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arxiv: 2605.24866 · v1 · pith:2LWTNEJXnew · submitted 2026-05-24 · ✦ hep-ph

Fermion Mass Hierarchies and the Exceptional Jordan Algebra

Bishnu Gupta Teli , Tejinder Pal Singh This is my paper

Pith reviewed 2026-06-30 00:07 UTC · model grok-4.3

classification ✦ hep-ph
keywords fermion mass hierarchiesexceptional Jordan algebraoctonionsspectral scalespower-law relationsneutrino mass orderingcharged lepton massesquark mass ratios
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The pith

The exceptional Jordan algebra embeds three fermion generations into spectral scales whose multiplicative consistency produces power-law mass relations.

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

The paper proposes that the structure of the exceptional Jordan algebra accounts for fermion mass hierarchies by embedding three generations into Hermitian elements whose eigenvalues act as intrinsic spectral invariants. These invariants form cubic ladder structures, and requiring consistency under multiplicative composition of the hierarchies leads directly to power-law relations linking the observed masses to the underlying scales. A logarithmic fit to six charged-fermion mass ratios at the Z scale reduces the overall residual compared with a rigid version of the same algebra, driven mainly by the top-to-charm ratio. The best-fit exponent stays near one, corresponding to square-root scaling. The same construction accommodates both normal and inverted neutrino mass orderings while remaining consistent with oscillation and cosmological data.

Core claim

Starting from the octonionic realization of one Standard Model generation in C tensor O, the three-generation structure is embedded into Hermitian Jordan elements of J3(OC) whose ordered eigenvalues define intrinsic spectral invariants. The ordered spectral scales generate cubic ladder structures in the symmetric representation Sym^3(3), and consistency of multiplicative hierarchy composition naturally leads to power-law relations between fermion masses and spectral scales. The construction is presented as a phenomenological spectral deformation that retains the cubic-ladder, minimal-chain, and Dynkin-reflection structure while allowing the relative normalization, hierarchy exponent, and cha

What carries the argument

Cubic ladder structures in Sym^3(3) generated by ordered eigenvalues of Hermitian elements in the exceptional Jordan algebra J3(OC).

If this is right

  • Power-law relations between fermion masses and spectral scales follow directly from multiplicative consistency of the cubic ladders.
  • The hierarchy exponent stays near the square-root regime p approximately 1 in the best fit to charged-fermion ratios.
  • Both normal and inverted neutrino mass orderings remain compatible with the same spectral framework and current data.
  • The approach functions as an effective spectral organization rather than a parameter-free derivation of all masses.

Where Pith is reading between the lines

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

  • If the spectral scales are taken as more fundamental than the masses, future high-precision measurements at different renormalization scales could test whether the power-law exponents remain stable.
  • The improvement concentrated in the top-to-charm ratio suggests the framework may be particularly sensitive to up-type quark hierarchies, offering a potential diagnostic for extensions that include additional generations or new physics thresholds.
  • Accommodation of both neutrino orderings implies that tighter cosmological bounds on the sum of neutrino masses could eventually favor one ordering within this algebraic structure.

Load-bearing premise

The three-generation structure can be embedded into Hermitian Jordan elements of J3(OC) such that the ordered eigenvalues define intrinsic spectral invariants whose multiplicative consistency produces the observed power-law mass relations.

What would settle it

A global fit to the six charged-fermion mass ratios at the Z scale that yields a higher unpenalized log-residual than the rigid point while forcing the hierarchy exponent far from one.

Figures

Figures reproduced from arXiv: 2605.24866 by Bishnu Gupta Teli, Tejinder Pal Singh.

Figure 1
Figure 1. Figure 1: FIG. 1. Minimal ladders in the weight diagram of the sym [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Fano plane encoding octonion multiplication. Each [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Weight diagram of the symmetric cubic representa [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
read the original abstract

