; and the charge selection. The script
prints each check’s name and verdict, and its README maps checks to theorems.
3 The Fold and Its Origin: Orientation Is Not a Static Label
The FCC lattice is centrosymmetric: its space group contains inversion, and every node is a center
of inversion [13]. The tetrahedral voids come in two orientations—cages pointing up and down the
stacking axis—and matter occupies them [1, 3].
Theorem 1 (Static orientation blindness). In a single-domain FCC crystal, the up-void and down-
void environments of any bulk node are exact inversion partners: with the two in-plane primitive
vectors a
1
, a
2
and close-packing sublattice offsets o
A
= 0, o
B
=
1
3
(a
1
+a
2
), o
C
=
2
3
(a
1
+a
2
), one has
o
A
+ o
C
= 2o
B
, so the two orientation classes map onto each other under inversion through every
node, exactly.
Statically, therefore, orientation is not a well-defined label on matter: any construction that
respects the crystal’s symmetry must identify the two void types. This is the geometric origin of
the predecessor’s C = P theorem (which holds for the involutive lift of spatial inversion) —inversion
exchanges the orientations, and the orientation-identified charge grading reverses—and it is why the
algebra built there is orientation-blind. It is also, in the language of the sections below, the fold.
3
4 The Worldline Makes Orientation Physical
In SSM, the “worldline” of a defect is the ordered sequence of verification commits by which the
growth front advances the defect’s site: a chronological order supplied by the front’s dynamics, used
here only as an ordering. No action, metric structure, or equation of motion for the worldline is
introduced or needed; every use below is combinatorial.
A defect does not sit in the static crystal; it threads the stack in time, at one layer in the past
and the next in the future. Assign the two orientation connections their time coordinates: the
up-connection at t = +δt, the down-connection at t = −δt.
Theorem 2 (Relaxation from P to P T ). Spatial inversion through the node maps the spatial part
of the up-connection onto that of the down-connection, leaving each at its own time; as four vectors,
they no longer coincide. The operator that pairs them on a worldline is full spacetime inversion P T .
The static identification of the two orientations is thus a P relation, and on a chronological worldline
it relaxes to a P T relation.
The statement proved here is an internal automorphism result, and it is stated as exactly that.
Define the actions on the orientation doublet (|+⟩, |−⟩): P is the unitary exchange of the two sheets,
σ
x
; T is the antiunitary reversal of the worldline’s commit order, complex conjugation in this basis;
P T is their composition. The unique Pauli generator odd under P and even under P T is σ
y
, and the
oriented out-of-plane growth supplies exactly this generator: on the worldline, the two orientations
are inequivalent under P alone. This yields a P -distinguishing pair of orientation states, not by itself
a chiral gauge theory: the projection that would select one orientation as the propagating one remains
open (Section 9). Whether and how this relaxation interacts with the hypotheses of the Nielsen–
Ninomiya theorem [14]—locality, hermiticity, translation invariance of a realized fermion action—is
not analyzed here, and no evasion of that theorem is claimed; the Ginsparg–Wilson mechanism [15, 16]
is cited only as the classical example that lattice chirality prohibitions have conditional boundaries,
its exact realization on the lattice being well established [16].
The consequence this paper builds on: on a worldline, orientation is a physical, unremovable label
on defect states. The static crystal has no such label; the worldline does. The next section asks what
algebra the labeled matter generates.
5 The Unfold: Orientation Assembles E
6
Take the 24 bond roots unchanged. Lift every one of the 24 matter weights twice, labeled by
orientation ε = ±1, into two new dimensions, with the three glue classes of D
∗
4
/D
4
placed at 120
◦
:
s 7→ (s, ±v
class(s)
), |v
k
| = 1.
Theorem 3 (Closure, with a competitor removed). The resulting 72 vectors all have squared length
2, are closed under the reflection in every one of their members, span rank 6, and have Cartan matrix
of determinant 3 with node degrees {1, 1, 1, 2, 2, 3} and marks (1, 1, 2, 2, 2, 3): by the classification of
root systems, the unique system that is irreducible, simply laced, of rank 6, with 72 roots and Cartan
determinant 3, is E
6
; those five properties are exactly what is verified, and they are the assumptions
under which uniqueness is asserted. The alternative doubling, by charge sector instead of glue class,
fails reflection closure.
