Koide’s lepton mass relation from cubic close packing
Raghu Kulkarni
SSMTheory Group, IDrive Inc, Calabasas, CA 91302, USA
raghu@idrive.com
Abstract
Koide’s relation 𝐾 = (𝑚
𝑒
+𝑚
𝜇
+𝑚
𝜏
)/(
√
𝑚
𝑒
+
√
𝑚
𝜇
+
√
𝑚
𝜏
)
2
= 2/3 holds for the charged-lepton
pole masses to about 10
−5
, has no accepted explanation, and fails for running masses. We show that a
localized family sector on a face-centered-cubic (FCC) lattice, under six explicit assumptions, obeys
it exactly. Along a ⟨111⟩ axis, each FCC site’s twelve bonds divide 6 : 3 : 3 between its own sheet
and the sheets above and below. A localized state couples to the normalized neighbor combination
in a sheet with strength
√
𝑛, which turns 6 : 3 : 3 into
√
6 :
√
3 :
√
3. ABC stacking makes the
three sheet types a cycle. The family operator is then a circulant with off-diagonal-to-diagonal ratio
1/
√
2, which is Koide’s value. No parameter is fitted to obtain 𝐾 = 2/3: the relation holds for
every phase at which the amplitudes are non-negative. The central hypothesis is that the three lepton
families span the three sheet types. The other assumptions are that the lattice is the vacuum of
the Selection–Stitch Model, that particles hop along bonds, that charged leptons are localized, that
masses are second-order self-energies, and that the stacking axis is oriented. Delocalized states give
𝐾 = 1/2 instead, hexagonal close packing gives no threefold spectrum, and an unoriented axis leaves
two leptons degenerate. Quarks, which in the model sit between sheets rather than on a node, give
𝐾 ≤ 0.614, so the relation is specific to charged leptons. The hierarchy requires one phase, fitted from
the electron and muon masses (𝛿 = 0.2222221); it gives 𝑚
𝜏
/𝑚
𝑒
= 3477.44, against 3477.37 ± 0.18
observed. If neutrinos are delocalized, the same geometry gives 𝐾
𝜈
= 1/2. This value was proposed
empirically in 2005; the construction supplies a reason for it, since couplings go as
√
𝑛 for localized
states and as 𝑛 for delocalized ones. The neutrino masses that follow are those implied by the earlier
proposal. No lepton sector that respects the stacking translation can mix, so the neutrino sector needs
an ingredient that breaks it.
1 Introduction
Koide observed in 1981 [1, 2] that the charged-lepton masses satisfy
𝐾 =
𝑚
𝑒
+ 𝑚
𝜇
+ 𝑚
𝜏
√
𝑚
𝑒
+
√
𝑚
𝜇
+
√
𝑚
𝜏
2
=
2
3
. (1)
With the Particle Data Group masses [3], 𝐾 = 0.666664 ± 0.000005, equal to 2/3 within its uncertainty.
The uncertainty, set by the tau mass, is about one part in 10
5
. The numerical precision and simple
geometric form of the relation have motivated numerous attempts at explanation, and no accepted theory
derives it. Two facts constrain an explanation. The relation is unit-free but not scheme-free. It holds
for pole masses, the masses of propagating particles. It fails for running masses at a common scale (at
𝑀
𝑍
[4], 𝐾 = 0.667929). Its origin should therefore concern physical particles, not the parameters of a
high-energy Lagrangian. And it has a simple geometry. Writing
√
𝑚
𝑘
= 𝑀
h
1 +𝑟 cos
𝛿 +
2𝜋𝑘
3
i
, (2)
Eq. (1) is exactly 𝑟 =
√
2: the three leptons are three points 120
◦
apart on a circle (Fig. 2a), and the vector
(
√
𝑚
𝑒
,
√
𝑚
𝜇
,
√
𝑚
𝜏
) makes 45
◦
with (1, 1, 1) [5, 6]. Equivalently, the amplitudes are the eigenvalues of
1
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 the ratio
√
2. The phase 𝛿 is a second,
independent number.
