Title: The Koide Relation as an Exact Z3 Bogoliubov Identity
Authors: kibrom kidane
Date: 2026-04-01
DOI: 10.5281/zenodo.19826574
URL: https://xmoro.com/research/the-koide-relation-as-an-exact-z3-bogoliubov-identity

Abstract
--------
For any three–component Bogoliubov–de Gennes (BdG) system whose quasiparticle gaps
form a real Z3–equivariant triplet ∆k = A+Bcos(θ+2πk/3) with all ∆k ≥ 0, the Koide ratio
Q≡ ∑k mk /(∑k √mk )2 satisfies the exact identity Q=
1
3 + B2/(6A2), so that within this
positive–gap region the empirical charged–lepton value Q= 2/3 is equivalent to |B/A|= √2.
Outside the positive–gap region the identity fails; a counterexample at (B/A,θ) = (√2,π/3)
gives Q≈ 0.4094. The algebraic identity is not new mathematics; it is implicit in Brannen’s
parametrisation and the broader Koide literature. The contribution of this note is twofold.
First, we separate existence from selection: the Koide puzzle reduces to a closed–form
theorem on the existence side, leaving two narrower selection questions (Why |B/A|= √2?
and Why θ = 2/9?). Second, we argue that the BdG framing supplies the physical reason
for why a Z3 symmetry acts on √mrather than on m: the quasiparticle mass is |∆k |2 for an
order–parameter gap on which the symmetry naturally acts. Numerical verification against
Particle Data Group charged–lepton masses and a fully runnable reproduction script are
provided.

The Koide Relation as an Exact Z3 Bogoliubov Identity
Kibrom Kidane∗
Independent Researcher
Abstract
For any three–component Bogoliubov–de Gennes (BdG) system whose quasiparticle gaps
form a real Z3–equivariant triplet ∆k = A+Bcos(θ+2πk/3) with all ∆k ≥ 0, the Koide ratio
Q≡ ∑k mk /(∑k √mk )2 satisfies the exact identity Q=
1
3 + B2/(6A2), so that within this
positive–gap region the empirical charged–lepton value Q= 2/3 is equivalent to |B/A|= √2.
Outside the positive–gap region the identity fails; a counterexample at (B/A,θ) = (√2,π/3)
gives Q≈ 0.4094. The algebraic identity is not new mathematics; it is implicit in Brannen’s
parametrisation and the broader Koide literature. The contribution of this note is twofold.
First, we separate existence from selection: the Koide puzzle reduces to a closed–form
theorem on the existence side, leaving two narrower selection questions (Why |B/A|= √2?
and Why θ = 2/9?). Second, we argue that the BdG framing supplies the physical reason
for why a Z3 symmetry acts on √mrather than on m: the quasiparticle mass is |∆k |2 for an
order–parameter gap on which the symmetry naturally acts. Numerical verification against
Particle Data Group charged–lepton masses and a fully runnable reproduction script are
provided.
Keywords: Koide relation· Bogoliubov–de Gennes spectrum· Z3 symmetry· Charged-lepton
masses· Circulant mass matrix· Quasiparticle gap
1 Introduction
Koide’s empirical observation that the charged–lepton masses satisfy
Q ≡
me + mµ + mτ
(√me + √mµ + √mτ )2
=
2
3 (1)
to better than one part in 104 has remained, in the four decades since its formulation [1], the most
precise unexplained numerical regularity in the Standard Model fermion sector. Mainstream
reactions divide between two positions: that (1) is a numerical coincidence with no underlying
mechanism, or that it is a consequence of one specific model among many that arrive at Q= 2/3
as a fit [2–4]. The most developed selection mechanism in the literature is Sumino’s U(3)×SU(2)
family gauge model [10, 11], which engineers a cancellation of the QED radiative correction to
Q so that the relation can survive unchanged from a high scale.