Explicit data for independent verification. With the generic functional of the verification
script, the six simple roots of the 72 are, in the coordinates (R
4
bond
⊕ R
2
class
),
4
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
),
with Cartan matrix
C =
2 −1 0 0 0 0
−1 2 −1 −1 0 0
0 −1 2 0 −1 0
0 −1 0 2 0 −1
0 0 −1 0 2 0
0 0 0 −1 0 2
, det C = 3,
the trivalent node at position 2, as printed by the linked script; the chain of root counts and the
class-plane geometry are shown in Figure 1.
D
4
⊂ F
4
⊂ E
6
E
6
: 72
F
4
: 48
D
4
: 24
class plane: ±v
1
, ±v
2
, ±v
3
+v
1
+v
2
+v
3
−v
1
−v
2
−v
3
120
◦
Figure 1: The embedding checked visually. (Left ): root counts of the chain—the 24 D
4
bonds,
the 48 of F
4
after the triality twist, the 72 of E
6
after the orientation unfold. (Right): the class
plane, with the three glue-class vectors +v
k
(orange) at the forced 120
◦
and their orientation partners
−v
k
(blue, dashed); each of the 24 short weights is lifted to (s, ±v
class(s)
).
Theorem 4 (The fold). The orientation flip ε → −ε is an automorphism of the 72 whose fixed roots
are exactly the 24 bonds. The orientation-symmetric part of e
6
is f
4
: the predecessor’s algebra is the
fold of this one, and its C = P identity is the folded shadow of the structure below. Explicitly: the flip
fixes the 24 bond roots individually and exchanges the remaining 48 in 24 two-cycles; the invariant
algebra is spanned by the 24 fixed root spaces, the 24 symmetric combinations E
α
+E
σ(α)
of exchanged
pairs, and the 4-dimensional flip-fixed Cartan subspace, totalling 24 + 24 + 4 = 52 = dim f
4
—the
count distinguishes fixed roots from invariant combinations, as it must.
The doubling datum is the worldline label of Theorem 2. The 120
◦
class geometry is not a choice;
it is the discriminant data of the lattice. The three nontrivial glue classes satisfy [1] + [2] + [3] = 0
in D
∗
4
/D
4
, so any additive placement must satisfy v
1
+ v
2
+ v
3
= 0. The discriminant quadratic
form gives all three classes norm 1, so the v
k
have equal unit length. Unit length and zero sum
force v
i
· v
j
= −
1
2
: the class-plane Gram matrix is the discriminant form of D
∗
4
/D
4
, realized in
two dimensions, unique up to rotation. The scan of alternatives was extended beyond the charge
competitor. Angle: three unit vectors in a plane with equal pairwise inner product t exist only at
t = −
1
2
—the 120
◦
configuration is forced by planarity before closure is even tested; configurations
attempted at t ∈ {+
1
2
, 0, −
1
3
, −
1
√
2
, −0.9} do not exist as three-unit-vector systems, and none closes.
5
Labeling: doubling keyed to charge sign, to the sign of the first component, or to index parity all fail
reflection closure; among the tested labelings, only the glue-class labeling closes. Partial doublings
fail on the root count (56 or 64, not 72). Within this widened space, the construction is the unique
survivor; uniqueness beyond it is not claimed. Nothing else enters (Figure 2).
f
4
(52)
orientation-blind
C = P ; octet; thirds only
e
6
(78)
orientation-aware
chiral pair; family-structured
worldline (Thm. 2)
unfold (Thm. 3)
orientation flip: fold (Thm. 4)
Figure 2: One geometric dichotomy, two algebras. The static crystal identifies the two void orienta-
tions (Theorem 1) and lives in f
4
; a defect worldline distinguishes them (Theorem 2) and lives in e
6
;
the flip that forgets the label is the fold (orange arrow).
Table 1 lists the 72 roots by block, and Table 2 shows where the 27 goes.
Table 1: The 72 roots of E
6
by block.
Block Count Role
D
4
roots (vacuum bonds) 24 so(8), color inside
matter weights, up-voids 24 the predecessor’s octet, lifted
matter weights, down-voids 24 the inversion partners
total 72 the E
6
root system
Table 2: The 27 under the three su(3) factors.
27 ⊃ Dim Content
(3,
¯
3, 1) 9 colored: the quark block
(
¯
3, 1, 3) 9 colored: the conjugate block
(1, 3,
¯
3) 9 singlets: the lepton block
6 Trinification, and What the Flip Does to It
Three notions are kept distinct throughout: the orientation label (the geometric ± of the void sheets),
chirality (a property of a realized fermion action, not derived here), and the two flip-swapped su(3)
factors (algebraic objects the orientation label acts on). No equivalence among the three is claimed;
the orientation label is the only one this paper derives.
