F4 to E6: Unfolding Vacuum Chirality and Trinification

From F

to E

: The Worldline Unfolds the Vacuum — Chirality,

Lepton Families, and Triniﬁcation

Raghu Kulkarni

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

raghu@idrive.com

August 2026

Abstract

The predecessor paper closed vacuum and matter into the exceptional algebra f

and left

three marks: charge conjugation equal to spatial inversion, the uncolored matter locked in a

horizontal octet rather than lepton families, and the integer charge unit for ±2/3 absent. This

paper explains all three with one geometric fact and its one physical exception. The fact: the two

tetrahedral void orientations of the lattice are exact inversion partners at every bulk node, so the

static crystal cannot tell them apart — orientation is not a label, and the algebra that results is

orientation-blind. The exception: on a chronological defect worldline the two orientations separate

in time, the inversion pairing relaxes from P to P T , and orientation becomes a physical label.

Doubling every matter weight by that label, with the three glue classes at 120

◦

in a new plane,

assembles exactly the 72-root system of E

— veriﬁed by reﬂection closure, rank, and Cartan

determinant, with the natural competitor (doubling by charge) failing closure. The orientation

ﬂip is an automorphism whose ﬁxed roots are exactly the bonds: f

is the orientation-symmetric

fold of e

, and the equality of charge conjugation with inversion is the folded statement. Deleting

the mark-3 node gives triniﬁcation, three mutually orthogonal su(3)’s; the ﬂip swaps two and

ﬁxes the third, which is color. The 27 branches as (3,

3, 1) ⊕ (1, 3,

3) ⊕ (

3, 1, 3); the color-singlet

block is nine states — lepton families — and folding it reproduces the predecessor’s octet plus

one singlet: the octet was the folded lepton bifundamental. One construction is also excluded:

under the lattice’s geometric parity and charge conjugation, both realized as automorphisms, the

admissible Cartan charges form a single ray, and no point on it extends the thirds; the integer

unit for ±2/3 is not a Cartan charge of e

. Three independent roads — the vector su(2) of the

worldline analysis, the chirality-symmetric triniﬁcation kinematics, and the charge fence — now

converge on a single open frontier: the selection of the chiral projection. Every claim carries a

status tag.

1 Introduction

What the predecessor left. The companion paper [4] derived, on the D

home of the face-

centered-cubic (FCC) vacuum [1], a charge quantized in thirds, the identity of charge conjugation

with spatial inversion, and the closure of vacuum and matter into f

, with an emergent second su(3)

whose 26-branching put three families of colored matter at the down-quark charge. It also left three

marks that this paper takes as its starting questions. 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 answer in one sentence. 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

the algebra of the orientation-aware worldline is e

, and the three marks are what folding e

down to

looks like from below.

Protocol and provenance. As throughout the series, the structure was computed ﬁrst and com-

pared with known physics afterward; comparisons are labeled as comparisons. No new substrate is

introduced: the lattice, its code, and matter as defects [2, 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

, its 27,

and triniﬁcation-type decompositions is in Refs. [11, 8, 10]; ﬁnite-order automorphisms in Ref. [9];

the exceptional Jordan route to adjacent structures in Ref. [12].

2 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 [7]. The tetrahedral voids come in two orientations — cages pointing up and down the

stacking axis — and matter occupies them [2, 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 close-packing sublat-

tice oﬀsets o

= 0, o

), o

), one has o

+ o

= 2o

, so the two orientation

classes map onto each other under inversion through every node, to machine precision.

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 pre-

decessor’s C = P theorem — 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. [veriﬁed; reconstructs an earlier result of the series in void

language]

3 The worldline makes orientation physical

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 a P relation that relaxes to a P T relation on a

chronological worldline, veriﬁed directly.

A chiral structure cannot survive a P identiﬁcation but is compatible with a P T one — the weak

interaction violates P while respecting CP T — so the prohibition is removed, not merely weakened.

That a lattice resists chirality is the content of the Nielsen–Ninomiya theorem [5]; the static obstruc-

tion above is a concrete geometric realization of it, and the worldline relaxation is, in the spirit of the

Ginsparg–Wilson construction [6], a route by which the prohibition is loosened rather than evaded.

