Koide's Lepton Mass Formula via 6:3:3 Bonded FCC Lattice

Koide’s lepton mass relation from the 6 : 3 : 3 bond split

of the face-centered-cubic lattice

Raghu Kulkarni

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

raghu@idrive.com

Abstract

Koide’s relation, K = (m

+ m

)/(

√

)

= 2/3, holds for the charged-

lepton pole masses to about 10

−5

and has no accepted explanation. It is equivalent to the amplitudes

√

being the eigenvalues of a 3 ×3 circulant with off-diagonal-to-diagonal ratio 1/

√

2. We show

that the face-centered-cubic (FCC) vacuum lattice of the Selection–Stitch Model supplies this ratio.

Along a ⟨111⟩axis each site’s twelve bonds divide 6 : 3 : 3 between its own sheet and the sheets above

and below; a localized state sees couplings

√

6 :

√

3 :

√

3, and ABC stacking cycles the three sheet

types. Given six stated assumptions, K = 2/3 follows exactly, with no ﬁtted parameter, as a bond

count: K is the total coordination divided by three times the in-sheet coordination. One phase, ﬁtted

to the electron and muon masses, gives m

= 3477.44, against 3477.37±0.18 observed. Among

the lattices and crystals surveyed, only FCC has this structure below six dimensions. Delocalized

states give K = 1/2, implying

∑

= 59.8 meV for delocalized neutrinos in the normal ordering;

states in tetrahedral voids, the model’s quarks, give K ≤ 0.614. On a straight column of sites, the

stacking step acts as the coordinate cycle that earlier work calls triality, which we show is a Weyl-

group element of D

, not its outer automorphism. Lepton mixing requires a neutrino sector that is

complex in the basis of the sheet types.

Keywords: Koide relation; charged-lepton masses; lepton generations; face-centered-cubic lattice; cir-

culant mass matrix; neutrino masses

1 Introduction

Koide observed [1, 2] that the charged-lepton masses satisfy

K =

+ m



√



. (1)

With the Particle Data Group masses [3], K = 0.666664 ±0.000005, equal to 2/3 within its uncertainty

of about one part in 10

. The relation has motivated many attempts at explanation [4], and no accepted

theory derives it.

Two facts constrain an explanation. First, the relation holds for pole masses, the masses of propagat-

ing particles, and fails for running masses at a common scale (K = 0.667929 at M

[5]). Its origin should

therefore concern physical particles rather than the parameters of a high-energy Lagrangian. Second, it

has a simple geometry. Writing

√

= M



1 + r cos



δ +

2πk



, (2)

Eq. (1) is exactly r =

√

2: the three leptons are three points 120

◦

apart on a circle, and the vector

(

√

) makes 45

◦

with (1,1,1) [6, 7]; it has also been recast as a null-cone condition and

as a level set of a scale-invariant functional [8]. Equivalently, the amplitudes

√

are the eigenvalues

of a 3 ×3 circulant whose off-diagonal-to-diagonal ratio is 1/

√

2. An explanation in these terms needs

a threefold cyclic symmetry on the generations and a reason for that ratio. The phase δ is a separate

number.

The circulant form is known [9], and so is the fact that the phase it needs is close to 2/9 [9]. What

has been missing is a structure that produces the ratio 1/

√

2: a physical reason why the off-diagonal

coupling is that fraction of the diagonal one. The contribution of this paper is such a structure, not the

circulant itself. We propose the face-centered-cubic lattice with ABC stacking (Fig. 1). Along a stacking

axis each site’s bonds split 6 : 3 : 3, a localized state’s couplings go as the square root of the number of

neighbors, and the stacking cycles the three sheet types:

6 : 3 : 3

localized, normalized coupling

−−−−−−−−−−−−−−−−−→

√

6 :

√

3 :

√

ABC circulant

−−−−−−−−→ K =

. (3)

The structure is speciﬁc. A survey of 54 lattices in two to eight dimensions, and of 12 other crystals

and close-packed stackings, ﬁnds no other structure below six dimensions that gives the ratio 1/

√

with a threefold cycle (Section 8, Table 2). Within this construction, Koide’s relation selects FCC. This

is a statement about the alternatives examined, not a uniqueness theorem, but it has a sharper form:

FCC is the root lattice A

, and in the series A

the number of sheet types is d while the bond ratio is

= 1/(d −1), so three families and Koide’s ratio are the same condition, d = 3.

The construction rests on six assumptions (Section 3), and two of them carry the physics. A2 iden-

tiﬁes the lepton families with the three symmetry sectors of the stacking. A4 builds the family operator

from the couplings a localized state sees; the operator is assumed, not projected from the lattice Hamil-

tonian, and Section 5 states exactly what this contains. Given the assumptions, no parameter is ﬁtted to

obtain K = 2/3. The hierarchy needs one more number, a phase, which is ﬁtted to data (Section 6).

The paper belongs to a series on the Selection–Stitch Model. Ref. [10] introduced the model’s

account of particle masses, and Ref. [11] extended it from three dimensions to four. In those papers a

mass is an integer, the number of classical bits generated in verifying the particle (Section 2). For the

charged leptons the integers are C

= 1, C

= 207 and C

= 3483, which give K = 0.66672. No integer

can give 2/3 exactly: with C

= 1 and C

= 207, the third would have to be 3480.997. The present

construction keeps the lattice but uses amplitudes instead of counts. Section 10 shows how the two

descriptions ﬁt together. The supports of the counted muon and tau, and the bond group that anchors the

electron, share the bond division that gives Koide’s ratio. The threefold coordinate cycle that Ref. [11]

calls triality, whose three bond groups it uses to deﬁne its generations, is the rotation about the stacking

axis used here. It is an automorphism of the code of Ref. [11] but, despite its name, not the outer triality

automorphism of D

. On a straight column of sites a step up the axis acts as this cycle, so the family

states of this paper are its charges. Taking these charges as the generations differs from the assignment

of Ref. [11], which uses engagement depths, and is offered as a proposal (Section 10.3).

Section 2 summarizes the model and Section 3 states the assumptions. Section 4 sets out the three-

fold symmetry, and Section 5 derives the relation and states what A4 assumes. Section 6 treats the

phase. Section 7 shows that delocalized states give K

= 1/2, the value proposed empirically by Li

and Ma [12], and that no sector respecting the stacking translation gives realistic lepton mixing. Sec-

tion 8 tests alternatives to each assumption and to the lattice, Section 9 explains why quarks do not

satisfy the relation, Section 10 relates the construction to the triality description of Ref. [11] and de-

rives a condition on lepton mixing, and Section 11 concludes. An interactive three-dimensional view

of the lattice, its sheets and the three lepton states of Section 5 accompanies the paper at https:

//raghu91302.github.io/ssmtheory/koide_fcc_leptons.html.

[111]

(a) the cubic FCC cell: atoms at corners and

face centers; (111) sheets cut across the body diagonal

[111]

(b) 2 × 2 × 2 cells seen along [1



10]: the (111)

sheets are edge-on, repeating A, B, C along [111]

[111]

sheets A, B, C (spacing exaggerated)

(d) seen down [111]: B and C sit over

the two kinds of hollow in A

sheet A sheet B sheet C

6 in own

sheet (A)

3 above (B)

3 below (C)

(e) one site and its 12 bonds: 6 : 3 : 3

Colors: sheet type along [111]

A layer index 0 (mod 3)

B layer index 1 (mod 3)

C layer index 2 (mod 3)

Sheet spacing

√

2/3

L; bond length L;

cubic cell edge

√

Figure 1: The face-centered-cubic lattice and its three sheet types along a ⟨111⟩ growth axis; colors mark

the sheet type, the layer index along [111] modulo 3. (a) The conventional cubic cell, with atoms at the

corners and face centers. The (111) sheets are the planes perpendicular to the body diagonal: the shaded

triangles are one B sheet and one C sheet, each holding six atoms of the cell, and the two A atoms lie at

the ends of the diagonal. (b) A block of 2 ×2 ×2 cells seen along [1

10], where the sheets are edge-on

and appear as bands repeating A, B, C along [111]. (c) The same lattice as a stack of triangular sheets

(true sheet spacing

2/3L, exaggerated in the drawing). (d) Seen down the growth axis, a B sheet sits

over one kind of hollow of an A sheet and a C sheet over the other; the three sheet types are the three

lateral positions, or registries, a sheet can take. (e) A site of type A and its twelve nearest neighbors:

six in its own sheet (solid bonds), three in the B sheet above and three in the C sheet below (dashed).