Theorem 5 (Three su(3)s, and the flip’s action). Deleting the mark-3 node of the affine E
6
diagram
leaves three mutually orthogonal A
2
s, realized explicitly on the construction. The orientation flip
permutes them by swapping two and fixing one. The fixed factor is color: matching the fold of the
27 against the predecessor’s verified 26-branching identifies the flip-fixed factor as su(3)
color
and the
diagonal of the swapped pair as the predecessor’s horizontal su(3), uniquely.
Parity fixing color and exchanging the two members of a chiral pair is exactly the action the
electroweak interaction requires of it (Figure 3).
6
su(3)
L
su(3)
R
su(3)
color
flip-fixed
orientation flip = parity
(3,
¯
3, 1)
quark block
(1, 3,
¯
3)
lepton block
(
¯
3, 1, 3)
conjugate block
27 = (3,
¯
3, 1) ⊕ (1, 3,
¯
3) ⊕ (
¯
3, 1, 3); fold of the lepton block = 3 ⊗
¯
3 = 8 ⊕ 1
Figure 3: Trinification on the unfolded algebra. The mark-3 deletion gives three orthogonal su(3)s.
The orientation flip swaps the chiral pair and fixes color. The 27 splits into three nine-state blocks;
the color-singlet block is the family-structured sector, and its fold is the predecessor’s octet plus
one singlet. The block names “quark,” “lepton,” and “conjugate” are physical interpretations of
representation-theoretic blocks, not derived Standard Model identifications.
7 The 27: The Family-Structured Sector, and the Octet Explained
Theorem 6 (Families). The 27-dimensional minuscule representation [17] branches under the
three factors as (3,
¯
3, 1) ⊕ (1, 3,
¯
3) ⊕ (
¯
3, 1, 3), nine states each. The color-singlet block is a bifun-
damental of the flip-swapped pair: a lepton-family-structured sector—three color-singlet families of
each orientation type. Folding the 27 under (color, diagonal-of-the-pair) reproduces the predecessor’s
26-branching (3,
¯
3) ⊕ (
¯
3, 3) ⊕ (1, 8) plus exactly one singlet.
The octet is therefore explained rather than excused: it was the folded shadow of the lepton
bifundamental, 3 ⊗
¯
3 = 8 ⊕ 1, locked to its diagonal because the orientation-blind algebra could
not tell the two chirality types apart. A family-structured color-singlet sector exists from the mo-
ment the crystal’s worldline remembers which void is which. Identifying it with the three observed
lepton generations would additionally require the SU(2)
L
× U(1)
Y
assignments, the chiral quantum
numbers, and a generation-dependent mass structure, none of which is derived here (the electric
charges themselves are derived in Section 8); Table 3 sets the construction against one Standard
Model generation, marking each quantity derived, structurally analogous, or absent.
Table 3: This construction against one Standard Model generation: derived, structurally analogous,
or absent.
Quantity SM Generation Here
color triplets SU (3)
c
triplets derived (flip-fixed factor)
electric charge ±
1
3
, ±
2
3
, 0, ±1 derived, unique under C, P (Section 8)
three families 3 generations structurally analogous (9-block)
SU (2)
L
doublets chiral doublets absent (kinematic home only)
hypercharge Y U(1)
Y
kinematic: Y = Q − T
3L
awaits the chiral
projection
chirality left projection absent (open; Section 9)
masses generation-dependent absent
7
8 The Electric Charge Is Selected by the Lattice’s C and P
The remaining mark is ±2/3. Standard E
6
embeddings contain the full Standard Model charge
spectrum [8, 5], but they admit many charge operators, and the question here is which one this con-
struction selects. Physical parity is the orientation flip (it exchanges the sheets); charge conjugation
is the extended spatial inversion, verified to be an automorphism of the 72 (it inverts space, fixes
e
4
, and reflects the class plane across the fixed class’s axis, swapping the two spinor classes). An
electric-charge operator must be P -even, C-odd, and commute with color.
On the Cartan subalgebra these three conditions leave exactly one ray. Normalized so that the
color-singlet states carry integer charge, it is the charge operator
Q =
2
3
ω
2
− ω
3
− ω
4
+ ω
5
+ ω
6
in the fundamental-coweight basis {ω
i
} dual to the simple roots above, up to an overall sign. On the
27 it gives the Standard Model trinification spectrum exactly: in the color-triplet block, +
2
3
once and
−
1
3
twice for each color; in the antitriplet block the conjugate values; and in the nine color-singlet
states −1 twice, 0 five times and +1 twice. The integer unit is therefore a Cartan charge of e
6
.