This circulant form is known; what has been missing is a structure that produces it. The contribution
of this paper is such a structure. FCC ABC stacking (Fig. 1) with localized nearest-neighbor hopping
gives the Koide circulant automatically, because
6 : 3 : 3
localized, normalized coupling
−−−−−−−−−−−−−−−−−−−−−−−→
√
6 :
√
3 :
√
3
ABC circulant
−−−−−−−−−−−→ 𝐾 =
2
3
. (3)
This paper states six assumptions about the vacuum lattice of the Selection–Stitch Model (Section 2)
and shows that a family sector satisfying them obeys Eq. (1) exactly (Sections 3–5). The central one
is that the lepton family space is spanned by the three sheet types of the stacking axis. Given it, the
derivation uses FCC geometry and standard quantum mechanics, and no parameter is fitted to obtain
𝐾 = 2/3. The two results should be kept apart: FCC geometry and localization give 𝐾 = 2/3, and a
single phase then gives the hierarchy. The ratio
√
2 arises from the coupling of a localized state to its
neighbors. It does not arise for delocalized states. The relation is therefore tied to the leptons being
localized particles, as the pole-mass fact suggests. The phase is fitted to data, not derived (Section 6). If
neutrinos are delocalized, the same construction gives 𝐾
𝜈
= 1/2, the value proposed empirically by Li
and Ma [7], and supplies a reason for that value (Section 7). The construction as written gives no lepton
mixing, and neither does any lepton sector that respects the stacking translation, so its neutrino sector
needs an ingredient that breaks it. Section 8 examines what fails when the key assumptions are dropped
or replaced, and Section 9 explains why the relation holds only for charged leptons.
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 [8]. The lattice carries a
quantum error-correcting code with one qubit on each bond [9]; particles are defects the code detects but
cannot repair.
Mass as verification cost. In the model a defect’s rest mass is the energy of the standing syndrome
information the code must carry to keep it, at a fixed energy per bit. This is the standing-information
postulate of Ref. [10]. Only mass ratios are predicted [8]. Counting the bits gives the muon-to-electron
ratio as the integer 207 [8]. This paper does not use those counts. Its masses come instead from the
family operator of Section 5; Section 8 explains why the counts cannot satisfy Koide’s relation exactly.
Leptons. A charged lepton is a small, localized defect: the electron is the single-bit defect of Ref. [8],
localized on the scale of one bond. This paper uses one fact from this picture: that charged leptons are
localized, not their internal structure. It models a charged lepton as a state localized at a site, that is, on
a node in a sheet (assumption A4 below), because a site belongs to exactly one sheet type (Fig. 1) and a
bond between sheets does not. Neutrinos, which are neutral and propagate rather than stay bound to a
site, are treated here as delocalized (assumption A4); Section 7 is conditional on that treatment. Quarks,
by contrast, sit in the tetrahedral voids between sheets [11], not on a node in a sheet; Section 9 uses this
contrast.
2
[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]
A
B
C
A
[111]
(c) the same lattice as a stack of triangular
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
√
2
L.
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
(vertical spacing,
2/3 𝐿, exaggerated). (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 registries.
(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.
Growth: stitch and lift. In the model the lattice is not static; it grows [11]. 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, which 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; A1 assumes this. Near where growth began, the lattice grows outward
in every direction. A region far from there grows at a front that moves one way, so its stacking axis has
a direction. This paper uses that direction only through assumption A6.
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
3
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 Δ𝐸
(2)
𝑖
=
Í
𝑛
|⟨𝑛|𝑉 |𝑖⟩|
2
/(𝐸
𝑖
− 𝐸
𝑛
); under
the additional assumption that the virtual-state denominators are independent of the family, this is
represented by a mass operator 𝑀 ∝ 𝑊
2
, where 𝑊 is the family operator of first-order couplings.