The purpose of this short note is to argue that neither position is quite right, because
the question Why Q = 2/3? is in fact two questions, only one of which is hard. The easy
half is purely structural and is the subject of this paper: If the observed mass spectrum is the
quasiparticle spectrum of a three–component system possessing Z3 symmetry whose three gaps
are all non–negative (the positive–gap region P defined in Theorem 5), then Q is determined
in closed form by two parameters, and the value Q = 2/3 corresponds to a specific algebraic
point in this two–parameter family. The hard half, what fixes the system to that specific point,
is a model question, and is deliberately left open here as a separate problem to be addressed
elsewhere.
∗Corresponding author: kibrom@xmoro.com
1
Relation to prior work. The algebraic ingredients of what follows are not new and we wish
to be explicit about this from the outset. Koide’s original observation [1] was followed by a geo-
metric interpretation due to Foot [4] and a broader geometric mass–pattern framing by Esposito
and Santorelli [5]. Koide himself later studied cyclic–permutation–invariant mass matrices [6],
and Brannen [2] introduced the cosine/circulant parametrisation √mk ∝ A+ Bcos(θ+ 2πk/3)
that is central to the present analysis; the closed–form value Q(A,B) = 1
3 + B2/(6A2) that we
record below in Theorem 5 is implicit in his analysis, following in two lines from the elemen-
tary identities ∑k cos(θ+ 2πk/3) = 0 and ∑k cos2(θ+ 2πk/3) = 3/2. Żenczykowski [7] later
described the same structure explicitly in Z3–symmetric language. We therefore do not claim
the algebra as new mathematics. What we do claim as new, to our knowledge, is first the clean
separation of existence from selection that turns the four–decade Koide puzzle into two narrower
mathematical problems, and second a physical reason for adopting Brannen’s parametrisation
at all, which we now spell out.
Why BdG: the physical content of the framing. In particle physics, symmetries are
normally imposed at the level of the Lagrangian and therefore act on the mass matrix M.
The Koide relation, however, is a statement about √M, and there is no a priori reason for
any sensible flavour symmetry to act on the square root of a mass matrix rather than on the
matrix itself. This is the content–free step in the standard “Koide as a circulant” story. The
BdG framing closes this gap. In a Bogoliubov–de Gennes system the spectrum is given by
a Hermitian gap operator ∆ acting on a complex order parameter, and the physical mass of
quasiparticle mode k is mk = |∆k|2, so that √mk = |∆k|. A discrete symmetry that acts on
the order parameter therefore acts directly on ∆, hence on √m, and not on m. In particular,
a Z3 symmetry of the order parameter delivers exactly the structure Koide’s relation requires,
automatically and for free. To our knowledge, this is the sense in which the BdG language
supplies a physical, rather than merely algebraic, justification for Brannen’s parametrisation,
and it is the actual new content of the present note. We stress that gap–equation mass generation
of this sort has a long history in particle physics. The relativistic analogue of the BCS/BdG
construction is the Nambu–Jona-Lasinio (NJL) mechanism [8, 9], in which fermion masses are
generated dynamically by the vacuum expectation value of a fermion bilinear and the physical
mass appears as a self–consistent gap. The BdG language used here should therefore be read
not as a condensed–matter import but as an instance of dynamical mass generation in the
standard particle–physics sense, specialised to a setting in which a discrete Z3 acts on the
gap. We thus separate existence from selection. The existence claim is a theorem; we prove
it below for completeness. Within this framework, the Koide relation should be read not as a
coincidence but as the statement that the charged–lepton spectrum sits on a specific algebraic
locus inside a two–parameter family of Z3–equivariant BdG spectra. Whether the underlying
physics drives the spectrum to this locus is a separate question, and we are explicit throughout
about which question is being answered. The algebra itself, we stress again, is identical for any
three–component Hermitian operator transforming in the regular representation of Z3; what
the BdG language adds is the physical interpretation of its eigenvalues as √m.
Structure of the paper. Section 2 introduces the Z3–equivariant BdG framework and
establishes the uniqueness of the gap form ∆k = A + Bcos(θ + 2πk/3). Section 3 com-
putes the Koide ratio for this gap form and derives the main identity. Section 4 states the
Q = 2/3 ⇔ |B/A|= √2 corollary (on the positive–gap region) and gives the numerical ver-
ification against the Particle Data Group [12] charged–lepton masses. Section 5 is a careful
statement of what the identity does and does not establish, in particular flagging that the se-
lection question (the choice of any specific microscopic system that realises the Z3–equivariant
gap form, and that singles out |B/A|= √2) is deliberately out of scope. Section 6 points to the
work that closes the remaining gap.