The selection is also unique. Of the eighteen Standard Model charge operators that can be built
from the two non-color su(3) factors, exactly two, ±Q, are odd under the geometric C and even
under the geometric P . The lattice’s own discrete symmetries thus do not obstruct the physical
charge; they choose it. The fractional part of Q comes from its ω
2
component, the coweight of the
mark-3 node, in agreement with the predecessor’s grading, and the integer part from the remaining
coweights.
In the Standard Model the same charge splits as Q = T
3L
+ Y , and that split requires knowing
which su(2) is the chiral weak group. The electric charge itself is fixed here; its split into weak isospin
and hypercharge waits on the chiral projection of Section 9.
9 The Single Frontier
One candidate can be dismissed at the outset: the three ⟨100⟩ axis pairs of the nearest-neighbor shell
close as an su(2), but the predecessor’s commutant theorem rules them out as an internal structure:
no su(2) commuting with color exists in the bond algebra. The weak structure’s kinematic home is
a chiral factor of the trinification pair. The relevant two-state system is the orientation doublet, not
a spatial rotation triple.
Three complementary consistency checks—not independent proofs, since all three rest on the
same orientation-doubling construction—now end at the same point. The worldline analysis reaches
a parity-violating but vector su(2): nothing forces the left projection. The trinification kinematics
contains both chirality types symmetrically: nothing selects one. The electric charge does not wait
on that selection (Section 8); what waits is its split into weak isospin and hypercharge. The chiral
projection P
L
is the series’ one open gate to the electroweak sector, and it is a dynamical question—
which orientation the code’s verification machinery couples to—not a further kinematic one. The E
7
and E
8
extensions are explicitly fenced off until that gate is settled.
10 Predictions and Falsifiers
The predecessor’s falsifiers are inherited unchanged: exact e/3 quantization with no millicharged
states, and the mark-3 counting rule. This paper adds two internal ones. The electric charge is fixed
8
uniquely, up to sign, by the geometric C and P (Section 8): a second admissible charge operator, or
a state of the 27 carrying a charge other than the one given there, would refute it. And the family
reading now stands on the unfold: if the flip-fixed factor proves not conjugate to the geometric color
A
2
, the lepton-family identification falls with it.
An external, discriminating consequence. Beyond these internal tests, the construction
makes one statement that separates it from generic E
6
trinification, where hypercharge is a freely
available Cartan direction [8, 18]. Here, the electric charge is already fixed by the geometric C and
P , so the weak-hypercharge normalization is not free: once the chiral projection selects the weak
group, Y = Q −T
3L
follows with no further choice. Two observable corollaries follow if that selection
is ever realized within the framework. First, because the two orientation sheets enter symmetrically
until the projection acts, the framework forbids a fourth chiral generation of the same trinification
type: the block structure of the 27 admits exactly the threefold family pattern and no more, so any
confirmed fourth sequential generation refutes the model. Second, because the charge is fixed and
hypercharge is not an independent Cartan direction, the construction predicts no light Z
′
from an
independent U(1) hypercharge extension below the scale at which the projection is resolved—distinct
from trinification models in which such a U (1) is generically present and its gauge boson need not be
heavy. Neither corollary is a numerical prediction, and both are contingent on the open projection;
they are stated so the construction carries at least one consequence that a competing E
6
model does
not share.
11 What Is Derived and What Is Not
Table 4 lists each claim of the paper with its type, status, source, and falsifier.
9
Table 4: Each claim, with its type, status, source, and falsifier.
Claim Type Status Source Falsifier
void environments are inversion partners math verified
(Thm.)
Sections 3 and
A.2
a distinguishing bulk
invariant
worldline relaxes P to P T math verified
(Thm.)
Section 4 a P -invariant ordering
σ
y
: P -odd, P T -even generator math verified Section 4 another odd generator
orientation×class doubling = E
6
math verified
(Thm.)
Section 5 any closure failure (script)
charge doubling fails math excluded Section 5 its closure
fold: fixed = bonds; 24 + 24 + 4 = 52 math verified
(Thm.)
Section 5 a miscounted invariant
class plane = discriminant form; 120
◦
forced
math derived
Sections 5 and
A.1
a second planar solution
trinification; flip swaps two, fixes one math verified
(Thm.)
Section 6 wrong flip action
flip-fixed = geometric color (not just
conjugate)
physics open Section 6 –
27 branching; family-structured 9-block math verified
(Thm.)