The amplitudes
√
𝑚
𝑘
are then the eigenvalues of 𝑊.
A6 (Orientation). The growth axis is oriented. A hop to the sheet above carries a phase 𝑒
𝑖 𝛿
and a hop
to the sheet below the phase 𝑒
−𝑖 𝛿
, with the sign fixed by the growth direction.
A1 is the model’s geometry. A2 is the central hypothesis of this paper. It takes the lepton family
space to be spanned by the three sheet types. It is made here, not derived from the model, and another
assignment exists in the series (Section 10). Section 4 states precisely what A2 assumes. A3 is the
minimal dynamics on a lattice. A4 follows the model’s picture of charged leptons as localized defects,
and treats neutrinos, which propagate, as delocalized (Section 2). A5 is our reading of the model’s
identification of mass with the cost of maintaining a defect: the energy of the virtual hops that maintain
it. In operator form, second-order perturbation theory gives 𝑀
(2)
eff
= 𝑊 𝐷
−1
𝑊
†
, with 𝐷 the operator of
virtual-state denominators. A5 takes 𝐷
−1
proportional to the identity on the relevant intermediate states,
so 𝑀 ∝ 𝑊𝑊
†
, which equals 𝑊
2
because the 𝑊 of Eq. (6) is Hermitian. 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
𝜈
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)
and each of a site’s twelve bonds falls in exactly one of the three classes. 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. It sends type A to B, B to C, and C to A. The hopping Hamiltonian of A3 commutes
with this translation. Any coupling between types is therefore unchanged by the cyclic shift A → B →
C. The coupling from A to B equals the couplings from B to C and from C to A. The same holds for the
4
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, so they respect the symmetry too. This is the threefold cyclic
symmetry on the generations, and it is why the family operator of Section 5 is a circulant.
Write 𝑃 for this translation acting on the sheet types, A → B → C → A. The physical family states
are the three characters of 𝑃, not the individual sheets. On the three-dimensional registry space, where
states differing by a full stacking period are identified, 𝑃
3
= 1, and the eigenvalues of 𝑃 are 1, 𝜔 and 𝜔
2
,
with 𝜔 = 𝑒
2𝜋𝑖/3
; these are the Fourier states of Section 5. Because 𝑃 is a lattice translation, the three
characters are Bloch-phase sectors of translation modulo the ABC stacking period. A2 amounts to two
hypotheses: that this registry space is the lepton family space, and that its three modes are 𝑒, 𝜇 and 𝜏.
Neither can be derived within the present model. Translation by one layer is an exact symmetry of the
code, so the registry is not a code label. 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: three Bloch-phase
sectors along the stacking axis that behave as distinct, approximately conserved species.
5 Derivation
Couplings (A3, A4). By A3 a state at a site couples with amplitude 𝑡 to each neighbor. The coupling of
a localized state (A4) to the normalized combination of its 𝑛 neighbors of one type, |𝑛
ℓ
⟩ = 𝑛
−1/2
Í
𝑗
|𝑗⟩,
is
⟨𝑛
ℓ
|𝐻|site⟩ =
1
√
𝑛
· 𝑛 𝑡 =
√
𝑛 𝑡, (5)
the collective enhancement familiar from superradiance [12]. By Eq. (4), 𝑛 = 6 within the own sheet
and 𝑛 = 3 to each adjacent sheet. For delocalized states (A4, neutrinos) the matrix elements between
uniform states on whole sheet types are instead the coherent counts 6, 3, 3. Both results were checked on
an explicit periodic FCC lattice (Appendix A).
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
h
√
6 𝐼 +
√
3
𝑒
𝑖 𝛿
𝑃 + 𝑒
−𝑖 𝛿
𝑃
−1
i
,
√
𝑚
𝑘
=
√
6 𝑡
h
1 +
√
2 cos
𝛿 +
2𝜋𝑘
3
i
. (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
√
3/
√
6 = 1/
√
2 is Koide’s 𝑟 =
√
2 in Eq. (2).
