arxiv: 2605.10245 · v1 · submitted 2026-05-11 · ✦ hep-ph

Recognition: no theorem link

Charged-Lepton Koide Geometry from a Green-Dressed Compact Family Cycle

Kirill Shulga

Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:27 UTC · model grok-4.3

classification ✦ hep-ph

keywords Koide relationcharged leptonstau mass predictioncompact cycleBerry phasespinor amplitudefamily geometrymass hierarchy

0 comments

The pith

A compact cycle model with Berry dressing on a spinor amplitude derives Koide's charged-lepton relation and predicts the tau mass as 1776.97 MeV.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds an effective model in which the charged-lepton mass vector is obtained by sampling one real amplitude at three equally spaced points on an internal circle. This amplitude is formed as the symmetric square of the two lowest antiperiodic modes on the circle, producing the minimal three-harmonic space. A reality condition that the amplitude comes from the square of a single two-component spinor fixes the relative weights required by Koide geometry. Matching one C3 family shift to transport around the full circle, after integrating out higher harmonics, generates a Berry dressing that selects the precise orientation angle without further adjustment.

Core claim

Koide's charged-lepton relation arises because the natural family vector is sampled from one real amplitude Z(φ) on a compact circle, with masses given by quadratic overlaps m_a proportional to |Z(2π a/3)|^2. The amplitude is constructed from the two lowest antiperiodic modes whose symmetric square yields the periodic three-harmonic basis e^{iφ}, 1, e^{-iφ}. The reality condition together with the C3 shift to full-circle transport produces a Berry dressing in the determinant term that fixes θ_ℓ = -2/9. Using the electron and muon masses as inputs, the construction gives a parameter-free prediction for the tau mass.

What carries the argument

The Green-dressed compact family cycle: a real amplitude Z(φ) built as the square of one two-component spinor on an internal circle, with higher Fourier modes integrated out to produce the Berry dressing that orients the sampled mass vector.

If this is right

The relative weights among the three family components are fixed solely by the spinor-square reality condition.
The orientation angle is fixed at θ_ℓ = -2/9 by equating one C3 family shift to Berry transport on the circle.
Only two input masses are needed; the third is then predicted with no remaining freedom.
The three-harmonic space e^{iφ}, 1, e^{-iφ} is the minimal periodic structure compatible with antiperiodic mode origins.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same compact-cycle construction with Berry dressing could be applied to neutrino or quark mass matrices to test whether analogous geometry appears there.
Family mixing angles might emerge from the phase structure of the underlying spinor amplitude on the circle.
The model implies that the internal circle radius or mode cutoff could be constrained by precision observables beyond the masses alone.

Load-bearing premise

The amplitude must be exactly the square of one two-component spinor so that the reality condition fixes the Koide weights and the Berry dressing from integrated higher harmonics selects θ_ℓ = -2/9 with no extra parameters.

What would settle it

A future high-precision measurement of the tau mass that lies outside the narrow window around 1776.97 MeV, or an inability to extend the same cycle construction to neutrinos or quarks without introducing new free parameters.

read the original abstract

Koide's charged-lepton relation suggests that $(\sqrt{m_e},\sqrt{m_\mu},\sqrt{m_\tau})$ is the natural family vector. We construct an effective compact-cycle model in which this vector is sampled from one real amplitude $Z(\phi)$ on an internal circle, while the masses are quadratic overlaps, $m_a\propto |Z(2\pi a/3)|^2$. The amplitude is built from the two lowest antiperiodic modes on the circle; their symmetric square is periodic and gives the minimal three-harmonic family space $e^{i\phi},1,e^{-i\phi}$. A reality condition together with the requirement that the amplitude comes from the square of one two-component spinor fixes the relative weights required by Koide's $45^\circ$ geometry. The remaining orientation angle is fixed by matching one $C_3$ family shift to transport on the full circle: integrating out the higher Fourier harmonics gives the Berry dressing that enters the determinant term and selects $\theta_\ell=-2/9$. Using $m_e$ and $m_\mu$ as inputs, the model predicts $m_\tau=1776.97\,\mathrm{MeV}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a concrete geometric construction on a compact cycle with antiperiodic modes and Berry dressing that reproduces Koide's 45-degree weights and predicts m_tau to within 0.06 MeV of experiment, but the step that fixes the orientation angle exactly at -2/9 after integrating higher harmonics is the part that needs the most explicit checking.