Section 7 wrong branching
octet = folded bifundamental math verified Section 7 fold mismatch
SM chiral charges on the block physics not claimed Section 7 –
admissible charge
Q =
2
3
ω
2
− ω
3
− ω
4
+ ω
5
+ ω
6
math verified Section 8 a second admissible ray
SM charges on the 27, unique under
geometric C, P , color
math verified Section 8
a second admissible
operator
chiral projection P
L
physics open Section 9 –
gauge dynamics for e
6
physics not claimed Section 12 –
imported: D
4
→ F
4
twist, geometric color physics imported [4] predecessor refutation
imported: stabilizer code [[192, 130, 3]] math imported [2] code invalidity
12 Applications and Future Directions
One inherited caveat is recorded: the predecessor’s one-loop coupling-ratio comparison uses unmodi-
fied Standard Model running; it is a bare-SM consistency check, not a model-specific prediction, and
any states this construction adds would alter it.
The unfold fixes the arena the series’ dynamical program now works in: the orientation doublet it
makes physical is the object on which parity violation, the charge unit, and the electroweak algebra
are derived downstream, and the lepton block it identifies is where the framework’s breaking sector
lives. The kinematics of this paper is, in that sense, load-bearing: each of its structures is consumed
by a registered continuation.
Three continuations are registered in the series’ pre-registration and are not claimed here. The
single frontier of Section 9—the selection of the chiral projection—is a dynamical question, and a
dynamical program on the code’s verification front is under way. The family count of the lepton
block awaits reconciliation with the defect-worldline state count. The electric charge is settled at
the algebraic level (Section 8); its split into weak isospin and hypercharge waits on the same chiral
projection. Each has a registered protocol with outcomes fixed in advance.
10
13 Conclusions
One dichotomy runs through this paper: the crystal has two kinds of voids, and the question of
whether anything can tell them apart is the question of the electroweak sector. Statically, nothing
can, and the physics is f
4
: charge conjugation is inversion, the leptons hide in an octet, and charge
stops at thirds. On a worldline, the label exists, and the physics is e
6
: the fold accounts for the
first mark, the unfolded bifundamental accounts for the second, and the lattice’s own C and P fix
the electric charge, the third, uniquely; what still waits at the chiral projection is its split into weak
isospin and hypercharge. The frontier is now single, named, and dynamical. That is the position a
continuation should leave its series in.
Funding
This research received no external funding.
Data availability
All computations are specified in the text. Python 3.12 (NumPy 2.4) scripts reproducing every
check—the inversion pairing and P → P T computations, and the full E
6
chain including the clo-
sure, the fold, trinification, the 27, and the charge selection—are available at https://github.com/
raghu91302/ssmtheory/raw/main/e6_scripts.zip (accessed on 23 September 2026; 26 labeled
checks, NumPy only).
Conflicts of interest
Author Raghu Kulkarni was employed by SSMTheory Group, IDrive Inc., Calabasas, USA. The
author declares that the research was conducted in the absence of any commercial or financial rela-
tionships that could be construed as a potential conflict of interest.
A Prior Results This Paper Stands On
This appendix makes this paper self-contained: it states, with the actual objects, everything imported
from the three cited works.
A.1 The D
4
→F
4
Construction [4]
This subsection reconstructs the predecessor in enough detail that a reader new to the series can
follow every step and check it by hand. The source is an archived preprint that has not been peer
reviewed, so nothing is left implicit.
Step 1—the lattice gives D
4
. An FCC node has twelve nearest neighbors, in the directions
1
√
2
(±1, ±1, 0) and the two permutations that put the zero in the first or second slot. Append the
growth front’s axis as a fourth coordinate (value 0 for an in-layer bond) and rescale by
√
2: the
twelve become twelve of the vectors ±e
i
± e
j
. Including the front-axis bonds completes the set to
all
4
2
· 4 = 24 sign-and-index combinations ±e
i
± e
j
(1 ≤ i < j ≤ 4), each of squared length 2.
11
These 24 vectors are exactly the root system of D
4
; the algebra they generate is so(8), of rank 4
(Figure 4). The passage from a three-dimensional crystal to a four-dimensional root system is explicit
and not a resemblance: the twelve FCC bond directions are the projections onto the spatial e
1
, e
2
, e
3
hyperplane of the twenty-four ±e
i
±e
j
once the growth-front axis e
4
is adjoined. The in-layer bonds
carry e
4
= 0; the twelve remaining D
4
roots, those with a nonzero fourth component, are the lift
steps of the surface-front growth (Appendix A.2), which are physical moves of the construction rather
than added dimensions. In this precise sense, the FCC geometry, together with its one dynamical
growth direction, generates D
4
—the fourth coordinate is the front’s advance, not a hidden spatial
axis.