The proof is two lines. Write the amplitudes as 𝑥
𝑘
= 𝐴[1 +
√
2 cos 𝜃
𝑘
] with 𝜃
𝑘
= 𝛿 + 2𝜋𝑘/3. Since
Í
𝑘
cos 𝜃
𝑘
= 0 and
Í
𝑘
cos
2
𝜃
𝑘
= 3/2 for any 𝛿,
𝑘
𝑥
𝑘
= 3𝐴,
𝑘
𝑥
2
𝑘
= 𝐴
2
3 + 2 ·
3
2
= 6𝐴
2
, 𝐾 =
Í
𝑘
𝑥
2
𝑘
(
Í
𝑘
𝑥
𝑘
)
2
=
6
9
=
2
3
. (7)
In terms of the signed eigenvalues the relation holds for every 𝛿; physical masses require non-negative
amplitudes, which holds for |𝛿| ≤ 𝜋/12 (modulo 2𝜋/3). At 𝛿 = 𝜋/12 the electron amplitude vanishes
(Fig. 2).
Proposition 2 (neutrinos). Under A1–A6, for which neutrinos are delocalized (A4), 𝐾
𝜈
= 1/2 for
every 𝛿
𝜈
, and all three amplitudes are non-negative. The coherent counts give a ratio of 1/2, so
√
𝑚
𝑘
= 𝑎[1 + cos(𝛿
𝜈
+ 2𝜋𝑘/3)] ≥ 0. The same two lines as in Eq. (7), with
√
2 replaced by 1, give
Í
𝑘
𝑥
𝑘
= 3𝑎 and
Í
𝑘
𝑥
2
𝑘
=
9
2
𝑎
2
, so 𝐾 = 1/2.
5
e
μ τ
1
√
2
(a) the Koide circle:
√
m
k
∝ 1 +
√
2
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 ∝
√
m
k
e
μ
τ
(b) localized family operator:
√
m
k
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
K
electron amplitude < 0
π/12: m
e
= 0
measured
(c) K = 2/3 localized (|δ| ≤ π/12), 1/2 delocalized
localized (
√
6
:
√
3
)
delocalized (6 :3)
Figure 2: (a) The Koide circle: the three amplitudes
√
𝑚
𝑘
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.
Proposition 3 (self-energy). The masses are the eigenvalues of 𝑊
2
, whose diagonal is 12𝑡
2
, the
coordination number. Every generation has the same total virtual hopping. The masses differ because of
the couplings between sheets. Without a phase, these couplings split the spectrum into one heavy lepton
and two degenerate light ones. The opposite phases of the excursions up and down lift the remaining
degeneracy. For every 𝛿 the eigenstates of 𝑊 are the characters of 𝑃, (1, 𝜔
𝑘
, 𝜔
2𝑘
)/
√
3 (Section 4): each
physical lepton has equal weight on the three sheet types, and the generations differ in the phases between
sheets.
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 𝑒
𝑖 𝛿
= 𝑒
−𝑖 𝛿
. A
diagonal rephasing of A, B, C changes individual link phases but leaves invariant the total phase around
the cycle A → B → C → A, which is 3𝛿. Keeping the hopping uniform, it can change 𝛿 only by 2𝜋𝑛/3,
which relabels the three 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.
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 = 0.2222221. 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
𝜏
/𝑚
𝑒
=
3477.44, or 𝑚
𝜏
= 1776.969 MeV. The measured value is 3477.37 ± 0.18 (1776.93 ± 0.09 MeV [3]), so
the difference is 0.4𝜎. This is a test of 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
7
. We make no use of that.