On the orientation doublet, parity is the exchange of the two sheets, and the unique generator odd

under P and even under P T is σ

; the oriented out-of-plane growth supplies exactly this generator,

so the worldline genuinely violates parity. [veriﬁed]

(52)

orientation-blind

C = P ; octet; thirds only

(78)

orientation-aware

chiral pair; lepton families

worldline label (Thm. 2)

unfold (Thm. 3)

orientation ﬂip: fold (Thm. 4)

Figure 1: One geometric dichotomy, two algebras. The static crystal identiﬁes the two void orienta-

tions (Theorem 1) and lives in f

; a defect worldline distinguishes them (Theorem 2) and lives in e

;

the ﬂip that forgets the label is the fold. Drawn to true aspect ratio.

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.

4 The unfold: orientation assembles E

The construction. 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

∗

placed at 120

◦

: s 7→ (s, ±v

class(s)

), |v

| = 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): the root system

is E

, uniquely among simply laced systems. The alternative doubling, by charge sector instead of

glue class, fails reﬂection closure.

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

is f

: the predecessor’s algebra

is the fold of this one, and its C = P identity is the folded shadow of the structure below.

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

∗

, so any additive placement must satisfy v

+ v

= 0. The discriminant quadratic

form gives all three classes norm 1, so the v

have equal unit length. Unit length and zero sum

force v

· v

= −

: the class-plane Gram matrix is the discriminant form of D

∗

, realized in two

dimensions, unique up to rotation. Nothing else enters (Figure 1). [both theorems veriﬁed; class

placement derived from the discriminant data; the sole tested competitor is excluded]

5 Triniﬁcation, and what the ﬂip does to it

Theorem 5 (Three su(3)’s, and the ﬂip’s action) Deleting the mark-3 node of the aﬃne E

di-

agram leaves three mutually orthogonal A

’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 elec-

su(3)

color

ﬂip-ﬁxed

orientation ﬂip = parity

(3,

3, 1)

quark block

(1, 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 2: 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 carries the lepton families, and its fold is the predecessor’s octet plus one

singlet. Drawn to true aspect ratio.

troweak interaction requires of it (Figure 2). [veriﬁed; the conjugator exhibiting the ﬂip-ﬁxed factor’s

equality with the geometric color A

, rather than its conjugacy class, is a stated reﬁnement]

6 The 27: lepton families, and the octet explained

Theorem 6 (Families) The 27-dimensional minuscule representation branches under the three fac-

tors as (3,

3, 1) ⊕ (1, 3,

3)⊕ (

3, 1, 3), nine states each. The color-singlet block is a bifundamental of the

ﬂip-swapped pair: three lepton families of each chirality 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 bifunda-

mental, 3 ⊗

3 = 8 ⊕ 1, locked to its diagonal because the orientation-blind algebra could not tell the

two chirality types apart. Lepton families exist from the moment the crystal’s worldline remembers

which void is which. [veriﬁed; “lepton” here means color-singlet family-structured matter — attaching

the Standard Model’s chiral quantum numbers to these states is not claimed]

7 A fence: the integer unit is not a Cartan charge

The remaining mark is ±2/3. We ask whether the unfolded algebra supplies it as a Cartan charge,

and the answer is no, for a symmetry reason, computed as follows. 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

, and reﬂects the class plane across the ﬁxed

class’s axis, swapping the two spinor classes). An electric-charge functional must be P -even, C-odd,

and commute with color. On the Cartan these three conditions admit exactly a one-parameter ray.

Direct scan shows no point of that ray extends the thirds: states with equal pairing against the ray

carry diﬀerent color types, so no Cartan functional reproduces the fractional pattern, let alone adds

an integer unit.