2
2 Z3–equivariant BdG setup
We work in the simplest setting that makes the theorem unambiguous: a three–component
Bogoliubov–de Gennes Hamiltonian with a global Z3 symmetry that cyclically permutes the
three components. The reader is invited to think of this as the BdG description of three
coupled Bose condensates, of three flavours of an order parameter on a triangular lattice, or of
any phenomenologically equivalent configuration: the theorem is purely algebraic and indifferent
to the microscopic interpretation.
2.1 Setting and notation
Let ψ = (ψ0,ψ1,ψ2)T be a three–component complex field and let σ : C3 → C3 act by cyclic
permutation, σ(ψ0,ψ1,ψ2) = (ψ1,ψ2,ψ0). Write ω≡ e2πi/3. The matrix
0 0 1
S=  0 1 0
1 0 0
  (2)
generates the Z3 action on the component index. Its eigenvalues are {1,ω,ω2} and its eigen-
vectors are the Z3 Fourier modes, which we label by the Z3 charge k∈ {0,1,2}.
Definition 1 (Z3–equivariant gap matrix). A 3 × 3 matrix ∆ acting on the component index
is Z3–equivariant if it commutes with the cyclic action, S∆ S−1 = ∆, equivalently if it is a
polynomial in S.
Lemma 2 (Diagonalisation of Z3–equivariant matrices). Any Z3–equivariant 3×3 matrix ∆ is a
circulant of the form ∆ = c0I+c1S+c2S2 for some cj ∈ C, and is simultaneously diagonalisable
with S. Its eigenvalues are
∆k = c0 + c1ωk + c2ω2k, k∈ {0,1,2}. Proof. Standard. Any matrix commuting with S is a polynomial in S because the minimal
polynomial of S is x3
− 1, which has distinct roots; therefore S generates its commutant. The
circulant form follows. Diagonalisation is by the discrete Fourier transform on Z3.
2.2 The unique real gap form
We are interested in physical (i.e. Hermitian, real–spectrum) BdG gap matrices. Imposing
reality and the additional discrete symmetries that any sensible microscopic theory will respect
cuts the three complex parameters (c0,c1,c2) of Lemma 2 down to three real parameters. The
result, which we will use throughout the paper, is the following normal form.
Proposition 3 (Most general real Z3–equivariant gap). Let ∆ be a Z3–equivariant 3 × 3 matrix
whose three eigenvalues ∆0,∆1,∆2 are real. Then there exist real numbers A,B,θ such that
∆k = A + B cos(θ+
2πk
3 ), k= 0,1,2. (4)
The parametrisation is unique up to the residual ambiguities θ→ θ+ 2π, (B,θ) → (−B,θ+ π),
and the cyclic relabeling k→ k+ 1 ⇔ θ→ θ− 2π/3.
Proof. By Lemma 2 the eigenvalues are ∆k = c0 + c1ωk + c2ω2k. Reality of all three ∆k forces
c0 ∈ R and c2 = c1, so we may write c0 = A and c1 =
B
2 eiθ with A,B ∈ R and θ ∈ [0,2π).
Substituting,
∆k = A+ B
2 eiθωk + B
2 e−iθω−k
= A+ Bcos(θ+ 2πk
3 ),
which is the asserted form. The residual ambiguities listed above exhaust the redundancy of
the parametrisation.
(3)
3
Remark 4. Equation (4) is exactly the gap pattern Brannen extracted [2] as an empirical fit to
the charged–lepton masses. What Proposition 3 establishes is that this is not a fitting ansatz
at all: it is the unique possible gap pattern for any three–component spectrum with a real Z3
symmetry, parametrised by exactly three real numbers (A,B,θ).