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 orientation comes from the direction in which the lattice
grew (A6). The value of the phase is not derived here. The phase used in this paper is the fitted one, and
its origin is left open.
6
7 Neutrinos: the origin of 𝐾
𝜈
= 1/2
This section depends on a step that is less established than the localization of charged leptons. The step is
that a neutrino, which propagates rather than staying bound to a site, is spread uniformly over each sheet
type (A4). Given it, by Proposition 2, neutrinos satisfy 𝐾
𝜈
= 1/2, with the non-negative amplitudes
√
𝑚
𝑘
= 𝑎
h
1 + cos
𝛿
𝜈
+
2𝜋𝑘
3
i
. (8)
The value 𝐾
𝜈
= 1/2 is not new. Li and Ma [7] proposed it empirically, from a conjectured complemen-
tarity between quark and lepton masses. In their normalization it reads 𝑘
𝜈
= 3/4, with 𝐾 =
2
3
𝑘. They
found it consistent with the normal ordering. A seesaw construction [13] gives neutrino masses whose
Koide ratio is 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. The couplings go as
√
𝑛 in the first case
and as 𝑛 in the second.
We take the two measured splittings from the global fit that includes the first JUNO data [14, 15]:
Δ𝑚
2
21
= 7.53×10
−5
eV
2
and Δ𝑚
2
31
= 2.529×10
−3
eV
2
(normal) or Δ𝑚
2
32
= −2.484×10
−3
eV
2
(inverted).
They fix both 𝑎 and 𝛿
𝜈
, and hence the absolute masses (script neutrino.py). Given 𝐾
𝜈
= 1/2, the
masses follow from the two splittings alone. Li and Ma’s value therefore gives the same numbers with
current data; they are consequences of 𝐾
𝜈
= 1/2, not independent predictions of this construction:
ordering 𝛿
𝜈
𝑚
1
(meV) 𝑚
2
(meV) 𝑚
3
(meV)
Í
𝑚
𝜈
(meV)
normal 0.332 0.80 8.71 50.30 59.8
inverted 1.041 49.08 49.84 0.00 98.9
In both orderings the spectrum is hierarchical and the lightest mass is fixed. The inverted solution
has 𝑚
3
≈ 0. It is exactly the minimal inverted spectrum, with
Í
𝑚
𝜈
= 98.9 meV. The normal solution
has 𝑚
1
= 0.80 meV and
Í
𝑚
𝜈
= 59.8 meV, 0.8 meV above the minimal normal spectrum (59.0 meV).
The value 𝐾
𝜈
= 1/2 therefore excludes quasi-degenerate neutrinos and fixes the sum for each ordering.
The normal-ordering solution is not excluded by current data (Fig. 3). The global oscillation fit
including the first JUNO data prefers the normal ordering, with Δ𝜒
2
= 9.4 when atmospheric neutrino
data are included [14]. DESI DR2 baryon acoustic oscillations, combined with the cosmic microwave
background, give
Í
𝑚
𝜈
< 64.2 meV (95%) in the standard cosmological model [16]. This bound lies
above 59.8 meV and below the minimal inverted sum. In that model the data in fact prefer sums below
the oscillation minimum. Any normal-ordering spectrum shares this tension, of about 3𝜎. It disappears
in models with evolving dark energy, where the bound relaxes to 163 meV. These masses are close to
the minimal normal spectrum, so present data cannot distinguish them from a spectrum with a massless
lightest neutrino. The sharp test is the sum. A measured value clearly above about 60 meV with the
normal ordering, or above 99 meV with the inverted ordering, would falsify 𝐾
𝜈
= 1/2, and with it the
construction as applied to neutrinos. Future cosmological surveys, at a resolution of order 15 meV, can
confirm a sum near 60 meV and exclude much larger ones. They cannot distinguish 59.8 meV from
the minimum. The beta-decay effective mass, about 8.9 meV, is below planned direct sensitivity. If
neutrinos were instead localized (𝐾 = 2/3), there would be no solution with non-negative amplitudes.