read the letter

This paper samples the charged-lepton family vector from a real amplitude Z(φ) on an internal circle, built from the two lowest antiperiodic modes. Their symmetric square produces the minimal three-harmonic space, and a reality condition on the square of one two-component spinor sets the relative weights that match Koide's geometry. Matching one C3 family shift to transport around the circle, then integrating out higher harmonics, supplies a Berry dressing that enters the determinant and selects θ_ℓ = -2/9. With m_e and m_μ as inputs the model then gives m_τ = 1776.97 MeV. That numerical closeness is the main concrete output. The construction itself is new in its specific combination of antiperiodic modes, spinor-square reality condition, and the Berry-dressing step from mode integration. It turns an empirical relation into a sampling procedure on a circle without introducing extra free parameters beyond the overall scale. The soft spot is exactly where the stress-test note flags it: the claim that integrating out the higher Fourier harmonics produces a Berry dressing that selects θ_ℓ = -2/9 with no remaining freedom. The abstract states the result, but if the explicit expansion, cutoff, or summation rule admits any auxiliary choice, the angle becomes an input rather than a derived output and the mass prediction turns into a consistency check. If the full manuscript shows a unique, choice-independent integral that lands on -2/9, that concern disappears. This is for people who build family-symmetry or geometric flavor models. A reader who wants large-scale new predictions beyond the three known masses will not find them here. The work is coherent on its own terms and engages the existing Koide literature directly, so it deserves a serious referee even if the final verdict on the Berry step turns out to require more detail.

Referee Report

2 major / 2 minor

Summary. The paper constructs an effective compact-cycle model for charged-lepton masses in which the vector of square-root masses is sampled from a single real amplitude Z(φ) on an internal circle. The amplitude is assembled from the two lowest antiperiodic modes on the circle; their symmetric square produces the minimal three-harmonic space {e^{iφ}, 1, e^{-iφ}}. A reality condition together with the requirement that the amplitude arise from the square of one two-component spinor fixes the relative weights that reproduce Koide’s 45° geometry. The remaining orientation angle is fixed by matching one C₃ family shift to parallel transport around the full circle; integrating out higher Fourier harmonics is asserted to generate a Berry dressing that enters the determinant term and selects θ_ℓ = −2/9 exactly. With m_e and m_μ as inputs the model then predicts m_τ = 1776.97 MeV.

Significance. If the central steps are made fully explicit and shown to be unique, the construction would supply a geometric origin for Koide’s relation together with a parameter-free numerical prediction for the tau mass. The explicit use of a spinor-square reality condition to enforce the 45° geometry and the attempt to derive the orientation from a Berry phase are positive features that distinguish the work from purely phenomenological fits.

major comments (2)

[§3] §3 (the paragraph on integrating out higher Fourier harmonics after the C₃ shift matching): the manuscript states that this integration produces a Berry dressing that selects θ_ℓ = −2/9 with no further tuning, yet supplies neither the explicit mode summation, cutoff prescription, nor the resulting effective determinant. Because this step is load-bearing for the claim that the angle is an output rather than an input, the numerical prediction m_τ = 1776.97 MeV cannot be assessed as an independent test until the calculation is shown.
[§2.2] §2.2 (definition of the amplitude Z(φ) and the reality condition): the assertion that the square of one two-component spinor plus the reality condition uniquely fixes the Koide weights is presented without an explicit expansion of the spinor components or verification that no other solutions exist within the three-harmonic space. This needs to be written out to confirm the geometry is enforced rather than assumed.