We develop a spectral framework for fermion mass hierarchies based on the exceptional Jordan algebra $J_3(\mathbb{O}_{\mathbb{C}})$. Starting from the octonionic realization of one Standard Model generation in $\mathbb{C}\otimes\mathbb{O}$, we embed the resulting three-generation structure into Hermitian Jordan elements whose eigenvalues define intrinsic spectral invariants. The ordered spectral scales generate cubic ladder structures in the symmetric representation $\mathrm{Sym}^3(\mathbf{3})$, and consistency of multiplicative hierarchy composition naturally leads to power-law relations between fermion masses and spectral scales. The construction should be viewed as a phenomenological spectral deformation of the rigid exceptional-Jordan framework discussed below: we retain the same cubic-ladder, minimal-chain, and Dynkin-reflection structure, but promote the relative normalization, hierarchy exponent, and charged-lepton octonionic phase to fitted spectral moduli. A global logarithmic fit to six charged-fermion mass ratios at $\mu=M_Z$ lowers the unpenalized log-residual relative to the rigid point, mainly through the top-to-charm ratio, while the individual ratios are not uniformly improved. The best-fit hierarchy exponent remains close to the square-root scaling regime, $p\simeq1$. In the neutrino sector, the framework accommodates both normal and inverted ordering while remaining consistent with oscillation data and current cosmological bounds on the total neutrino mass. Thus, the proposal is an effective spectral organization of fermion hierarchies, not a parameter-free replacement for the broader rigid construction discussed below.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a spectral framework for fermion mass hierarchies based on the exceptional Jordan algebra J_3(O_C). It starts from the octonionic realization of one SM generation in C⊗O, embeds the three-generation structure into Hermitian Jordan elements whose ordered eigenvalues define spectral invariants, and generates cubic ladder structures in Sym^3(3). Multiplicative consistency of these scales is argued to produce power-law relations between masses and spectral scales. The construction is a phenomenological deformation of a rigid framework: the cubic-ladder, minimal-chain and Dynkin-reflection structures are retained, but relative normalization, hierarchy exponent p and charged-lepton octonionic phase are promoted to fitted moduli. A global logarithmic fit to six charged-fermion mass ratios at μ=M_Z lowers the unpenalized log-residual (mainly via top/charm), with best-fit p≃1. The framework accommodates both normal and inverted neutrino orderings consistent with oscillation data and cosmological bounds on Σm_ν. The proposal is presented as an effective spectral organization rather than a parameter-free derivation.

Significance. If the algebraic embedding can be shown to enforce the power-law form and the observed ratios without the fitted moduli, the work would supply a novel algebraic origin for SM flavor structure rooted in the exceptional Jordan algebra. The retention of the cubic-ladder and Dynkin-reflection structures while allowing a controlled deformation is a conceptually clean approach, and the ability to treat both neutrino hierarchies within the same spectral setup is a positive feature. At present the significance is that of a new phenomenological parametrization organized by J_3(O_C) invariants rather than a derivation of the hierarchies from the algebra alone.

major comments (2)
  1. [Abstract] Abstract, paragraph beginning 'Starting from the octonionic realization': the claim that 'consistency of multiplicative hierarchy composition naturally leads to power-law relations' is not accompanied by a derivation showing that the power-law form is forced by the eigenvalues of the Hermitian Jordan elements or the Sym^3(3) representation. Instead the hierarchy exponent p is explicitly promoted to one of three fitted spectral moduli in the phenomenological deformation. This makes the power-law an input chosen to improve the fit rather than a necessary algebraic output.
  2. [Abstract] Abstract, sentence on the global logarithmic fit: the reported lowering of the unpenalized log-residual is achieved by fitting the three moduli (p, normalization, phase) directly to the same six charged-fermion mass ratios that are subsequently compared to the model. The manuscript should clarify whether this constitutes an independent test of the spectral framework or reduces to a post-fit residual, and whether any cross-validation or out-of-sample check was performed.
minor comments (1)
  1. [Abstract] Abstract: the distinction between the 'rigid exceptional-Jordan framework discussed below' and the present phenomenological deformation is introduced without a concise recap of the rigid case; a one-sentence summary of the rigid construction would improve standalone readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We agree that the abstract requires clarification on both the status of the power-law relations and the fitted nature of the comparison to data. We will revise the abstract to address these points while preserving the manuscript's framing as a phenomenological spectral organization.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph beginning 'Starting from the octonionic realization': the claim that 'consistency of multiplicative hierarchy composition naturally leads to power-law relations' is not accompanied by a derivation showing that the power-law form is forced by the eigenvalues of the Hermitian Jordan elements or the Sym^3(3) representation. Instead the hierarchy exponent p is explicitly promoted to one of three fitted spectral moduli in the phenomenological deformation. This makes the power-law an input chosen to improve the fit rather than a necessary algebraic output.

    Authors: We agree that the abstract phrasing risks implying a derivation of the power-law form from the Jordan-algebra eigenvalues or Sym^3(3) structure. The manuscript already states that the construction is a phenomenological deformation in which p is one of the fitted moduli; the cubic-ladder and multiplicative consistency motivate but do not enforce the power-law without this deformation. We will revise the abstract to remove any suggestion of necessity and to state explicitly that the power-law is an adopted feature of the effective framework. revision: yes

  2. Referee: [Abstract] Abstract, sentence on the global logarithmic fit: the reported lowering of the unpenalized log-residual is achieved by fitting the three moduli (p, normalization, phase) directly to the same six charged-fermion mass ratios that are subsequently compared to the model. The manuscript should clarify whether this constitutes an independent test of the spectral framework or reduces to a post-fit residual, and whether any cross-validation or out-of-sample check was performed.