The conclusion is a fence with content: under the lattice’s own geometric C and P , the up-type charge

unit cannot be a Cartan charge of e

. In the Standard Model the same unit enters through T

of the

chiral weak group — available only after the left projection selects a chirality. The fence therefore

points where everything else in this paper points. [derived negative; veriﬁed]

8 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. [excluded by the commutant theorem]

The single frontier. Three independent roads 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 charge fence shows

the integer unit waits on exactly that selection. The chiral projection P

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

and E

extensions are explicitly fenced

oﬀ until that gate is settled. [open; three convergent statements veriﬁed]

9 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. Any lattice construction

that produces a ±2/3 charge as a symmetric Cartan functional under the geometric C and P above

refutes the fence of Section 7. And the family reading now stands on the unfold: if the ﬂip-ﬁxed

factor proves not conjugate to the geometric color A

, the lepton-family identiﬁcation falls with it.

[falsiﬁers stated]

10 What is derived and what is not

statement status

up/down void environments are inversion partners veriﬁed (theorem)

worldline relaxes the pairing from P to P T veriﬁed (theorem)

generator: P -odd, P T -even; parity violated veriﬁed

orientation×class doubling closes as E

veriﬁed (theorem)

charge-based doubling fails closure competitor excluded

ﬂip is an automorphism; ﬁxed roots = bonds; fold = f

veriﬁed (theorem)

class-plane inner products = discriminant form of D

∗

derived

triniﬁcation realized; ﬂip swaps two factors, ﬁxes color veriﬁed (theorem)

ﬂip-ﬁxed factor equal (not just conjugate) to geometric color open (reﬁnement)

27 = (3,

3, 1) ⊕ (1, 3,

3) ⊕ (

3, 1, 3); lepton block veriﬁed (theorem)

octet = folded lepton bifundamental (+ one singlet) veriﬁed

Standard Model chiral charges attached to the lepton block not claimed

C realized as an automorphism; admissible charges = one ray veriﬁed

integer unit is not a Cartan charge under geometric C, P derived negative

chiral projection P

open (single frontier)

, E

extensions fenced

11 Conclusion

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

: 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

: the fold explains the ﬁrst mark,

the unfolded bifundamental explains the second, and the charge fence shows the third waits — with

the other two roads — at the chiral projection. The frontier is now single, named, and dynamical.

That is the position a continuation should leave its series in.

Declaration of competing interest

The author declares that he has no known competing ﬁnancial interests or personal relationships

that could have appeared to inﬂuence the work reported in this paper.

Data availability

All computations are speciﬁed in the text. Python scripts reproducing every check — the inversion

pairing and P → P T computations, and the full E

chain including the closure, the fold, triniﬁca-

tion, the 27, and the charge fence — are available at github.com/raghu91302/ssmtheory/raw/main/

ew scripts.zip and .../e6 scripts.zip (NumPy; the generator classiﬁcation also SciPy).

References

[1] R. Kulkarni, “A 67%-rate CSS code on the FCC lattice: [[192, 130, 3]] from weight-12 stabilizers,”

arXiv:2603.20294 (2026).

[2] R. Kulkarni, “Matter as incomplete crystallization,” Phys. Open 27, 100423 (2026).

doi:10.1016/j.physo.2026.100423.

[3] R. Kulkarni, “The mass–energy–information equivalence: a bottom-up identiﬁcation

of the particle spectrum via FCC lattice error correction,” Phys. Open (2026).

doi:10.1016/j.physo.2026.100414.

[4] R. Kulkarni, “From D

to F

: color, matter, and charge from one triality twist,” Zenodo preprint

(2026), submitted for publication. doi:10.5281/zenodo.22164400.

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

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

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

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

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

(1976).

[8] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer,

New York (1999). doi:10.1007/978-1-4757-6568-7.

[9] V. G. Kac, Inﬁnite-Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge

(1990). doi:10.1017/CBO9780511626234.

[10] J. C. Baez, “The octonions,” Bull. Am. Math. Soc. 39, 145 (2002). doi:10.1090/S0273-0979-01-

00934-X.

[11] R. Slansky, “Group theory for uniﬁed model building,” Phys. Rep. 79, 1 (1981).

doi:10.1016/0370-1573(81)90092-2.

[12] I. Todorov and M. Dubois-Violette, “Deducing the symmetry of the standard model from the

automorphism and structure groups of the exceptional Jordan algebra,” Int. J. Mod. Phys. A

33, 1850118 (2018). doi:10.1142/S0217751X1850118X.