24 roots of D
4
(schematic projection)
±e
i
± e
j
Figure 4: The 24 roots of D
4
, the bond directions of the vacuum in their four-dimensional home. All
have equal length; the apparent shells are an artifact of projecting four dimensions to two. These
are the “long” roots that survive untouched into F
4
and E
6
.
Step 2—the defect gives three eights. A node trapped in a tetrahedral void carries the
weight systems of so(8), which by triality are three inequivalent eight-dimensional representations:
8
v
= {±e
i
}, (8 vector weights, squared length 1)
8
s
= {(±
1
2
, . . . , ±
1
2
) : even # of minus signs},
8
c
= {(±
1
2
, . . . , ±
1
2
) : odd # of minus signs},
A total of 24 short vectors in all, each of squared length 1. A worked membership check:
(
1
2
,
1
2
,
1
2
,
1
2
) ∈ 8
s
(zero minus signs, even), while (
1
2
,
1
2
,
1
2
, −
1
2
) ∈ 8
c
(one, odd).
Step 3—glue classes. Modulo the D
4
lattice, the short weights fall into the three nontrivial
cosets of D
∗
4
/D
4
∼
=
Z
2
× Z
2
: [v] = 8
v
, [s] = 8
s
, [c] = 8
c
. The group law gives [v] + [s] + [c] = 0,
and the discriminant quadratic form assigns all three the same norm; these two facts—zero sum,
equal norm—are all that Section 5 needs, and they force the three class vectors to sit at 120
◦
(Figure 5). The two-dimensional realization is canonical, not a convenient picture: the discriminant
group D
∗
4
/D
4
∼
=
Z
2
× Z
2
carries a Q/2Z-valued discriminant quadratic form under which the three
nontrivial classes are equinormal and sum to zero, and any faithful real representation of three
equinormal, zero-sum vectors is the 120
◦
configuration in a plane, unique up to an overall rotation
and scale. No two-dimensional freedom beyond that rotation and scale is introduced; the plane and
its 120
◦
angle are fixed by the discriminant form of the lattice itself, which is why the construction
has no free angular parameter.
12
[v]
[s] [c]
120
◦
120
◦
[v] + [s] + [c] = 0, equal norms ⇒ 120
◦
Figure 5: The three glue classes of D
∗
4
/D
4
. Their vanishing sum and equal discriminant norm force
the mutual 120
◦
geometry that this paper lifts into a new plane to build E
6
; the angle is not a
modeling choice but a property of the lattice. The colors only distinguish the three labeled classes.
Step 4—the twist closes bonds and matter as F
4
. Triality is the order-3 outer automor-
phism of so(8) that cyclically permutes 8
v
→ 8
s
→ 8
c
. The trapped defect realizes one such twist
geometrically—its three valence directions single out the cyclic pairing. The predecessor’s central
theorem is that the 24 long roots and the 24 short weights together close under reflection as the
48-root system of F
4
. Reflection in a root α acts by s
α
(x) = x − 2
(x,α)
(α,α)
α; a worked case: reflecting
the short weight e
1
in the long root e
1
+ e
2
gives e
1
− 2
1
2
(e
1
+ e
2
) = −e
2
, again a short weight,
as closure requires. That every one of the 48 × 48 reflections lands back in the 48 is the nontrivial
content, verified as check 1 of this paper’s script. The resulting F
4
has two root lengths in ratio
√
2 : 1 and simple roots α
1
= e
2
−e
3
, α
2
= e
3
−e
4
, α
3
= e
4
, α
4
=
1
2
(e
1
−e
2
−e
3
−e
4
), with Cartan
matrix
2 −1 0 0
−1 2 −2 0
0 −1 2 −1
0 0 −1 2
,
the double bond (the −2) marking the long/short transition.