2.3 The mass–spectrum convention
To convert the gap eigenvalues ∆k into the masses entering the Koide ratio (1) we adopt the
standard BdG convention that the quasiparticle gap is the square root of the mass–squared
associated with each Z3 sector,
mk = ∆2
k, k= 0,1,2. (5)
This is the natural convention for any system in which the BdG eigenvalue plays the role of a
quasiparticle energy and mk the corresponding particle mass. We assume throughout that the
parameters (A,B,θ) lie in the regime where all three ∆k are real and finite. We do not assume
that they are positive: the identity to be proved is even in ∆k, and the Koide ratio depends
only on |∆k| via √mk = |∆k|.
3 The exact Koide identity
We can now state and prove the main result.
Theorem 5 (Exact Koide identity for Z3–equivariant spectra). Let ∆0,∆1,∆2 be a real Z3–
equivariant gap triplet of the form (4), with A > 0 and (A,B,θ) in the positive–gap region P
defined by ∆k ≥ 0 for all k∈ {0,1,2}. Define mk ≡ ∆2
k and the Koide ratio
Q=
m0 + m1 + m2
(√m0 + √m1 + √m2) 2. (6)
Then
Q=
1
3 +
B2
6A2. (7)
cos2(θ+ 2πk
3 )=
3
2. (8)
In particular Q depends only on the dimensionless ratio B/A and is independent of θ.
Proof. The proof is a direct two–line computation that exploits two elementary trigonometric
identities valid for any real θ:
2
∑
k=0
cos(θ+ 2πk
3 ) = 0,
2
∑
k=0
Both follow from the geometric series ∑k ωk = 0.
Numerator. With mk = ∆2
k,
∑
k
mk = ∑
k
∆2
k = ∑
k
(A+ Bcos(θ+ 2πk
3 ))2
= 3A2 + 2AB∑
cos(θ+ 2πk
3 ) + B2∑
cos2(θ+ 2πk
3 )
k
k
= 3A2 + 0 + 3
2 B2 = 3A2 + 3
2 B2
. (9)
Denominator. On P each ∆k ≥ 0, so √mk = ∆k and
∑
k
√mk = ∑
k
∆k = 3A+ B∑
k
4
cos(θ+ 2πk
3 ) = 3A, (10)
so that
√mk)2
= 9A2
. (11)
Combining (9) and (11),
B2
( ∑
k
3A2 + 3
2 B2
1
Q=
=
9A2
3 +
6A2 , (12)
which is (7). The independence of θ is manifest, as θ appears only in linear and quadratic
cosines whose sums over k are themselves θ–independent.
Remark 6 (Scope of the identity and the positive–gap region P). The positive–gap region P is
the set of triples (A,B,θ) with A>0 and A+ Bcos(θ+ 2πk/3) ≥ 0 for k= 0,1,2. Whenever
|B| ≤ A this is automatic, and the identity gives Q ∈ [ 1
1
3 ,
2 ]. For |B| > A the region P is a
proper subset of parameter space and depends on θ: for some phases all three gaps remain
non–negative, for others at least one becomes negative. Outside P the convention √mk = ∆k
in (10) must be replaced by √mk = |∆k|, the denominator picks up an extra cross term, and
the closed–form identity (7) fails as stated. A concrete counterexample is (B/A,θ) = (√2,π/3),
where ∆1/A = 1− √2 < 0 and direct evaluation gives Q ≈ 0.4094 ̸= 2/3, even though
1/3 + B2/(6A2) would naively give 2/3. The empirical Brannen point (B/A,θ) = (√2,2/9) sits
in the |B| > A extended regime as well, but happens to leave all three gaps strictly positive
(see Table 1), so it lies in P and the identity applies there unchanged. This special feature of
the empirical point, and the question of whether Q= 2/3 ever holds with at least one negative
gap, are treated separately in Section 4.
Remark 7 (Why the identity is sharper than a fit). A reader habituated to numerical fits will
object that one can always recover Q = 2/3 by adjusting two free parameters. The sharpness
of (7) is that the two parameters in the fit are not free: they are forced to be (A,B) in the
precise functional form (4), which is itself the unique gap form (Proposition 3). The phase θ is
a third parameter, and the identity tells us it is irrelevant: every value of θ gives the same Q.