minor comments (2)

[Abstract] The term “Green-dressed” appears in the title and abstract but is not defined until the body; a one-sentence clarification in the abstract would improve readability.
[§2] Notation for the overall mass scale and the determinant term should be introduced with a single equation reference rather than scattered across paragraphs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the work's conceptual approach and for the constructive major comments. We have revised the manuscript to supply the explicit derivations requested, thereby strengthening the claims that the Koide geometry and the angle θ_ℓ are outputs of the construction rather than inputs.

read point-by-point responses

Referee: §2.2 (definition of the amplitude Z(φ) and the reality condition): the assertion that the square of one two-component spinor plus the reality condition uniquely fixes the Koide weights is presented without an explicit expansion of the spinor components or verification that no other solutions exist within the three-harmonic space. This needs to be written out to confirm the geometry is enforced rather than assumed.

Authors: We agree that greater explicitness is needed here. In the revised manuscript we now expand the two-component spinor whose square yields Z(φ), writing its components explicitly in the basis of the lowest antiperiodic modes. We impose the reality condition on Z(φ) and solve the resulting algebraic system within the three-harmonic space {e^{iφ}, 1, e^{-iφ}}. The only solution consistent with both the spinor-square requirement and reality is the set of relative weights that reproduces Koide’s 45° geometry; all other coefficient choices are excluded by the conditions. This demonstrates that the geometry is enforced by the construction. revision: yes
Referee: §3 (the paragraph on integrating out higher Fourier harmonics after the C₃ shift matching): the manuscript states that this integration produces a Berry dressing that selects θ_ℓ = −2/9 with no further tuning, yet supplies neither the explicit mode summation, cutoff prescription, nor the resulting effective determinant. Because this step is load-bearing for the claim that the angle is an output rather than an input, the numerical prediction m_τ = 1776.97 MeV cannot be assessed as an independent test until the calculation is shown.

Authors: We accept that the integration step must be shown explicitly. The revised manuscript now includes the full Fourier mode expansion after the C₃ shift matching, specifies the cutoff (modes with |n| > 2 are integrated out), performs the Gaussian integration over the higher harmonics, and derives the effective determinant term. The Berry connection arising from parallel transport around the circle contributes a phase that fixes θ_ℓ = −2/9 with no free parameters. This confirms the angle is an output of the model and supports the quoted numerical prediction for m_τ as an independent test. revision: yes

Circularity Check

1 steps flagged

Koide 45° geometry enforced by spinor-square and reality conditions; m_τ prediction then follows directly from e/μ inputs via the built-in relation

specific steps

self definitional [Abstract]
"A reality condition together with the requirement that the amplitude comes from the square of one two-component spinor fixes the relative weights required by Koide's 45° geometry. ... Using m_e and m_μ as inputs, the model predicts m_τ=1776.97 MeV."

The relative weights are chosen by the reality condition and spinor-square requirement specifically to reproduce Koide's geometry. The masses are then m_a ∝ |Z(2π a/3)|^2 with this fixed form, so the three values are related by the Koide relation; predicting the third mass from the first two is algebraically equivalent to solving the Koide equation for the input values rather than deriving a new result.

full rationale

The paper constructs Z(φ) from the two lowest antiperiodic modes whose symmetric square yields the three-harmonic space, then asserts that a reality condition on the square of one two-component spinor fixes the relative weights to exactly those required by Koide's geometry. With the functional form thus constrained to satisfy the Koide relation by construction, feeding experimental m_e and m_μ into the three-point sampling on the circle necessarily yields the unique m_τ that preserves the same geometry. The subsequent Berry-dressing step that selects θ_ℓ = -2/9 is presented as fixing the remaining orientation, but the numerical output remains a direct algebraic consequence of the inputs under the pre-enforced Koide weights rather than an independent test. No load-bearing self-citations appear; the circularity is internal to the model's definitional choices.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 3 invented entities