This 6 : 3 : 3 split, and the cycle A → B → C that the stacking produces, are the geometric inputs of the

derivation.

2 The Selection–Stitch Model in brief

The Selection–Stitch Model (SSM) is a proposal that the vacuum is a crystal. Its elements, and the parts

of it this paper uses, are the following.

The vacuum lattice and its code. The vacuum is taken to be a face-centered-cubic lattice of nodes

joined by bonds of one length L, each node having twelve nearest neighbors [10]. The lattice carries

a quantum error-correcting code with one qubit on each bond [13]. The code is a set of local checks

on these qubits. A defect is a local pattern that fails some of the checks, and a particle is a localized,

persistent syndrome pattern [10].

Mass as veriﬁcation cost. In the model a defect’s rest mass is the energy dissipated in verifying it,

kT ln 2 per classical bit [10]. The number of bits, C

= E

×C

, the number of edges in the defect’s

footprint times the number of checks able to detect it, is the defect’s veriﬁcation cost. Only mass ratios

are predicted [10]. Counting bits gives m

= 207 [10] and m

= 3483 [11]. This paper does not

use those counts. Its masses come from the family operator of Section 5; Section 10 shows how the two

are related.

Four dimensions. Ref. [11] treats spacetime as the four-dimensional lattice D

, in which each node

has 24 nearest neighbors. The 12 neighbors that share the node’s time coordinate are its twelve nearest

neighbors in the FCC lattice of its time slice. The 24 neighbors fall into three groups of eight. Each group

forms the corners of a regular four-dimensional solid called a sixteen-cell, and a cyclic permutation of

coordinates, which Ref. [11] calls triality, permutes the three groups. Despite the name it is not the

outer triality automorphism of D

(Section 10.2). Within one time slice, each group contributes four of a

node’s twelve FCC bonds; we call these four bonds a triality component. In Ref. [11] the three charged

leptons differ in how much of this structure their path through spacetime (their worldline) touches, called

its engagement depth: one sixteen-cell for the electron, the spatial parts of all three for the muon, and all

three in full for the tau. Within a time slice, this cycle permutes the three spatial coordinates cyclically;

we use the direction (x,y,z) 7→ (z,x,y), the inverse of the one written in Ref. [11], which generates the

same group. This paper uses this structure only in Sections 4, 7, 8 and 10 and Appendix B.

Leptons and quarks. A charged lepton is a small, localized defect; the electron is the single-bit defect

of Ref. [10], one bond across. This paper uses only that fact, that charged leptons are localized, and not

their internal structure. It models a charged lepton as a state on a node, in a sheet (A4 below), because

a node always belongs to exactly one sheet type (Fig. 1), whereas a bond may join two. Neutrinos are

neutral and propagate rather than staying bound, so they are treated as delocalized (A4); Section 7 is

conditional on that treatment. Quarks sit in the tetrahedral voids between sheets [14], not on a node in a

sheet; Section 9 uses this contrast.

Growth: stitch and lift, and the observable patch. In the model the lattice is not static; it grows [14].

A stitch places a new node in the plane of a sheet, so repeated stitches build ﬂat triangular sheets. A

lift places a new node above the center of a triangle and starts the next sheet. Each new sheet can sit

over either of two kinds of hollow in the sheet below. When every new sheet continues the sequence A

→ B → C, the stack is FCC with the ABC stacking of Fig. 1, which A1 assumes. The direction along

which the sheets pile up is the growth axis. Near where growth began, the lattice grows outward in every

direction. A region far from there grows at a front, the moving edge of the lattice, that advances one

way, so its growth axis has a direction; this paper uses that direction only through A6. By an observable

patch we mean a region the size of the observable universe; A1 takes the growth axis to be the same

throughout it.

3 Assumptions

The derivation rests on six assumptions. The ﬁrst two concern the lattice and how the families sit on it.

The next three concern single-particle quantum mechanics on the lattice. The last concerns the growth

that produced the lattice.

A1 (Lattice). The vacuum is an FCC lattice with nearest-neighbor bonds of length L, stacked in the

ABC sequence along a growth axis, one of its four ⟨111⟩ directions. Within an observable patch

the growth axis, and with it the stacking sequence, is uniform.

A2 (Families). The lepton family space is spanned by three basis states, one for each sheet type A, B,

C of the growth axis (the sites whose layer index is 0,1,2 modulo 3). The family operator acts on

this three-dimensional space, and the physical leptons e, µ, τ are its eigenstates.

A3 (Hopping). A single particle moves by hopping along bonds, with one amplitude t per bond.

A4 (Localization). Charged leptons are localized: the family basis state of type k is a state at a site

of type k, and its amplitude into basis state l is its coupling to the normalized combination of its

neighbors of type l. Neutrinos are delocalized: their basis state of type k is spread uniformly over

the sites of type k.

A5 (Mass). A lepton’s rest mass is its second-order self-energy, the energy of a virtual hop from its

state and back. Second-order perturbation theory gives ∆E

(2)

∑

|⟨n|V |i⟩|

/(E

−E

); under

the additional assumption that the virtual-state denominators are independent of the family, this is

represented by a mass operator M ∝ W

, where W is the family operator of ﬁrst-order couplings.

The amplitudes

√

are then the eigenvalues of W (when non-negative; see Theorem 1).

A6 (Orientation). The growth axis is oriented. A hop to the sheet above carries a phase e

iδ

and a hop

to the sheet below the phase e

−iδ

, with the sign ﬁxed by the growth direction.

A1 is the model’s geometry. A2 is the central hypothesis. It is made here, not derived from the

model; Section 4 states precisely what it assumes, and Section 10 compares it with another assignment

in the series. A3 is the minimal dynamics on a lattice. A4 follows the model’s picture of charged

leptons as localized defects and of neutrinos as propagating. Its couplings are those a state localized

in the plane sees; Section 5 shows that they are not obtained by projecting the lattice Hamiltonian onto

three single-site states. A5 reads the model’s identiﬁcation of mass with the cost of maintaining a defect

as the energy of the virtual hops that maintain it. Its form follows second-order perturbation theory,

(2)

eff

= W D

−1

†

, with D the operator of virtual-state denominators. A5 takes D

−1

proportional to the

identity, so M ∝ WW

†

, which equals W

for the Hermitian W of Eq. (6). This is a structural analogy,

not a derivation. A6 expresses that, far from where growth began, the lattice grew in one direction

(Section 2). For charged leptons, the phase δ and the overall scale t are the only parameters. Neutrinos

have a phase δ

and a scale of their own, which the two measured mass splittings ﬁx (Section 7).

4 The stacking axis and its threefold symmetry

Along the growth axis the FCC lattice is a stack of triangular sheets, and every site belongs to exactly

one sheet type. A site’s twelve bonds divide by sheet,

12 = 6

|{z}

own sheet

+ 3

|{z}

sheet above

+ 3

|{z}

sheet below

. (4)

In ABC stacking the sheets above and below a site of type A are of types B and C, so a hop up and a hop

down lead to different types (Fig. 1).

The three types are related by a symmetry of the lattice. Each of the three bonds from a site to the

sheet above is a lattice vector. Translation by such a bond maps the lattice onto itself and raises every

layer index by one, sending A to B, B to C and C to A. The hopping Hamiltonian of A3 commutes with

this translation, and so do the A4 couplings built from it, so every coupling between types is unchanged

by the cyclic shift: the coupling from A to B equals those from B to C and from C to A, and likewise for

the couplings within a type and to the type below. The phases of A6 depend only on whether a hop goes

up or down, which the translation preserves. This is the threefold cyclic symmetry on the generations,

and it makes the family operator of Section 5 a circulant.

Write P for this translation acting on the sheet types, A → B → C → A. We call the three-

dimensional space spanned by the sheet types the registry space. On it, states that differ by a full

stacking period count as the same, so P

= 1, and the eigenvalues of P are 1, ω and ω

, with ω = e

2πi/3

The corresponding eigenstates, the characters of P, are the Fourier states of Section 5. They, not the

individual sheets, are the physical family states. Because P is a lattice translation, each character carries

a Bloch phase, the phase a crystal state gains under a lattice translation.