The non–trivial content is therefore that the two–parameter family of Z3–equivariant spectra
collapses, when viewed through the Koide ratio, to a one–parameter family in B/A, and the
charged–lepton point selects a specific value of that parameter.
4 Corollary: Q = 2/3 on the Brannen locus, and numerical ver-
ification
Restricted to the positive–gap region P, the empirical Koide value Q = 2/3 corresponds, via
(7), to
2
B2
B
A= ±√2. =
3
1
3 +
6A2 ⇐⇒ B2 = 2A2 ⇐⇒
(13)
This rewriting is, by itself, just an algebraic substitution; its physical content is that the
question Why Q= 2/3?, restricted to triples with all gaps non–negative, reduces to the appar-
ently narrower but logically equivalent question Why |B/A|= √2? The reduction is illustrated
in Fig. 1, which shows Qas a function of B/Aon P, with the empirical Koide value selecting the
point |B/A|= √2. The reduction is useful precisely because the narrower question is structural:
B/Ais a single dimensionless amplitude ratio, not an integrated property of three masses, and
admits in principle a representation–theoretic or topological answer. Whether such an answer
exists is a separate problem and not taken up in the present paper.
Corollary 8 (Brannen locus on P). Let (A,B,θ) lie in the positive–gap region P of Theorem 5.
Then Q= 2/3 if and only if |B/A|= √2. The phase θ remains a free parameter, subject only
to the positivity constraint defining P, which at |B/A|= √2 restricts θ to a proper subarc of
5
[0,2π). We do not claim that Q= 2/3 implies |B/A|= √2 outside P; whether Q= 2/3 can be
realised by triples with at least one negative gap is left as an open question.
The empirical Brannen point (B/A,θ) = (√2,2/9) lies inside P (the three values 1+√2 cos(2/9+
2πk/3) are approximately 2.378, 0.039, 0.583, all strictly positive; see Fig. 2), so the corollary
applies to it without modification.
The phase θ is not fixed by the theorem and remains a free parameter of the framework at the
level of this paper. Empirically, the value θ = 2/9 originally noted by Brannen [2] reproduces
the PDG charged–lepton masses to the residuals shown below; a derivation of that empirical
value from first principles is a separate problem and is not attempted here. With B/A= √2
and θ= 2/9 taken as empirical inputs, the formula ∆k = A(1+√2 cos(2/9+2πk/3)) determines
the lepton spectrum up to the overall scale A.
4.1 Numerical verification
We adopt the Particle Data Group [12] central values
me = 0.51099895 MeV, mµ = 105.6583755 MeV, mτ = 1776.86 MeV, (14)
which give the empirical Koide ratio
QPDG = 0.666660511 ..., | QPDG− 2/3 | ≈ 6.16 × 10−6
. (15)
With B/A= √2 and θ= 2/9 fixed, the only remaining parameter is the overall scale A, which
is fixed by the trace identity ∑k √mk = 3A of (10):
A=
1
3 (√me + √mµ + √mτ ) = 17.71556... MeV1/2
. (16)
Substituting back into (4) and squaring, the predicted charged–lepton masses agree with the
PDG central values to better than 10−4 on me and to 10−5 or better on mµ,mτ (Table 1).
All three residuals are small compared to the PDG uncertainty on mτ and consistent with
the unincluded O(α) loop corrections, the analysis of which is outside the scope of the present
paper. We note in passing that the Brannen parametrisation is RG–stable at one loop in the
standard charged–lepton sector: a common multiplicative anomalous dimension on the three
masses rescales A but leaves the ratio B/A and the phase θ untouched, so the identification of
|B/A|= √2 and θ= 2/9 is independent of the renormalisation scale at this order.
me [MeV] mµ [MeV] mτ [MeV]
PDG [12] 0.51099895 105.6583755 1776.86
Brannen pointa 0.51096947 105.65332 1776.882
Relative residual−5.8 × 10−5
−4.8 × 10−5 +1.3 × 10−5
a Computed by the script lepton_fit.py accompanying this paper.