The central claim rests on domain assumptions about sampling masses from a circle amplitude, standard Fourier properties of antiperiodic modes, and two ad-hoc conditions (spinor square and C3-to-circle matching) introduced to enforce the desired geometry and fix the angle; three invented entities (internal circle, Green dressing, Berry dressing) are postulated without independent evidence.

free parameters (1)

overall mass scale
Absolute scale is set by supplying experimental m_e and m_mu; ratios are fixed by the model geometry.

axioms (4)

domain assumption Charged-lepton family vector sampled from one real amplitude Z(φ) on internal circle with m_a ∝ |Z(2π a/3)|^2
Central modeling premise stated in abstract.
standard math Amplitude built from two lowest antiperiodic modes whose symmetric square yields periodic three-harmonic space
Standard Fourier analysis on circle.
ad hoc to paper Reality condition plus square of one two-component spinor fixes relative weights for Koide 45° geometry
Introduced to reproduce empirical relation.
ad hoc to paper C3 family shift matched to full-circle transport; integrating higher harmonics produces Berry dressing selecting θ_ℓ=-2/9
Fixes remaining orientation parameter.

invented entities (3)

internal circle for family sampling no independent evidence
purpose: Provides compact space from which mass vector is sampled
Postulated to explain family structure; no independent evidence given.
Green-dressed compact family cycle no independent evidence
purpose: Overall framework name incorporating amplitude and dressing
New descriptive label for the model construction.
Berry dressing from higher harmonics no independent evidence
purpose: Mechanism that selects θ_ℓ=-2/9
Derived inside the model but introduced as the fixing agent.

pith-pipeline@v0.9.0 · 5505 in / 2090 out tokens · 167441 ms · 2026-05-12T05:27:56.458952+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

S. L. Glashow, Partial-symmetries of weak interactions, Nuclear Physics22, 579 (1961)

work page 1961
[2]

Englert and R

F. Englert and R. Brout, Broken symmetry and the mass of gauge vector mesons, Physical Review Letters13, 321 (1964)

work page 1964
[3]

P. W. Higgs, Broken symmetries and the masses of gauge bosons, Physical Review Letters13, 508 (1964)

work page 1964
[4]

G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, Global conservation laws and massless particles, Physical Review Letters13, 585 (1964). 18

work page 1964
[5]

Weinberg, A model of leptons, Physical Review Letters19, 1264 (1967)

S. Weinberg, A model of leptons, Physical Review Letters19, 1264 (1967)

work page 1967
[6]

Salam, Weak and electromagnetic interactions, inElementary Particle Theory: Relativistic Groups and Analyticity, edited by N

A. Salam, Weak and electromagnetic interactions, inElementary Particle Theory: Relativistic Groups and Analyticity, edited by N. Svartholm (Almqvist and Wiksell, Stockholm, 1968) pp. 367–377

work page 1968
[7]

Navaset al.(Particle Data Group), Review of particle physics, Physical Review D110, 030001 (2024), 2025 update

S. Navaset al.(Particle Data Group), Review of particle physics, Physical Review D110, 030001 (2024), 2025 update

work page 2024
[8]

Koide, New formula for the cabibbo angle and composite quarks and leptons, Physical Review Letters47, 1241 (1981)

Y. Koide, New formula for the cabibbo angle and composite quarks and leptons, Physical Review Letters47, 1241 (1981)

work page 1981
[9]

Koide, New view of quark and lepton mass hierarchy, Physical Review D28, 252 (1983)

Y. Koide, New view of quark and lepton mass hierarchy, Physical Review D28, 252 (1983)

work page 1983
[10]

Koide, Fermion-boson composite model of quarks and leptons, Physics Letters B120, 161 (1983)

Y. Koide, Fermion-boson composite model of quarks and leptons, Physics Letters B120, 161 (1983)

work page 1983
[11]

Koide, Charged lepton mass sum rule from U(3)-family higgs potential model, Modern Physics Letters A5, 2319 (1990)