A2, with the translation symmetry, amounts to two hypotheses: that the registry space is the lepton

family space, and that its three characters are e, µ and τ. Neither can be derived within the present

model. Translation by one layer is an exact symmetry of the FCC code [13], so the registry by itself is

not a code label. (Section 10.2 shows that on a straight column of sites the registry character coincides

with a charge under the triality cycle of Ref. [11], an order-three automorphism of the D

code of that

work. This gives A2 a counterpart in the code but does not derive it.) For a particle with no internal

structure, the three characters are one band of the lattice folded into the ABC cell. And a defect’s internal

orientation leaves the family label positional (script koide A2 tests.py). A2 is therefore a valley-type

hypothesis. In crystals such as graphene and silicon, electrons in different minima, or valleys, of one

energy band behave as separate, approximately conserved species. A2 assumes the same of the three

Bloch-phase sectors of the stacking.

5 Derivation

Couplings (A3, A4). By A3 a state at a site couples with amplitude t to each neighbor. Its coupling to

the normalized combination of its n neighbors of one type, |n

ℓ

⟩ = n

−1/2

∑

|j⟩, is

⟨n

ℓ

|H|site⟩ =

√

·nt =

√

n t, (5)

the collective enhancement familiar from superradiance [15]. By Eq. (4), n = 6 within the own sheet and

n = 3 for each adjacent sheet. For delocalized states, spread uniformly over whole sheet types, the matrix

elements are instead the bond counts 6,3,3, because all sites of one type add in phase. Appendix A

checks both on an explicit lattice.

Family operator (A2, A5, A6). By the translation symmetry of Section 4, the family operator is a

circulant in the three types. With P the cyclic shift of Section 4 and the phases of A6,

W = t

√

6I +

√



iδ

P + e

−iδ

−1

i

√

1 +

√

2cos



δ +

2πk



. (6)

Theorem 1 (Koide’s relation). Under A1–A6, the charged-lepton masses satisfy K = 2/3 for every δ

at which the three amplitudes are non-negative. The ratio

√

6 = 1/

√

2 is Koide’s r =

√

2 in Eq. (2).

The proof is two lines. Write the amplitudes as x

= A[1 +

√

2cosθ

] with θ

= δ + 2πk/3. Since

∑

cosθ

= 0 and

∑

cos

= 3/2 for any δ ,

∑

= 3A,

∑

= A



3 + 2 ·



= 6A

, K =

∑

(

∑

)

. (7)

For signed eigenvalues the relation holds at every δ . Physical masses need non-negative amplitudes,

which holds for |δ | ≤π/12 (modulo 2π/3); at δ = π/12 the electron amplitude vanishes (Fig. 2).

The relation in terms of bond counts. The same result follows from traces. The inter-sheet terms of

W are traceless, so the amplitude sum sees only the in-sheet bonds, while the mass sum sees all of them:

∑

√

= TrW = 3

√

∑

= TrW

= 3zt

, K =

, (8)

with n

= 6 the in-sheet coordination and z = 12 the total coordination. Koide’s relation, K = 2/3, is

therefore the statement z = 2n

: half of a site’s bonds lie in its own sheet and half leave it, 6 = 3 + 3.

The phase drops out because traces do not depend on it. This also restates Koide’s geometry on the

lattice. The vector (

√

) makes 45

◦

with (1,1,1) because its democratic component is the

in-sheet coupling, which cannot distinguish the families, and its transverse component is the inter-sheet

coupling, which does; the two have equal squared norm, 18t

each, exactly when the in-sheet bonds

equal the out-of-sheet bonds.

Proposition 2 (neutrinos). Under A1–A6, for which neutrinos are delocalized (A4), K

= 1/2 for

every δ

, and all three amplitudes are non-negative. The bond counts 6,3,3 give a ratio of 1/2, so

√

= a[1 + cos(δ

+ 2πk/3)] ≥ 0. The same two lines as in Eq. (7), with

√

2 replaced by 1, give

∑

= 3a and

∑

, so K = 1/2.

Proposition 3 (self-energy). The masses are the eigenvalues of W

, whose diagonal is 12t

, the co-

ordination number. Every generation has the same total virtual hopping; the masses differ because of

the couplings between sheets. Without a phase, these split the spectrum into one heavy lepton and two

degenerate light ones, and the opposite phases of the excursions up and down lift the remaining degen-

eracy. For every δ the eigenstates of W are the characters of P, (1,ω

−k

,ω

−2k

√

3 for the label k of

Eq. (6) with P|A⟩= |B⟩ (Section 4): each physical lepton has equal weight on the three sheet types, and

the generations differ only in the phases between sheets (Fig. 3).

Proposition 4 (the orientation is required). If the stacking axis is not oriented, two of the three

masses are degenerate. Hermiticity gives a hop down the conjugate phase of the hop up it reverses.

Without an oriented axis, inversion maps each hop up onto a hop down with the same amplitude, so

the phase e

3iδ

around the cycle must be real. A diagonal rephasing of A, B, C changes individual link

phases but leaves the total phase 3δ around the cycle A →B →C →A unchanged; keeping the hopping

uniform, it can shift δ only by 2πn/3, which relabels the characters. Inversion symmetry therefore leaves

δ ≡ 0 or π/3 (mod 2π/3). The eigenvalues of W are then t(

√

6 + 2

√

3) and, twice, t(

√

6 −

√

3); or,

twice, t(

√

6 +

√

3) and t(

√

6 −2

√

3). Either way two coincide, so a non-degenerate spectrum requires

the oriented axis of A6: an oriented, rather than inversion-symmetric, family coupling.

Lemma 5 (the couplings of A4). Decompose the hopping into waves of in-plane momentum q, with

in-sheet coupling ε(q) and coupling f (q) to an adjacent sheet, in units of t. Then |f (q)|

= 3 + ε(q)

at every q, and the root-mean-square values of ε and f over the in-plane Brillouin zone are exactly

√

6 and

√

3. The mean of |f |

is the number of bonds to one adjacent sheet, 3, and the mean of ε

is the number of in-sheet bonds, 6, by Parseval’s theorem; the identity is checked on a ﬁne grid of q

by koide projection.py. A state localized in the plane couples to all in-plane momenta with equal

weight, so these root-mean-square values are the couplings it sees, and A4 takes them as the entries of

W .

What A4 assumes. W is therefore assumed, not projected from the lattice Hamiltonian. The as-

sumption is smaller than it looks. For any three orthonormal single-site family states, the second-order

operator H

restricted to them has trace 3zt

whatever its off-diagonal elements, so the mass sum rule

of Eq. (8) holds rigorously. What A4 adds is the amplitude sum rule alone,

∑

√

= 3

√

t. Two re-

sults show that this one number is not supplied by an exact projection. First, no three single-site family

states reproduce W . At the ﬁtted phase WW

†

would need an off-diagonal element of 11.0t

against a

diagonal of 12t

, while two FCC sites share at most four neighbors; the true second-order operator for

such states, with virtual hops through sites outside the family, gives K between 0.333 and 0.352, with

μ τ

√

(a) the Koide circle:

√

∝ 1 +

√

cos(δ + 2πk/3)

0.0 0.2 0.4 0.6 0.8 1.0

phase δ

0.0

0.5

1.0

1.5

eigen-amplitude ∝

√

(b) localized family operator:

√

vs δ

0.0 0.2 0.4 0.6 0.8 1.0

phase δ

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

electron amplitude < 0

π/12: m

= 0

measured

localized (

√

)

delocalized (6 : 3)

Figure 2: (a) The Koide circle: the three amplitudes

√

are the projections of three points 120

◦

apart

on a circle of radius

√

2 centered at 1, rotated by δ . (b) The three eigen-amplitudes of the localized

family operator, Eq. (6), as functions of the phase; the measured leptons sit at δ = 0.2222 (dotted), and

the electron amplitude vanishes at δ = π/12 (dashed). (c) K against δ: 2/3 for localized states wherever

all amplitudes are non-negative (|δ |≤π/12; beyond it, with

√

m ≥0, the relation fails, dotted), and 1/2

for delocalized states at every δ.

an amplitude sum of 9.5t to 10.4t against the 7.35t that Koide’s relation requires. The relation needs

the three amplitudes to interfere far more strongly than single-site states allow: at the ﬁtted phase the

electron’s in-sheet amplitude +2.45t and out-of-sheet amplitude −2.35t nearly cancel, which is why it

is so light (Fig. 3). Second, waves within a sheet do give an exact projection, a circulant with entries

ε(q) and f (q). By the identity, it has Koide’s ratio only on the contours ε = (1 ±

√

7)t. On the ﬁrst

the phase it supplies is at most 0.021, and on the second it lies outside the physical range |δ | ≤ π/12;

the observed hierarchy would need an added twist along the stacking axis, as in A6. At zero in-plane

momentum the same projection gives (6,3), the delocalized operator of Proposition 2, so the neutrino

result is an exact projection. Deriving A4 from a projection is, with A2, the main open problem (script

koide projection.py).