Allowing a negative one, as in Brannen’s neutrino version of Koide’s formula [17], gives 𝑚
1
= 0.37 meV
and
Í
𝑚
𝜈
= 59.3 meV.
Lepton mixing. The construction as written gives no 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. Dirac neutrinos then do not mix at all. This holds at
every in-plane momentum, and also for neutrinos on the null bonds of the four-dimensional lattice, since
7
10
−1
10
0
10
1
10
2
lightest neutrino mass (meV)
40
60
80
100
200
400
∑m
ν
(meV)
DESI DR2 + CMB (ΛCDM): ∑m
ν
< 64.2 meV
m
1
= 0.80 meV, ∑ = 59.8 meV
IO: m
3
= 0, ∑ = 98.9 meV
normal ordering
inverted ordering
K
ν
= 1/2, normal ordering
K
ν
= 1/2, inverted ordering (above the DESI bound)
Figure 3: The sum of neutrino masses against the lightest mass for the two orderings, with the measured
splittings. The normal-ordering solution (𝑚
1
= 0.80 meV,
Í
𝑚
𝜈
= 59.8 meV) lies below the DESI DR2
bound; the inverted-ordering solution (𝑚
3
= 0,
Í
𝑚
𝜈
= 98.9 meV) lies above it.
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
unmixed state would be pure tau, while the observed 𝜈
3
is about half muon and half tau. A different
choice of neutrino basis does not help. Realistic mixing requires an ingredient that breaks the stacking
translation in the neutrino sector, and that ingredient could change 𝐾
𝜈
(script koide mixing nogo.py).
The neutrino masses above therefore depend on it as well as on delocalization. The charged-lepton result
of Section 5 depends on neither.
8 Robustness and alternative constructions
Within the alternatives examined here, dropping or replacing the assumptions that fix the result loses it.
This is robustness against those alternatives, not a uniqueness theorem. A3, the minimal lattice dynamics,
is not varied. The computations behind this section are in alternatives.py and koide model.py.
Without localization (A4). Delocalized charged leptons give the coherent 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
4
lattice sliced along a coordinate axis has the same 2 : 1 ratio (12 : 6 :
6). But its slices alternate between two cosets, 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 three triality components of four bonds
each, and each component splits its bonds 2 : 1 : 1 by layer change, the same proportions as
6 : 3 : 3. A defect that couples only through one component therefore 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 (koide A2 tests.py).
Masses as counts instead of A5. The integer verification counts of Ref. [8] give 𝑚
𝜇
/𝑚
𝑒
= 207. Integer
counts cannot satisfy Koide’s relation exactly: with 𝐶
𝑒
= 1 and 𝐶
𝜇
= 207 it would need 𝐶
𝜏
=
8
3480.997. Nor would an approximate match carry much information, since 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.
A lattice direction as the Koide vector. The vector (
√
𝑚
𝑒
,
√
𝑚
𝜇
,
√
𝑚
𝜏
) makes 45
◦
with (1, 1, 1), but
no direction of a cubic lattice makes exactly 45
◦
with [111]. The condition reduces to 𝑆
2
=
2(𝑢
2
+𝑢𝑣 +𝑣
2
), 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
6
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
4
against the triality axis have squared cosines in {0, 1/3, 2/3, 1}, never 1/2. The root
angles of 𝐸
6
, 𝐸
7
and 𝐸
8
are only 60
◦
, 90
◦
and 120
◦
(and 180
◦
for opposite roots).
Other origins for the phase. Berry phases on the natural cones of the lattice carry factors of 𝜋, 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 per-bond energy-times-time phase
gives three times the fitted value per step. Growth treated as a discrete relabeling of sheets produces
only multiples of 2𝜋/3.
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 [11], between two sheets, with no nearest nodes in a sheet of its own
(Table 1).