Step 5—color, charge, and the three marks. Inside F
4
sits the commuting pair su(3)
color
×
su(3)
horiz
, with color built geometrically from the tetrahedron’s three skew-edge pairs (the tetrahedral
bond angle is arccos(−
1
3
) = 109.47
◦
). The 26 of F
4
branches as (3,
¯
3) ⊕ (
¯
3, 3) ⊕ (1, 8): the last piece
is the horizontal octet of mark (ii). Electric charge is the grading label set by the coweight of the
mark-3 node, and the glue structure quantizes it in thirds—an output, not an input. facts complete
the inheritance: charge conjugation, realized as the extended spatial inversion, comes out equal to
parity for the involutive lift of inversion, mark (i); and, because its label is Z
3
-valued, the charge is
fixed only modulo one, mark (iii). This paper re-derives (i) internally (the fold theorem), supplies at
the level of e
6
the integer part that (iii) left open (Section 8), and re-obtains the 26-branching as the
fold of the 27 (Theorem 5), so only the geometric realization of the twist and the color identification
are imported on trust. A last predecessor result, used in Section 9: no su(2) commuting with color
exists in the bond algebra, which is why the weak sector’s kinematic home is the orientation doublet
rather than a spatial triple. Table 5 collects the lattice-to-algebra correspondences used above.
13
Table 5: The geometric dictionary: each lattice object and the algebraic object it becomes.
Lattice Object Algebraic Object
12 bond directions + front axis 24 roots ±e
i
± e
j
of D
4
front (growth) coordinate the fourth axis e
4
trapped defect’s states the three eights 8
v
, 8
s
, 8
c
defect’s three valence directions the cyclic triality pairing
tetrahedron’s skew-edge pairs the color su(3)
tetrahedral bond angle 109.47
◦
charge quantized in thirds
glue classes of D
∗
4
/D
4
the families [v], [s], [c]
discriminant form (norm 1, sum 0) the 120
◦
class geometry
A.2 Matter as a Trapped Node [1]
This source is published and independently verifiable [1]. It is summarized in enough detail that the
one fact this paper actually uses—the inversion pairing of the two void orientations—is unambiguous,
and so a reader can see how “matter” arises as a lattice object rather than an added ingredient.
How the crystal grows. The vacuum is grown by two kinematic operations on an entanglement
graph: a planar stitch that adds a node within the current close-packed sheet, and a rare out-of-plane
lift (probability P
lift
= e
−3
) that starts the next sheet. Growth is confined to a moving surface front.
The simulation of the source shows that this dynamics spontaneously selects the face-centered-cubic
packing: the sheets stack in the ABC order of FCC and never in the ABAB order of hexagonal close
packing, and the measured interlayer spacing is h =
p
2/3 L for bond length L—the value forced
by close packing. The result is a single-domain FCC crystal with coordination number twelve, the
densest lattice packing in three dimensions.
Voids, and why matter is a trapped node. Between the close-packed sheets, the FCC
structure leaves interstitial voids of two shapes: octahedral voids (six surrounding nodes) and tetra-
hedral voids (four surrounding nodes). The tetrahedral voids are the small ones, and a node caught
in a tetrahedral void during growth cannot relax into the lattice—it is over-coordinated relative to
the bulk and sits as a persistent defect. That trapped node is matter in the framework: not a field
added to the vacuum, but an incompletely crystallized site of the vacuum itself (Figure 6). Its three
unsatisfied valence directions toward the tetrahedron’s faces are the origin of the three colors, and
the tetrahedral bond angle arccos(−
1
3
) = 109.47
◦
is the geometric origin of the charge thirds.
vacuum
nodes
trapped node = matter
three valence bonds → three colors; angle 109.47
◦
→ charge thirds
Figure 6: Matter as an incompletely crystallized site. Four vacuum nodes (black) form a tetrahedral
void; a node trapped at its center (red) cannot relax into the twelve-coordinated bulk and persists as
a defect—this is what the framework calls a matter particle. Its three valence directions toward the
tetrahedron give the three colors, and the tetrahedral angle arccos(−
1
3
) gives the charge quantized
in thirds.
14
The two void orientations, and the fact this paper uses. Tetrahedral voids come in
two orientations: an up-tetrahedron, whose apex points along the growth direction, and a down-
tetrahedron, whose apex points against it (Figure 7). These are genuine geometric opposites. Point
inversion through any bulk node maps the bond set to itself and carries every up-void to a down-
void; because that inversion is an exact symmetry of the infinite crystal, no static bulk observable
can distinguish an up-void from a down-void. This single fact is the entire geometric input of
Theorem 1, and it is why this paper’s orientation label can be switched on only by something that
breaks inversion—the chronological worldline of Appendix A.3.
up: apex ↑ growth down: apex ↓ growth
point inversion
Figure 7: The two tetrahedral-void orientations. Point inversion through a bulk node exchanges them
and is an exact crystal symmetry, so no static observable distinguishes them (Theorem 1). Only the
time-oriented worldline, which breaks inversion, can make the label physical.