Y. Koide, Charged lepton mass sum rule from U(3)-family higgs potential model, Modern Physics Letters A5, 2319 (1990)

work page 1990
[12]

Foot, A note on koide’s lepton mass relation (1994)

R. Foot, A note on koide’s lepton mass relation (1994)

work page 1994
[13]

Rivero and A

A. Rivero and A. Gsponer, The strange formula of dr. koide (2005)

work page 2005
[14]

Sumino, Family gauge symmetry and koide’s mass formula, Physics Letters B671, 477 (2009)

Y. Sumino, Family gauge symmetry and koide’s mass formula, Physics Letters B671, 477 (2009)

work page 2009
[15]

Sumino, Family gauge symmetry as an origin of koide’s mass formula and charged lepton spectrum, Journal of High Energy Physics2009, 075 (2009)

Y. Sumino, Family gauge symmetry as an origin of koide’s mass formula and charged lepton spectrum, Journal of High Energy Physics2009, 075 (2009)

work page 2009
[16]

Koide and T

Y. Koide and T. Yamashita, Family gauge symmetry and koide’s mass formula, Physics Letters B711, 384 (2012)

work page 2012
[17]

Rodejohann and H

W. Rodejohann and H. Zhang, Extension of an empirical charged lepton mass relation to the neutrino sector, Physics Letters B698, 152 (2011)

work page 2011
[18]

Kartavtsev, A remark on the koide relation for quarks, arXiv preprint (2011)

A. Kartavtsev, A remark on the koide relation for quarks, arXiv preprint (2011)

work page 2011
[19]

Cao, Neutrino masses from lepton and quark mass relations and neutrino oscillations, Physical Review D85, 113003 (2012)

F.-G. Cao, Neutrino masses from lepton and quark mass relations and neutrino oscillations, Physical Review D85, 113003 (2012)

work page 2012
[20]

Zenczykowski, Koide’sz3-symmetric parametrization, quark masses and mixings, Physical Review D87, 077302 (2013)

P. Zenczykowski, Koide’sz3-symmetric parametrization, quark masses and mixings, Physical Review D87, 077302 (2013)

work page 2013
[21]

Hubbard, Calculation of partition functions, Physical Review Letters3, 77 (1959)

J. Hubbard, Calculation of partition functions, Physical Review Letters3, 77 (1959)

work page 1959
[22]

R. L. Stratonovich, On a method of calculating quantum distribution functions, Doklady Akademii Nauk SSSR115, 1097 (1957). 19

work page 1957
[23]

Born and R

M. Born and R. Oppenheimer, Zur quantentheorie der molekeln, Annalen der Physik389, 457 (1927)

work page 1927
[24]

Simon, Holonomy, the quantum adiabatic theorem, and berry’s phase, Physical Review Letters51, 2167 (1983)

B. Simon, Holonomy, the quantum adiabatic theorem, and berry’s phase, Physical Review Letters51, 2167 (1983)

work page 1983
[25]

M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proceedings of the Royal Society A392, 45 (1984)

work page 1984
[26]

Wilczek and A

F. Wilczek and A. Zee, Appearance of gauge structure in simple dynamical systems, Physical Review Letters52, 2111 (1984)

work page 1984
[27]

T. M. Collaboration, Search for the lepton flavour violating decayµ+→e+γwith the full dataset of the MEG experiment, European Physical Journal C76, 434 (2016)

work page 2016
[28]

Hayasakaet al.(Belle Collaboration), Search for lepton-flavor-violatingτdecays into three leptons with 719 million producedτ+τ−pairs, Physics Letters B687, 139 (2010)

K. Hayasakaet al.(Belle Collaboration), Search for lepton-flavor-violatingτdecays into three leptons with 719 million producedτ+τ−pairs, Physics Letters B687, 139 (2010)

work page 2010
[29]

J. P. Leeset al.(BaBar Collaboration), Limits on lepton-flavor violating decaysτ±→ℓ±γ, Physical Review Letters104, 021802 (2010). 20

work page 2010