6 The charged-lepton phase

Determination from data. Apart from the overall scale, the phase is the one parameter of A1–A6. The

electron and muon masses ﬁx it at δ = 0.2222220. This lies inside the physical range of Theorem 1 and

close to its edge, which is why the electron is much lighter than the muon. Eq. (6) then gives m

3477.44, or m

= 1776.969 MeV, against the measured 3477.37±0.18 (1776.93±0.09 MeV [3]), a dif-

ference of 0.4σ.

This tests K = 2/3 itself: with the phase ﬁxed by two masses, the relation determines

the third. The ﬁtted value equals 2/9 to eight parts in 10

, as Brannen observed [9]; the construction

here does not explain that value, and we make no use of it.

Origin. By Proposition 4 the phase must be odd under reversal of the stacking axis, so its origin must

distinguish up from down. In the model that is the direction in which the lattice grew (A6). Equivalently,

δ is a Bloch twist: the phase 3δ gained around one stacking period acts like a ﬂux through the three-

cycle. The value of δ is not derived here. The phase at which the operator gives exactly the integer

muon count 207 of Ref. [11] is 0.222242, within 2 ×10

−5

of the ﬁt (Section 10).

We use the PDG 2024 masses throughout. Ref. [11] quotes the CODATA ratio m

= 3477.23; the two values differ by

less than one standard deviation, and every deviation quoted in this paper uses 3477.37.

electron

0°

240°

120°

−

2 0 2 4 6

amplitude

in-sheet

between sheets

total

√

+2.45

2.35

+0.10

= 1

muon

0°

120°

240°

−

2 0 2 4 6

amplitude

+2.45

1.03

+1.42

= 206.77

tau

0°

−

2 0 2 4 6

amplitude

+2.45

+3.38

+5.83

= 3477.44

lepton phase 0° lepton phase 120° lepton phase 240°

Figure 3: The three charged leptons on the FCC lattice, drawn to scale. Top: a block of 2 ×2 ×2 cubic

cells (corners, face centers and nearest-neighbor bonds), stood on a corner so that the ⟨111⟩ stacking

axis is vertical. Three of its (111) sheets, A, B and C, are shaded in the colors of Fig. 1; sites off these

sheets are gray. A lepton is an eigenstate of the family operator of Eq. (6) at the ﬁtted phase: equal

weight on one site of each sheet type (outlined sites, joined by bonds between sheets). Its phase on each

sheet, from the eigenvector, is given by the ﬁll color of the site and by an arrow in the sheet. The tau

has the same phase on all three sheets; the electron and the muon advance by 240

◦

and 120

◦

per sheet.

Bottom: each lepton’s amplitude

√

m in units of t, the sum of the in-sheet coupling

√

6t, the same for

all three, and the coupling between sheets 2

√

3t cos(δ + 2πk/3), which depends on those phases. For

the electron the two nearly cancel, which is why it is light. The tau ratio is the prediction of Section 6.

7 Neutrinos: the origin of K

= 1/2

This section rests on a step less established than the localization of charged leptons: that a neutrino,

which propagates rather than staying bound to a site, is spread uniformly over each sheet type (A4).

Given it, Proposition 2 gives K

= 1/2, with the non-negative amplitudes

√

= a



1 + cos



2πk



. (9)

The value K

= 1/2 is not new. Li and Ma [12] proposed it empirically, from a conjectured complemen-

tarity between quark and lepton masses; in their normalization it reads k

= 3/4, with K =

k, and they

found it consistent with the normal ordering. A seesaw construction [16] gives neutrino masses with a

Koide ratio close to 1/2. What the present construction adds is a reason for the value: the geometry that

gives 2/3 for localized states gives 1/2 for delocalized ones, because the couplings go as

√

n in the ﬁrst

case and as n in the second. Section 5 shows that the delocalized operator is an exact projection of the

lattice Hamiltonian.

We take the two measured splittings from the global ﬁt that includes the ﬁrst JUNO data [18, 19]:

∆m

= 7.53 ×10

−5

and ∆m

= 2.529 ×10

−3

(normal) or ∆m

= −2.484 ×10

−3

(in-

verted). They ﬁx both a and δ

, and hence the absolute masses (script neutrino.py). The masses are

consequences of K

= 1/2, so Li and Ma’s value gives the same numbers; they are not independent

predictions of this construction (Table 1).

ordering δ

(meV) m

(meV)

∑

(meV)

normal 0.332 0.80 8.71 50.30 59.8

inverted 1.041 49.08 49.84 0.00 98.9

Table 1: Neutrino masses for K

= 1/2 (Proposition 2), with the phase δ

and scale ﬁxed by the two

measured mass splittings.

In both orderings the spectrum is hierarchical and the lightest mass is ﬁxed. The inverted solution,

with m

≈ 0 and

∑

= 98.9 meV, is the minimal inverted spectrum. The normal solution has m

0.80 meV and

∑

= 59.8 meV, 0.8 meV above the minimal normal spectrum (59.0 meV). The value

= 1/2 therefore excludes quasi-degenerate neutrinos and ﬁxes the sum for each ordering.

Comparison with data. The global oscillation ﬁt including the ﬁrst JUNO data prefers the normal

ordering, with ∆χ

= 9.4 when atmospheric neutrino data are included [18]. DESI DR2 baryon acoustic

oscillations combined with the cosmic microwave background give

∑

< 64.2 meV (95%) in the

standard cosmological model [20]. This bound lies above 59.8 meV and below the minimal inverted

sum, so the normal-ordering solution is not excluded (Fig. 4). In that model the data prefer sums below

the oscillation minimum, a tension of about 3σ shared by every normal-ordering spectrum; it disappears

in models with evolving dark energy, where the bound relaxes to 163 meV.

Tests. The sum is one sharp test. A value clearly above about 60 meV with the normal ordering, or

above 99 meV with the inverted ordering, would falsify K

= 1/2. Planned cosmological surveys, with a

forecast resolution of about 16 meV [21], can conﬁrm a sum near 60 meV and exclude much larger ones,

but cannot distinguish 59.8 meV from the minimum. With the measured mixing angles (NuFIT 6.0 [17]),

the beta-decay effective mass is about 8.9 meV, far below the direct limit of 0.45 eV [22]. If neutrinos

are Majorana particles, the effective mass for neutrinoless double-beta decay is 1.0 to 4.3 meV for any

Majorana phases, below the current limit of 28 to 122 meV [23]; a signal above 4.3 meV with the

normal ordering would falsify K

= 1/2. If neutrinos were instead localized (K = 2/3), no solution

would have non-negative amplitudes. Allowing a negative one, as in Brannen’s neutrino version of

Koide’s formula [24], gives m

= 0.36 meV and

∑

= 59.3 meV.

Lepton mixing. The construction as written gives no realistic lepton mixing, and the limitation is

general. If the families are the characters of the stacking translation P, any sector whose mass operator

respects P has those characters as its mass eigenstates, so Dirac neutrinos do not mix. This holds at

every in-plane momentum. It also holds for neutrinos that move along the light-like (null) bonds of the

four-dimensional lattice of Ref. [11], since two null steps make one stacking translation. A Majorana

mass can pair only the characters k and −k. It gives one unmixed neutrino and a degenerate, maximally

mixed pair, which the data exclude (the pair would also contradict ∆m

= 0): the unmixed state would

be pure tau, while the observed ν

is about half muon and half tau. A different choice of neutrino basis

does not help. Realistic mixing therefore requires an ingredient that breaks the stacking translation in

the neutrino sector, and that ingredient could change K

(script koide mixing nogo.py). The neutrino

masses above depend on it as well as on delocalization. The charged-lepton result of Section 5 depends

on neither. Section 10.4 adds a second condition on that ingredient: in the basis of the three sheet types,

it must be complex.

−1

lightest neutrino mass (meV)

100

200

400

∑m

(meV)

DESI DR2 + CMB (ΛCDM): ∑m

< 64.2 meV

= 0.80 meV, ∑ = 59.8 meV

IO: m

= 0, ∑ = 98.9 meV

normal ordering

inverted ordering

= 1/2, normal ordering

= 1/2, inverted ordering (above the DESI bound)

Figure 4: The sum of neutrino masses against the lightest mass for the two orderings, with the measured

splittings. The normal-ordering solution (m

= 0.80 meV,

∑

= 59.8 meV) lies below the DESI DR2

bound; the inverted-ordering solution (m

≈ 0,

∑

= 98.9 meV) lies above it.