What else the source derives (context only). Beyond the two facts used here—matter as
a trapped node and the inversion pairing—the source reports several results that this paper does
not import but lists for orientation: the charge thirds from the tetrahedral bond angle; the three
colors from the skew-edge pairs; the confinement of the trapped node behind an L/
√
3 metric wall
(its linear-confinement statement); the proton-to-electron mass ratio from the defect geometry; and
the physical bond scale fixed by matching the black-hole entropy in the companion calibration. No
theorem of the present paper depends on any of these; they are named so the reader can separate
the broader program of the source from the single geometric fact this paper rests on.
A.3 The Vacuum Code [2]
This source [2] is a quantum error-correction paper. Its results are summarized as stated there;
the framework reading that this paper places on top of the code—the verification front and the
worldline—is then flagged separately as this paper’s own, so that nothing is attributed to the code
paper that it does not claim.
Where the qubits and checks live. On the FCC lattice (coordination K = 12), the code
places one physical qubit on every edge. There are two families of stabilizer checks, both of uniform
weight twelve: a Z-check at each vertex, acting on the twelve edges incident to that node, and an
X-check at each octahedral void, acting on the twelve edges of that void (Figure 8). Because every
edge has exactly two endpoints and lies in exactly two octahedra, each vertex check and each void
check share an even number of qubits; the CSS commutation condition H
X
H
T
Z
= 0 over GF(2) then
holds identically, which is what makes the assignment a valid CSS code rather than a mere labeling.
15
Z-check: vertex,
its 12 edges
X-check: octahedral
void, its 12 edges
Figure 8: The two stabilizer families of the vacuum code. Qubits sit on the FCC edges; each vertex
carries a weight-12 Z-check on its incident edges (left), and each octahedral void a weight-12 X-check
on its edges (right). Every edge meets two vertices and two octahedra, so Z- and X-checks always
overlap evenly and the code is CSS.
The code parameters. Computational verification in the source confirms CSS validity and
finds k = 2L
3
+ 2 logical qubits, so the parameters are [[192, 130, 3]] on the L = 4 torus (encoding
rate 130/192 = 67.7%) and [[648, 434, 3]] at L = 6. The minimum distance d = 3 is proven by
exhaustive elimination of all weight-≤2 candidates together with explicit weight-3 codewords, so
every single-qubit error is detectable and correctable. The high rate is traced to a structural surplus:
the lattice has 3L
3
edges but only L
3
−2 independent stabilizer constraints, leaving k = 2L
3
+2 logical
degrees of freedom. The source also supplies a minimum-weight perfect-matching decoder adapted
to the FCC geometry, reports a tenfold coding gain at physical error rate p = 10
−3
(and 63× at
p = 5 × 10
−4
), and discusses neutral-atom and photonic implementations. All of these statements
are rebuilt from the lattice by this paper’s script, so nothing here rests on the preprint’s assertion.
What this paper adds on top (not from the code paper). The present work does not use
the code’s rate, decoder, distance, or error thresholds. It uses only the check geometry above—that
vertex and octahedral checks tile the FCC edges with the stated even overlaps—and it introduces one
ingredient of its own: a chronological order on the commits of those checks along the crystal’s growth
front (Appendix A.2), which it calls the defect’s worldline (Section 4). This ordering is not a result
of the code paper and is not claimed to be; it is this paper’s proposal, and its only used properties are
that it exists and that it is odd under time reversal. Its sole role is to break the inversion symmetry
of Appendix A.2: statically, the two void orientations are exchanged by P and are indistinguishable,
but along a time-ordered worldline the surviving relation is P T , and a P -odd, P T -even structure—
the seed of the paper’s chirality—becomes admissible (Figure 9). No metric, action, or equation of
motion for the worldline is introduced; everything downstream is representation theory of the objects
in Appendix A.1.
static crystal
up
down
P
indistinguishable
on the worldline
t
up (earlier)
down (later)
P T
ordered ⇒ P alone broken
Figure 9: The one new ingredient, and its only job. The code paper supplies the check geometry; this
paper adds a time order on their commits. In the static crystal (left) inversion, P relates the two void
orientations exactly. On the time-ordered worldline (right) one sheet is visited before the other, the
surviving relation is PT , and a P -odd, P T -even structure—the seed of chirality—becomes admissible.
16
Everything else in the paper is then representation theory of the objects defined in Appendix
A.1, with the verification script reproducing every numbered claim.
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18