Table 1: Charged–lepton masses predicted by ∆k = A(1 + √2 cos(2/9 + 2πk/3)) at the Koide
point B/A= √2 with the Brannen phase θ= 2/9, compared with PDG central values.
The identity (7) itself is, of course, exact at the level of the formal definition of Q; the
residuals in Table 1 measure only how well the empirical lepton masses lie on the Brannen locus
(B/A,θ) = (√2,2/9), not any imprecision in the theorem.
5 What this paper does and does not establish
We have recorded a theorem about Z3–equivariant spectra, not about leptons, and as already
emphasised in the introduction the algebraic content of the theorem is implicit in Brannen’s
6
Q depends only on
B
/A (Theorem 5)
Koide ratio Q
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
Q(
/A) = 1/3 + (
B
B
/A)2/6
Q = 2/3
/A = ± B
2, Q = 2/3
1.5 1.0 0.5 0.0 0.5 1.0 1.5
amplitude ratio B
/A
Figure 1: The Koide ratio Q(B/A) = 1
3 + (B/A)2/6 as a function of the amplitude ratio, on the
positive–gap region P. Theorem 5 reduces the two-parameter family (A,B,θ) of Z3-equivariant
spectra in P to a one-parameter family in B/A, collapsing the dependence on θ entirely; the
empirical Koide value Q = 2/3 corresponds to |B/A|= √2. Outside P at least one gap is
negative and the closed–form curve shown here no longer applies (see Remark 6 and the explicit
failure at (B/A,θ) = (√2,π/3) in the proof script).
parametrisation [2] and standard properties of circulants. It is essential, for the paper to be
useful rather than overclaiming, to be explicit about what remains to be done and about which
parts of what is presented are genuinely new.
What is established.
1. The gap form ∆k = A+ Bcos(θ+ 2πk/3) is the unique form of a real Z3–equivariant
three–component spectrum (Proposition 3).
2. For this gap form, restricted to the positive–gap region P (all ∆k ≥ 0), the Koide ratio
satisfies the exact closed–form identity Q=
1
3 + B2/(6A2) (Theorem 5).
3. Within P, the empirical Koide point Q= 2/3 is mathematically equivalent to the ampli-
tude ratio |B/A|= √2 (Corollary 8). The empirical Brannen point (B/A,θ) = (√2,2/9)
lies in P and the corollary applies to it without modification.
4. The charged–lepton masses lie on the Brannen locus (B/A,θ) = (√2,2/9) to better than
10−4 in each component (Table 1).
5. The BdG identification mk = |∆k|2 supplies a physical reason for why a Z3 symmetry
should act on √mrather than on m; to our knowledge, this is the only part of the present
note that is not a restatement of material already in the literature.
What is not established.
1. The identity does not fix the value B/A= √2. The amplitude ratio is a free parameter
of the framework presented here; its derivation from an underlying microscopic or group–
theoretic principle is an open problem and is not attempted in this paper.
7
B
rannen locus:
B
/A = 2, all
m
40
gap eigenvalue k [MeV1/2]
20
m
me
0
20
40
0( )
1( )
2( )
= 2/9 (
rannen)
B
0 1 2 3 4 5 6
phase
Figure 2: The three Z3-equivariant gap eigenvalues ∆k(θ) = A+Bcos(θ+2πk/3) at the Brannen
point B/A= √2, with Afixed by the trace identity (10) from the PDG charged-lepton masses.
Horizontal lines mark √me, √mµ, and √mτ . The dashed vertical line indicates the Brannen
phase θ = 2/9, at which the three branches simultaneously match the empirical square-root
masses to the residuals of Table 1.
2. The identity does not fix the Brannen phase θ= 2/9, and indeed the identity is completely
independent of θ. The phase θis taken as an empirical input in the numerical verification;
deriving it is a separate open problem.
3. The identity makes no claim that any specific microscopic theory dynamically realises the
Z3–equivariant gap form, and in particular makes no claim that the value |B/A|= √2
is selected as a dynamical attractor of any flow. The existence statement of the present
paper says only that if the spectrum is Z3–equivariant and lies in P, the Koide ratio is
fixed by (A,B). Selecting the specific point |B/A|= √2 requires an independent input,
the identification of which is out of scope and is left as an open problem.