8 Robustness and alternative constructions

This section tests what happens when each assumption is dropped or replaced, and when FCC is

replaced by other geometries. Within the alternatives examined, the result is lost. This is robustness

against those alternatives, not a uniqueness theorem. A3, the minimal lattice dynamics, is kept except for

one test of second-neighbor hopping. The computations are in alternatives.py, koide model.py,

koide A2 tests.py, koide projection.py, koide mei counts.py and koide geometry survey.py.

Without localization (A4). Delocalized charged leptons give the bond counts as couplings and K = 1/2

(Proposition 2). The

√

n of Eq. (5), and with it Koide’s

√

2, is a property of localized states.

Without ABC stacking (A1). Hexagonal close packing (ABAB) has the same split (4), but the sheets

above and below a site are of the same type, so there are only two types and no threefold spectrum.

The four-dimensional D

lattice sliced along a coordinate axis has the same 2 : 1 ratio (12 : 6 : 6),

but its slices alternate between two interleaved sublattices, so again there is no three-cycle.

Without the orientation (A6). Inversion symmetry restricts the phase to δ ≡ 0 or π/3 (mod 2π/3),

and both values make two leptons degenerate (Proposition 4). No quantity built from undirected

bonds alone can supply an odd phase.

A single triality component. The twelve bonds of a site fall into the three triality components of Sec-

tion 2, four bonds each. Each component splits its bonds 2 : 1 : 1 by layer change, the same propor-

tions as 6 : 3 : 3, so a defect that couples through only one component has couplings

√

2 : 1 : 1 and

the same ratio 1/

√

2. K = 2/3 holds exactly whether its orientation follows the bond, rotates with

the layer, or stays ﬁxed. The three components are permuted by the rotation about the stacking

axis; Section 10.2 uses this.

Second-neighbor hopping. Each site has six second neighbors, three in each adjacent sheet. Hopping

to them adds to the inter-sheet couplings and, in the root-mean-square couplings of A4, raises

K by (t

/t)

/3. The measured precision of 10

−5

therefore requires t

/t below about 5 ×10

−3

Masses as counts instead of A5. The integer veriﬁcation counts of Refs. [10, 11] give m

= 207

and m

= 3483, and so K = 0.66672. Integer counts cannot satisfy Koide’s relation exactly:

with C

= 1 and C

= 207 it would need C

= 3480.997 (Section 10). Nor would an approximate

match carry much information. K is insensitive to the masses (|dK/d ln m| ≈0.1), and any values

within ±0.2% of the measured masses satisfy it to 10

−3

. Koide’s relation needs amplitudes, not

counts.

Three further tests, of a lattice direction as the Koide vector, of other threefold algebraic structures

in place of A2, and of geometric and dynamical origins for the phase, also fail; they are collected in

Appendix B.

Other geometries and FCC comparison. The construction needs four things along a stacking axis:

exactly three sheet types, cycled by a lattice translation; the same neighbor counts (n

−

) at every

site, for its own type and the types above and below; n

= n

−

> 0; and n

= 2n

, which gives the

ratio 1/

√

2. When the ﬁrst three hold, K = (1 + 2n

)/3. The survey tests these conditions on 54

lattices in two to eight dimensions, along every stacking axis with Miller indices up to 6 in two and

three dimensions, 4 in four, 3 in ﬁve, and 2 in six to eight: about a million axes in all. It also tests

eight crystals with several sites per cell, HCP among them (Miller indices up to 3), and the close-packed

stacking sequences 4H, 6H, 9R and 15R. It assigns a neighbor two or more sheets away to its type, which

is generous to the alternatives. As a control, it ﬁnds the FCC result both from the lattice and from the

cubic cell with a four-site basis. Table 2 summarizes the results.

structure dim. axes with 3 sheet types best result

FCC 3 the four ⟨111⟩ (6,3, 3): K = 2/3

simple cubic 3 ⟨111⟩ (0,3, 3): no own-sheet neighbors

BCC 3 ⟨111⟩ (2,3, 3): K = 4/3

rhombohedral, 14 other angles 3 1 to 4 K = 1/3, 1/2, 1 or none

hexagonal, tetragonal, orthorhombic, 3 0 to 4 K = 1/3 or no own-sheet

monoclinic, triclinic (12 lattices) neighbors

square, triangular, rectangular (two 2 0 to 4 no own-sheet neighbors

aspect ratios), centered rectangular, oblique

, D

, A

∗

4 Z

only K = 4/3

, A

, D

(n = 5–8); 5–8 7 lattices K = 2/3 for Z

, E

and D

only

HCP; 4H, 6H, 9R, 15R stackings 3 none registries not cycled

diamond, honeycomb, kagome, wurtzite, 2, 3 ABC graphite K = 1/3

pyrochlore, graphite (AB, ABC) (stacking axis)

Table 2: The family construction on other geometries. “Best result” is the outcome closest to Koide’s

among the axes with three sheet types; the counts are (n

−

), and K is z/3n

, the value for signed

amplitudes, which can exceed 1.

Below six dimensions only FCC passes, and only along its four ⟨111⟩ axes. Many lattices have

no axis along which the sheets repeat every three layers. Among those that do, simple cubic has no

neighbors in a site’s own sheet, BCC has too few, and other angles of the rhombohedral family give K =

1/3, 1/2 or 1 or have no own-sheet neighbors. The four-dimensional D

lattice has no such axis; only its

three-dimensional slice, which is FCC, has one. A threefold cycle can still be deﬁned on D

, since two

null steps make one stacking translation, so each half-layer carries one registry. Sorting all 24 neighbors

this way with equal, Euclidean weights gives counts 6 : 9 : 9 and K = z/3n

= 4/3, because the twelve

null bonds all leave the registry. Two weightings suggested by the Lorentzian signature of Ref. [11]

restore the FCC result: weighting each bond by its interval, which is zero for a null bond, removes every

null bond, and giving future and past null bonds opposite signs cancels their sum. Either leaves 6 : 3 : 3

and K = 2/3. This shows that the FCC ratio is compatible with these Lorentzian prescriptions on D

not that Lorentzian time requires it: a null interval does not by itself make a hopping amplitude or a

self-energy vanish, a signed cancellation needs a speciﬁed operator, and the two prescriptions need not

agree once phases and second-order terms are included. A Hamiltonian formulation gives one reason to

expect the spatial count, since the Hamiltonian acts within a time slice and the null bonds, which join

slices, enter the evolution between slices rather than the operator whose eigenvalues are the masses; this

is the content of the rule of Ref. [11] that null bonds do not enter the sector counts of a defect whose

footprint lies within a time slice. Whether the null-bond terms of the second-order operator vanish,

cancel or renormalize the spatial couplings is a calculation left open. In the close-packed stackings other

than ABC, some sites have sheets of the same type above and below, with counts (6,0, 6) or (6, 6,0), so

no translation cycles the three registries. The crystals with several sites per cell have unevenly spaced

layers or a period other than three along most axes; ABC graphite has three sheet types along its stacking

axis, but no neighbors between sheets, so K = 1/3. Two six-dimensional lattices also pass: Z

, along an

axis such as (0,0, 0,1,1,1), with the FCC counts (6,3,3), and D

, with (30, 15,15). Neither can serve

as the three-dimensional space of the model or as a slice of its four-dimensional spacetime. The survey

covers standard lattices and crystals and axes with small Miller indices. It is not a proof that no other

geometry can give the relation.