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 1: 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 an added in-layer coupling of any strength.
In the direct analog of the family operator, with couplings
√
𝑛 to these nodes and masses from 𝑊𝑊
†
(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, with no in-layer coupling, gives 𝐾 ≤ 1/2. The relation
therefore holds only for particles on a node, in a sheet: the charged leptons.
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
𝑍
[4]; neither is 2/3. The
construction 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).
9
10 Relation to another generation assignment
Within the Selection–Stitch series, the generations have also been associated with the glue classes of 𝐷
4
in the 𝐹
4
→ 𝐸
6
construction [18]. There the three glue classes sit at 120
◦
and are permuted by triality,
so that assignment also has a threefold cycle. But nothing in it splits the three by an odd phase, so the
triality images stay degenerate. A2 is a different assignment, and reconciling the two is left open. The
assignment made here uses only the FCC geometry.
11 Conclusion
FCC geometry provides a structural realization of Koide’s relation under a minimal family assignment.
Along a stacking axis, each site’s bonds split 6 : 3 : 3. A localized state’s normalized coupling turns
this into
√
6 :
√
3 :
√
3. ABC stacking makes the family operator a circulant with Koide’s ratio,
so 𝐾 = 2/3 holds exactly, and no parameter is fitted to obtain it. The difficult physics lies in the
assumptions, chiefly the identification of the three ABC translation-character sectors with the observed
lepton families (A2), and the mass operator 𝑀 ∝ 𝑊
2
(A5). A2 cannot be derived within the present
model; it is a valley-type hypothesis. The relation holds only for particles on a node, in a sheet: particles
between sheets, which the model identifies with quarks, do not produce it, consistent with its absence
for quarks. Given the assumptions, Koide’s relation is exact rather than approximate. Delocalized states
give 𝐾 = 1/2, hexagonal close packing gives no threefold spectrum, and an unoriented axis leaves two
leptons degenerate.
Two results should be kept apart. The Koide ratio is parameter-free. The observed hierarchy
additionally requires one phase, fitted from two masses. With it, the relation gives the third mass within
0.4𝜎 of its measured value. The origin of the phase is left open. A structural distinction also follows
for the model as a whole. Integer verification counts reproduce mass ratios approximately but cannot
give Koide’s relation exactly. Coherent family amplitudes give it identically. The family structure may
therefore come from amplitudes even where approximate mass ratios come from counts.
If neutrinos are delocalized, the same geometry gives 𝐾
𝜈
= 1/2, a value proposed empirically
before; the construction supplies a reason for it. The masses that follow from it (𝑚
1
= 0.80 meV and
Í
𝑚
𝜈
= 59.8 meV for the normal ordering) are those implied by the earlier proposal, and they are
consistent with current data. The neutrino result depends on delocalization and on a further ingredient
that produces lepton mixing. Since any sector respecting the stacking translation gives no mixing, that
ingredient must break it.
A Numerical check of the couplings
Script derive sqrt.py builds a periodic FCC lattice of the even-sum integer points of a 6
3
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 with
|𝛿| ≤ 𝜋/12, and 1/2 for the delocalized ones at every phase, 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).
10
Data availability
The scripts reproducing every number are archived at https:/ /github.com/raghu91302/ssmthe
ory/blob/main / k oide_ccp_v2_scripts.zip: derive sqrt.py (Appendix A), koide model.py
(the fitted phase, the tau test, and the count and phase checks of Section 8), alternatives.py (the alter-
natives of Section 8), neutrino.py (Section 7, including the mixing check), koide mixing nogo.py
(the general mixing result of Section 7), koide A2 tests.py (the status of A2 in Section 4 and the tri-
ality check of Section 8), and koide quark voids.py (Section 9). make fig fcc.py, make fig2.py
and make fig3.py regenerate Figs. 1–3. All use numpy; some also use scipy, and the figure scripts
matplotlib.
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11