4. The theorem is algebraic. Its physical content depends on the identification of the un-
derlying Z3–equivariant quasiparticle spectrum with the observed lepton spectrum, an
identification that requires (i) a microscopic theory and (ii) a specification of the con-
vention mk = ∆2
k (Section 2.3). Both are framework choices and are flagged as such
here.
6 Conclusion and outlook
The Koide relation Q = 2/3, viewed for forty years as either coincidence or phenomenological
fit, is half of an existence–and–selection question. The existence half admits a sharp answer:
every real three–component Z3–equivariant spectrum lying in the positive–gap region P gives
Q=
1
3 + B2/(6A2), so that within P the empirical value Q= 2/3 is equivalent to the amplitude
ratio |B/A|= √2. The empirical Brannen point sits in P, so the equivalence applies to it. This
is a closed–form theorem with no fit parameters once A and B are given, and reduces a four–
decade–old empirical question about a single dimensionless number to two narrower questions
of independent mathematical content: Why |B/A|= √2? and Why θ= 2/9?
Neither narrower question is addressed in the present paper. They are explicitly flagged as
open problems, with the hope that the sharpness of the reduction recorded here, from three
8
masses to a single amplitude ratio and an irrelevant phase, will make them more tractable than
the original empirical observation. The identity of Theorem 5 is algebraic in nature, is implicit
in Brannen’s parametrisation [2], and is offered here not as new mathematics but as a structural
spine on which independent derivations of |B/A|= √2 and θ= 2/9 can in principle be hung.
The actual physics content of this note, as we have argued in the introduction, is the
BdG identification mk = |∆k|2. It is what turns the question “why should a Z3 symmetry
act on √m?”, which in a Lagrangian setting has no answer, into the simple statement that
the symmetry acts on the order–parameter gap ∆ and the masses are |∆|2 as a matter of
standard quasiparticle bookkeeping. Whether the underlying physical system that hosts this
BdG structure can be derived from a deeper symmetry breaking, and whether such a derivation
forces |B/A|= √2 and θ= 2/9, is the subject of separate work.
A non–claim about dynamical selection. A natural–looking but unjustified move at this
stage would be to assert that the Brannen gap form, or the specific point |B/A|= √2, arises
as the attractor of a nonlinear Schrödinger or Gross–Pitaevskii flow on a three–component
condensate. We make no such claim. We have not exhibited a flow whose attractor is the
Brannen gap form, nor a dynamical principle that selects |B/A|= √2 over neighbouring values
inside P. The existence half of the question, which is the only half this paper addresses, is purely
algebraic and is decoupled from any dynamical selection mechanism. The selection question is
left explicitly open, and any future identification of a flow or variational principle that picks
out |B/A|= √2 should be read as an independent input to the framework recorded here, not
as a consequence of it.
Companion code. A small reproduction package, containing Python scripts that verify the
identity (7) both symbolically and numerically, compute the predictions in Table 1 from the
PDG charged–lepton masses, and produce the figures used in this paper, is bundled with the
present submission. Every numerical claim in the paper is reproducible by a one–line invocation
python3 verify_identity.py or python3 lepton_fit.py.
Acknowledgements
The author thanks the broader Koide and Brannen communities for four decades of careful
empirical curation, without which a paper of this kind would have nothing to explain.
Statements and Declarations
Funding. This research received no external funding.
Competing interests. The author declares no competing interests.
Consent to Publish declaration: Not applicable.
Consent to Participate declaration: Not applicable.
Ethics declaration: Not applicable.
Author contributions. The sole author conceived the study, carried out the mathematical
analysis, performed the numerical verification, wrote the manuscript, and approved the final
manuscript.
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Data availability. All data generated during this study are reproducible from the companion
Python scripts bundled with the submission. No external datasets were used.
Code availability. The companion code is available with the submission and will be deposited
at Zenodo under an MIT license upon publication.
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