The family count and the ratio are one integer. The speciﬁcity of FCC has a sharper form. FCC is

the root lattice A

: a site’s twelve bonds are the twelve roots of su(4). Slicing along ⟨111⟩ grades su(4)

by the u(1) that singles out su(3), so the adjoint splits as 8 + 1 + 3 +

3 and the roots as 6 : 3 : 3. The

in-sheet bonds are the roots of su(3), the up and down bonds are the 3 and

3, and the three sheet types

are the three cosets of the triangular lattice A

in its dual, A

∗

∼

, the center of SU(3). The family

label of A2 is therefore the Z

center charge of the in-sheet SU(3). The same construction on the root

lattice A

in d dimensions, sliced along the coweight that singles out A

d−1

, has d(d −1) in-sheet bonds,

d up and d down, and d sheet types, since the glue group A

∗

d−1

is Z

(Table 3). The number of

families is d, and the bond ratio is n

= 1/(d −1). Koide’s relation needs two things of this series:

three families, which is d = 3, and the ratio n

= 1/2 that gives 1/

√

2, which is again d = 3. The two

conditions are the same condition. Within this series the observed number of generations ﬁxes Koide’s

ratio, and both are the dimension of space. The six-dimensional lattices that pass the survey lie outside

the series and tie no three-cycle to a family count.

d bonds in-sheet : up : down sheet types n

three families and 1/

√

2 2 : 2 : 2 2 1 no

3 6 : 3 : 3 3 1/2 yes

4 12 : 4 : 4 4 1/3 no

5 20 : 5 : 5 5 1/4 no

Table 3: The root lattice A

sliced along the coweight that singles out A

d−1

. The number of sheet types

equals d, and the bond ratio n

= 1/(d −1) equals Koide’s 1/2 only at d = 3, where A

is FCC.

9 Why the relation is speciﬁc to charged leptons

Koide’s relation concerns a family: three particles identical in every quantum number except mass. In

the Standard Model only the fermion generations form families: the charged leptons, the neutrinos, and

the up-type and down-type quarks. The relation holds for the charged leptons only; Section 7 treats the

neutrinos. The construction explains why it fails for quarks.

The derivation needs a particle that sits on a node, in a sheet. Such a particle has nearest neighbors

in its own sheet, and they supply the diagonal coupling

√

6 of the family operator. In the model a quark

instead sits in a tetrahedral void [14], between two sheets, with no nearest nodes in a sheet of its own

(Table 4).

In the direct analog of the family operator, with couplings

√

n to these nodes and masses from WW

†

nearest nodes

position own sheet adjacent sheets K

on a node, in a sheet (charged lepton) 6 3 and 3 2/3 exactly

tetrahedral void, between sheets (quark) 0 3 and 1 ≤ 0.614

octahedral void, between sheets 0 3 and 3 ≤ 1/2

Table 4: Nearest nodes of a particle on a node and of one in each kind of void, by layer along the stacking

axis, and the Koide ratio each gives in the family operator. For the tetrahedral void the bound holds at

every phase and for any added in-layer coupling (scanned to 8t; K → 1/3 as the coupling grows).

(A5), the unequal 3 : 1 split ﬂattens the spectrum. Even an added in-layer coupling of any strength

leaves K ≤0.614 at every phase. An octahedral void, which has no nearest nodes in its own layer, gives

K ≤ 1/2; it would reproduce the node result only with an in-layer coupling of exactly

√

6t, which its

geometry does not supply. The relation therefore holds for particles on a node, in a sheet, the charged

leptons, and fails for particles in tetrahedral voids, the model’s quarks.

This agrees with observation. The up-type and down-type quark triplets give K = 0.849 and 0.731

with PDG masses [3], and 0.887 and 0.743 with running masses at M

[5]; neither is 2/3. The con-

struction accounts for the absence of 2/3, not for these values, which also shift with the choice of mass

deﬁnition (script koide quark voids.py).

10 Relation to the triality description

Ref. [11] describes the charged leptons by their engagement depth in the triality structure of the four-

dimensional lattice, and this paper describes them as the characters of the stacking translation. This

section shows that the two descriptions share their bond structure and that, on a straight column of

sites, their threefold symmetries coincide. It identiﬁes the family states with R-charges on a straight

column of sites, derives from that identiﬁcation a condition on lepton mixing, and compares the inte-

ger counts of Ref. [11] with the masses of the family operator. The numbers are computed by scripts

koide triality bridge.py, koide triality operators.py and koide mei counts.py.

10.1 The shared bond structure

Theorem 1 uses one property of the bonds: half of a state’s bonds lie in its own sheet, z = 2n

(Eq. (8)).

Each lepton of Ref. [11] is carried by a cluster of bonds around a node. The muon’s cluster has this

property, and so do the spatial edges of the tau’s. The muon’s cluster, a node and its twelve neighbors,

has 18 edges of each kind. The tau’s tube section, the 24-cell, has 30 of each, besides its 60 null bonds

(48 shell edges and 12 spokes), which lie in no ﬁxed-time sheet. The electron’s support is a single edge,

which is either in a sheet or not; the triality component that contains it, and that anchors the electron in

Ref. [11], splits its four bonds 2 : 1 : 1, two within the sheet and two leaving it. The equal division that

gives K = 2/3 here is therefore present in the support of the muon, among the spatial edges of the tau,

and in the component that anchors the electron (script koide mei counts.py).

10.2 The triality cycle is the rotation about the stacking axis

Ref. [11] calls triality the coordinate cycle (x

) 7→ (x

). We use its inverse, π :

) 7→ (x

), which generates the same group of order three. Within a time slice

π is R : (x, y,z) 7→ (z,x,y) (Section 2). The direction ﬁxes only the sign convention in Proposition 6.

The cycle and Lie-theoretic triality. In Lie theory, triality is the outer automorphism of D

of order

three that permutes the vector and the two spinor representations of Spin(8), equivalently the three non-

zero classes of D

∗

. A permutation of coordinates need not be this automorphism. The following

three statements are checked by explicit computation (script koide triality operators.py).

(a) The cycle permutes three groups of bonds. π maps each of the three sixteen-cells of D

roots, the

perfect matchings {01|23}, {02|13}and {03|12}of the coordinate pairs, onto the next, and on the

FCC slice it is the rotation R about [111]. Facts (i)–(iv) below develop this.

(b) The cycle is a symmetry of the code and of its defects. On the L = 4 code of Ref. [11], rebuilt

from its deﬁnition and found to have n = 1536 physical and k = 1282 logical qubits (its dis-

tance, 3, is taken from Ref. [11]), π permutes the 1536 qubits in 512 orbits of three, maps every

Z-stabilizer onto a Z-stabilizer and every X-stabilizer onto an X-stabilizer, cycles the three bond

classes, and carries the syndrome of every single-edge defect, the electron of Ref. [10], onto the

syndrome of its image. It is therefore an automorphism of the code as an operator on its qubit,

stabilizer and single-defect spaces. Proposition 6 itself uses only (a); statement (b) is what makes

the charge of Proposition 6 a label that the code respects.

the Weyl group W (D

): it is an inner automorphism, and it ﬁxes each class of D

∗

. The full

symmetry group of the D

root system, W (F

) of order 1152, does contain elements that permute

the three classes cyclically, but none of them is a symmetry of the code. The code places its X-

stabilizers on one class, the odd-sum integer points, and the root-system symmetries that preserve

this choice are exactly the 384 signed permutations, W (B

). A genuine triality element sends an

X-stabilizer center such as (1,0,0, 0) to a half-integer point, where the code has no stabilizer.

Ref. [11] (its Section 6.1) identiﬁes the cycle with the outer automorphism and lists that identiﬁcation

among its open problems (its Section 16); statement (c) settles it in the negative. By (c), nothing in

this paper, and none of the counts of Ref. [11], rests on the representation theory of Spin(8). In the

assignment proposed in Section 10.3, the three generations would be labeled by charges under a Weyl

group element that the code shares with the lattice, not by the three eight-dimensional representations.

As in Ref. [11], we use the name triality for the coordinate cycle of (a) and (b), and not for the outer

automorphism.

Four facts about R follow from the bond vectors.

(i) R is the threefold rotation about the stacking axis. It ﬁxes the [111] direction and every layer

index (x + y + z)/2, so it maps each sheet onto itself and commutes with the stacking translation P on

the registry space.

(ii) The twelve bonds form a 3 ×3 grid. A triality component consists of the four bonds that share

their zero coordinate. Each component has two bonds in the sheet, one up and one down, and R permutes

the three components cyclically. In particular the three up-bonds, (0,1,1), (1,0,1) and (1, 1,0), lie one

in each component, and R permutes them.

(iii) The stacking period is an orbit of R. The shortest lattice vector along the axis is (2,2, 2), since

(1,1,1) has an odd coordinate sum and is not a lattice vector. It is a sum of three up-bonds in only one

way up to order: one up-bond from each triality component,

(2,2,2) = (0,1,1) + (1, 0,1) + (1,1, 0). (10)

(iv) On a straight column, a step is a rotation by R. Call a straight column a chain of sites, one

per sheet, joined by up-bonds and returning to its lateral position after three sheets, as in the lepton

states of Fig. 3. By (iii) its three bonds come one from each component, so the column is labeled

by the cyclic order (c

) in which it uses them. There are six such columns from a site, in two

classes of three; proper rotations preserve the class and reﬂections exchange them, so the classes are

the two chiralities of the column. Moving the base of the column one sheet up relabels its sequence as

) 7→ (c

). For one chirality this is the relabeling c 7→ c + 1 by R, and for the other it is

c 7→c −1.

Proposition 6 (family states as charges of the cycle). On a straight column, the character of P with

label k as in Eq. (6) is an eigenstate of the relabeling by R with eigenvalue ω

±k

, the sign set by the

chirality of the column. The state has amplitude ω

−kn

on sheet n, and by (iv) the bond it uses there is

of component c

±n. The tau, the trivial character, is therefore the R-symmetric state, with the same

amplitude and phase on all three components, and the electron and the muon carry the two nontrivial

R-charges. Script koide triality bridge.py checks (i)–(iv) and Proposition 6 on the bond vectors.

10.3 The generations as charges of the cycle

Ref. [11] labels the generations by engagement depth, a geometric label that does not change under

translation. Under that labeling a map between the two sets of generations that respected P would have

to carry the phases 1, ω,ω

onto 1, 1,1, which no map can do. Proposition 6 offers a different label

from the same structure: the R-charge of a superposition of the defect’s three anchorings, the three

components it can couple through. A single anchoring is not an R-charge state; the three R-charge states

are its Fourier combinations. With that label the two descriptions use one threefold symmetry, and on a

straight column the registry character of A2 and the R-charge are the same quantum number, up to the

chirality-dependent conjugation k ↔−k, which ﬁxes the tau and exchanges the two nontrivial charges.

The two constructions use different generation assignments. Ref. [11] takes the generations to be the

three engagement depths and not the three members of the triality orbit: since the orbit members have

identical veriﬁcation cost, it argues, the triality label cannot carry the mass hierarchy (its Section 4.2; see

also its Section 6.3). For a charged edge it identiﬁes the three anchorings as one electron (H2c), and for

a null worldline it takes the three anchorings themselves as the three neutrino ﬂavors. The present work

assigns the generations to the symmetry characters: the three R-charge states, the Fourier combinations

of the anchorings. The equal-cost argument does not bear on these states, because their masses come

from the family operator W , whose inter-sheet couplings and phase δ split the three charges, and not

from the cost of a single anchoring. In this assignment the number of generations is the order of R, and

engagement depth describes the states rather than labeling them. A lepton’s amplitude is the in-sheet

coupling plus the inter-sheet coupling, and the R-charge decides whether they add or cancel (Fig. 3). The

tau, symmetric over the three components, engages them coherently, and its two contributions add. In the

electron they nearly cancel. The order of engagement, e < µ < τ, is the order of 1+

√

2cos(δ +2πk/3).

Three limits apply. First, the identiﬁcation holds on straight columns. A state spread laterally

off the column is not covered, and A4’s localized state couples to all in-plane momenta (Lemma 5).

Second, it does not compute the integer counts of Ref. [11] from the charges; it gives the counts a role

as descriptions of the three states. Third, it does not explain why the two nontrivial charges, which are

complex conjugates, have different masses. That is still the work of the phase δ. It changes sign under a

twofold rotation that reverses the axis, and that proper rotation preserves the chirality of a column (fact

(iv)), so the chirality cannot supply it. The identiﬁcation is therefore a proposal for unifying the two

descriptions; it is not derived in either.

10.4 A condition on lepton mixing

A2 has a direct consequence for mixing. Work in the basis of the three sheet types, in which P acts as a

cyclic permutation; on a straight column this is the basis of the three anchorings, on which the relabeling

by R acts the same way. In it the charged-lepton states are the Fourier vectors (1,ω

−k

,ω

−2k

√

3, with

k as in Eq. (6). At the ﬁtted phase the trivial character is the heaviest state, the tau, and the electron and

the muon are the conjugate pair k = 1,2.

Proposition 7. If the neutrino mass matrix is real in the basis of the three sheet types, then |U

|= |U

µi

for every neutrino mass state i. The electron and muon vectors are complex conjugates of each other,

and the eigenvectors of a real symmetric matrix can be chosen real. The overlaps ⟨e|ν

⟩ and ⟨µ|ν

⟩ are

then complex conjugates and have equal modulus.

The data contradict this: |U

≈ 0.022 while |U

µ3

lies between about 0.45 and 0.55, the octant

of θ

being unresolved [17]; any real mass matrix misses one of the two by more than 0.2. A realistic

neutrino sector must therefore be complex in that basis. This sharpens the result of Section 7, that

mixing needs an ingredient breaking the stacking translation in the neutrino sector: that ingredient must

also carry phases.

Standard lepton-ﬂavor symmetries do not avoid this. The rotation R and the twofold rotations about

the coordinate axes generate the tetrahedral group A

, and adding the twofold rotations about the face

diagonals gives the octahedral group S

. Both are point groups of the FCC lattice, and both are stan-

dard ﬂavor groups for leptons [25, 26, 27, 28]. A residual Z

×Z

⊂ S

in the neutrino sector, gen-

erated by a twofold rotation about a coordinate axis and one about a face diagonal and acting in the

three-dimensional vector representation, gives tribimaximal mixing, but only if the electron is the trivial

character. With the tau there, as at the ﬁtted phase, every entry of the electron row satisﬁes |U

≥1/6.

More generally, by Proposition 7 no residual symmetry drawn from the full cubic group O

can give

a real mass matrix that ﬁts; a numerical scan of the 91 subgroups of O

generated by two elements

conﬁrms that none matches every entry of the measured |U|

to within 0.2 (the remaining 7 of its 98

subgroups contain these and allow fewer mass matrices). Point group operations act by real matrices, so

they cannot supply the phases on their own. We do not examine whether the required phases are related

to leptonic CP violation.

10.5 The counts as rounded masses

In Ref. [11] a mass is an integer count of bits, 1, 207 and 3483, while the masses here are in general

irrational. Integers cannot satisfy Koide’s relation exactly. With C

= 1 and C

= 207 = 9 ×23, the

relation reads (1 + 3

√

23 +

√

)

(208 +C

), and the term in

√

23 cannot cancel for any integer

. The solution near the measured value is C

= 3480.997. Equivalently, the family operator of Eq. (6)

gives 1 : 206.77 : 3477.44 at the ﬁtted phase, so the muon count 207 is its muon mass rounded to a

whole bit. The phase that makes the muon exactly 207, δ = 0.222242, lies within 2 ×10

−5

of the ﬁt

(Section 6) and gives a tau of 3480.997, two bits below the count 3483. Because K is insensitive to the

masses (Section 8), this is a consistency check, not independent evidence for the relation.

10.6 A spectral–geometric dictionary

The two descriptions use different mathematics: amplitudes and phases here, counts of edges, squares

and cells in Ref. [11]. Traces connect the two languages, as in the trace formulas that relate a spectrum

to closed paths. The traces of the family operator are exact sums over closed walks among the sheets:

∑

√

= 3

√

∑

= 3zt

∑

3/2



√

6 + 18

√

3cos3δ



. (11)

The ﬁrst two count bonds (Eq. (8)). In the third, the term in δ comes only from walks that wind once

around the cycle A →B → C →A, so 3δ is the ﬂux through that cycle. By Eq. (10), the straight winding

walk uses one bond of each triality component, so the ﬂux is carried by walks that visit all three. Three

traces ﬁx three eigenvalues, so the three masses are equivalent to two bond counts and one ﬂux. The

inverse is a Fourier transform over the generations,

√

t =

∑

√

and

√

t e

iδ

∑

√

−k

with k the character of each lepton. Applied to the integer counts of Ref. [11], 1, 207 and 3483, it

returns the bond ratio n

= 0.5000865, against 1/2 for FCC and 0.499997 for the measured masses.

Read through this transform, the geometric counts of Ref. [11] are, to within 10

−4

in the bond ratio,

the squared eigenvalues of the family operator built on the 6 : 3 : 3 split. The rules of Ref. [11] take

the same form. The number of objects modulo time reversal T (its hypothesis H3) is the dimension of

the T -even subspace, Tr(1 + T )/2. The sector counts are closed-walk counts too: the planar squares

of a cluster are its closed four-step walks whose two diagonals join second neighbors, divided by eight,

□

∑

, with A and G the nearest- and second-neighbor matrices of the cluster;

this gives 6 for the muon and 72 for the tau. The dictionary is heuristic in two respects. The inversion

applies to any three numbers, so its returning 6 : 3 : 3 restates the agreement with Koide’s relation rather

than adding evidence for it. And the second-neighbor ﬁlter on the walks is imposed by hand, not derived

from a hopping operator.

11 Conclusion

FCC geometry gives a structural realization of Koide’s relation. Along a stacking axis each site’s bonds

split 6 : 3 : 3, a state localized in the plane sees couplings

√

6 :

√

3 :

√

3, and ABC stacking makes the

family operator a circulant with Koide’s ratio. K = 2/3 then holds exactly, with no parameter ﬁtted: the

amplitude sum counts the in-sheet bonds, the mass sum counts all bonds, and K = z/3n

= 2/3 because

half of a site’s bonds lie in its sheet. Among 54 lattices and 12 other crystals and stackings only FCC

has this structure below six dimensions, and in the root-lattice series A

the number of families and the

Koide ratio are ﬁxed by the same integer: d = 3 gives three sheet types and the bond ratio 1/2. One

ﬁtted phase gives the hierarchy and the tau mass within 0.4σ of its measured value.

The physics lies in the assumptions. A2 identiﬁes the lepton families with the three symmetry

sectors of the stacking; like the valleys of a crystal band, the sectors are assumed to act as separate

species. A4 builds the family operator from the root-mean-square couplings of a localized state rather

than from a projection of the lattice Hamiltonian; the mass sum rule is rigorous, and what A4 supplies

is the amplitude sum rule. A5 takes the masses to be the squared eigenvalues of that operator. Deriving

these together, for instance from a three-dimensional subspace bound to a defect whose effective mass

matrix has Koide’s ratio, is the main open problem. The origin of the phase is also open, and so is lepton

mixing: any sector that respects the stacking translation gives no realistic mixing, so the ingredient that

produces it must break that translation, and in the basis of the three sheet types it must be complex.

Four consequences follow. Particles in tetrahedral voids between sheets, which the model identiﬁes

with quarks, do not give the relation, consistent with its absence for quarks. Delocalized states give

K = 1/2; if neutrinos are delocalized, the construction supplies a reason for a value proposed empiri-

cally before, with

∑

= 59.8 meV for the normal ordering, and here the family operator is an exact

projection. And within the model, integer veriﬁcation counts cannot give Koide’s relation exactly, while

coherent amplitudes give it identically. The counts of the earlier work share the equal division of bonds

(for the tau, among its spatial bonds); the muon count is the operator’s mass at the ﬁtted phase rounded

to a whole bit, and the tau count lies two bits above the exact value. Finally, the threefold symmetries

of the two descriptions are tied together. The coordinate cycle that the earlier work calls triality, whose

bond groups it uses to deﬁne the generations, is the rotation about the stacking axis, and the stacking

period is the sum of one bond from each of the bond groups it permutes. The cycle is an automorphism

of the earlier work’s code, checked as an operator on its qubits, stabilizers and single-edge defects. It

is an element of the Weyl group of D

, not the outer triality automorphism, which the code does not

have, so the construction does not rest on the representations of Spin(8). On a straight column a step

up the axis is a rotation by the cycle, the family states are its charges, and the tau is the symmetric one.

Taking the generations to be these charges, in place of the engagement depths of the earlier work, is

proposed as a uniﬁcation of the two descriptions. A neutrino mass matrix real in the basis of the sheet

types gives |U

| = |U

µi

|, a consequence of A2 that the data exclude, so lepton mixing requires phases

that the lattice’s point-group symmetries do not supply.

A Numerical check of the couplings

Script derive sqrt.py builds a periodic FCC lattice of the even-sum integer points of a 6

box (108

sites), with a hopping amplitude t = 1 on each of the twelve nearest-neighbor bonds. The stacking axis

is [111], and a site’s sheet type is its layer index (x + y + z)/2 modulo 3. For a site of type A, the

script forms the normalized combination of its neighbors in each sheet type. It then evaluates the matrix

element of the hopping Hamiltonian between the site and that combination. It obtains

√

6 for the site’s

own sheet and

√

3 for each adjacent sheet. It also forms the three states spread uniformly over each

whole sheet type and evaluates the 3 ×3 matrix of the Hamiltonian between them, obtaining 6 on the

diagonal and 3 off it. Finally it builds the family operator of Eq. (6) from each set of couplings and

evaluates K from its eigenvalues, scanning the phase. It obtains 2/3 for the localized couplings at every

phase of a ﬁne scan with |δ | ≤ π/12, and 1/2 for the delocalized ones at every phase of a ﬁne scan,

both to machine precision. It also evaluates the localized spectrum at δ = 0 and δ = π/3, the two values

allowed by inversion symmetry, and ﬁnds a degenerate pair in each (Proposition 4).

B Further alternatives

The following tests extend Section 8. None produces the structure of Section 5.

A lattice direction as the Koide vector. The vector (

√

) makes 45

◦

with (1, 1,1), but

no direction of a cubic lattice makes exactly 45

◦

with [111]. The condition reduces to S

2(u

+uv +v

), which has no nonzero integer solution because 2 is inert in the Eisenstein integers.

With other threefold structures in place of A2. The speciﬁc constructions examined here do not pro-

duce the required 1/

√

2. Operators built by counting root pairs in E

give rational off-diagonal

ratios. The Killing form of gl(3) on a family sector has its null cone at K = 1/3. Roots and

weights of D

against the plane rotated by its outer triality automorphism have squared cosines

in {0, 1/3,2/3,1}, never 1/2. The root angles of E

, E

and E

are only 60

◦

, 90

◦

and 120

◦

(and

180

◦

for opposite roots).

Other origins for the phase. Berry phases, the geometric phases a spin-

direction gains when carried

around a cone (half the solid angle per loop, a third of that per step), carry factors of π for the

natural cones about the stacking axis (bond, cube-axis, body-diagonal and 45

◦

), and none equals

the ﬁtted value. The cone at the tetrahedral angle gives exactly 2π/9 per step, which differs from

the ﬁtted value, close to 2/9, by the factor π. A dynamical phase, energy times time over

accumulated per bond gives three times the ﬁtted value per step. Growth treated as a discrete

relabeling of sheets produces only multiples of 2π/3.

Data availability

The scripts reproducing every number are archived at https://github.com/raghu91302/ss

mtheory/blob/main/koide_ccp_physo_scripts.zip: derive sqrt.py (Appendix A),

koide model.py (the ﬁtted phase, the tau test, the count check of Section 8, and the lattice-

direction test and phase checks of Appendix B), alternatives.py (the threefold structures and

Berry phases of Appendix B), neutrino.py (Section 7, including the mixing check, the effective

masses, the minimal sums and the signed variant), koide mixing nogo.py (the general mixing

result of Section 7), koide A2 tests.py (the status of A2 in Section 4 and the triality check

of Section 8), koide quark voids.py (Section 9), koide geometry survey.py (the survey

of other geometries in Section 8), koide triality bridge.py (Sections 10.2, 10.3 and 10.4),

koide triality operators.py (statements (a)–(c) of Section 10.2), koide projection.py (the

status of A4 in Section 5 and the second-neighbor bound of Section 8), and koide mei counts.py

(Sections 8 and 10). make fig fcc.py, make fig2.py, make fig leptons.py and make fig3.py

regenerate Figs. 1–4 respectively, and koide fcc leptons.html, in the same repository, is the inter-

active view cited in Section 1, which runs in any web browser at https://raghu91302.github.i o

/ssmtheory/koide_fcc_leptons.html. All use numpy; some also use scipy, and the ﬁgure scripts

matplotlib.

CRediT authorship contribution statement

Raghu Kulkarni: Writing – review & editing, Writing – original draft, Visualization, Validation, Soft-

ware, Methodology, Formal analysis, Conceptualization.

Declaration of competing interest

The author declares no known competing ﬁnancial interests or personal relationships that could have

inﬂuenced the work reported in this paper.

Funding

This research did not receive any speciﬁc grant from funding agencies in the public, commercial, or

not-for-proﬁt sectors.

Declaration of generative AI and AI-assisted technologies in the writing

process

During the preparation of this work the author used Claude (Anthropic) to check calculations, write and

run the veriﬁcation scripts, prepare ﬁgures, and edit the text for clarity. After using this tool, the author

reviewed and edited the content as needed and takes full responsibility for the content of the publication.

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