    Authors: The reported improvement is a post-fit residual obtained by fitting the three moduli directly to the six mass ratios. The work is presented as an effective spectral organization rather than a predictive derivation, so the comparison is not an independent test. No cross-validation or out-of-sample checks were performed, given that the number of observables is comparable to the number of parameters. We will add explicit language in the abstract and main text clarifying the post-fit character of the residual. revision: yes

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The model rests on the octonionic realization of one generation, the embedding into J3(OC), and the assumption that eigenvalue ordering produces multiplicative consistency; three quantities are introduced as free parameters to match data.

free parameters (3)
  • hierarchy exponent p = ~1
    Promoted to fitted spectral modulus; best-fit value near 1
  • relative normalization
    Fitted spectral modulus adjusted to mass ratios
  • charged-lepton octonionic phase
    Fitted spectral modulus
axioms (3)
  • domain assumption Octonionic realization of one Standard Model generation in C⊗O
    Starting point for the three-generation embedding
  • domain assumption Eigenvalues of Hermitian Jordan elements define intrinsic spectral invariants
    Central to generating the cubic ladder
  • ad hoc to paper Consistency of multiplicative hierarchy composition produces power-law relations
    Invoked to obtain the mass-scale relations before fitting

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discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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Reference graph

Works this paper leans on

31 extracted references · 16 canonical work pages · cited by 1 Pith paper · 13 internal anchors

  1. [1]

    Ref. [14]

    Together with T+ = 1 2(← −e4 −i ← −e5), T − = 1 2(← −e4 +i ← −e5),(10) which arises from the quaternionic triple containinge 7, one verifies [T3, T±] =±T ±,[T +, T−] = 2T3,(11) satisfying thesu(2) commutation relations. The eigenspace splitting, therefore, organizes (ν, e−),(u k, dk) (12) into weak doublets. The remaining abelian generator arises from a C...

  2. [2]

    The cubic invariant (determi- nant) is defined by det(A) = 1 3 tr (A#)◦ A ,(B11) whereA # =A ◦ A −(trA)A+σ(A)1

    + Sc(x3x† 3) .(B10) In the main text, the off-diagonal entries (x f , yf , zf) cor- respond to (x 1, x2, x3). The cubic invariant (determi- nant) is defined by det(A) = 1 3 tr (A#)◦ A ,(B11) whereA # =A ◦ A −(trA)A+σ(A)1. In terms of com- ponents, this becomes det(A) =α 1α2α3 −α 1 Sc(x3x† 3)−α 2 Sc(x2x† 2)−α 3 Sc(x1x† 1) + 2ℜ x2(x1x3) .(B12) In the comple...

  3. [3]

    A general element ofJ 3(OC) may then be written as A=   ξ x y † x† α z y z † β   , ξ, α, β∈C, x, y, z∈O C

    Spin(9)-Adapted Decomposition Choose the primitive idempotent e11 =   1 0 0 0 0 0 0 0 0   . A general element ofJ 3(OC) may then be written as A=   ξ x y † x† α z y z † β   , ξ, α, β∈C, x, y, z∈O C. Equivalently, A= ξ ϕ † ϕV , ϕ= x y ,V= α z z† β ∈J 2(OC). The corresponding Peirce decomposition is J3(OC) ∼= Ce11 ⊕J 2(OC)⊕O 2 C. Under the Spin(9) s...

  4. [4]

    Equivalently, detA=ξdetV − Q 9(ϕ,V, ϕ), where detV=αβ−Sc(zz †), and Q9(ϕ,V, ϕ) =βSc(xx †) +αSc(yy †)−2ℜ(y(xz))

    Spin(9) Structure of the Cubic Jordan Norm For the Hermitian elementA, the cubic Jordan norm [17] is detA=ξαβ−ξSc(zz †)−αSc(yy †)−βSc(xx †)+2ℜ(y(xz)). Equivalently, detA=ξdetV − Q 9(ϕ,V, ϕ), where detV=αβ−Sc(zz †), and Q9(ϕ,V, ϕ) =βSc(xx †) +αSc(yy †)−2ℜ(y(xz)). Using the decompositionV=t1 2 +W, Q9(ϕ,V, ϕ) =t h Sc(xx†) + Sc(yy†) i +u h Sc(yy †)−Sc(xx †) i...

  5. [5]

    Symmetric Sector and Canonical Spread For the symmetric Jordan element used in the main text, Af =   rsf xf y† f x† f rsf zf yf z† f rsf   , the determinant reduces to detA f =r 3s3 f −rs f δ2 f + 2τf , with δ2 f = Sc(xf x† f) + Sc(yf y† f) + Sc(zf z† f), and τf =ℜ(y f(xf zf)). 18 This reproduces the invariant structure used in the main text and pro...

  6. [6]

    However, this decomposition does not by itself determine the effective phase parame- ters appearing in the phenomenological fit

    Interpretation and Limitations The Spin(9)-adapted decomposition clarifies the alge- braic role of the octonionic triple product in the spectral construction: the invariant τf =ℜ(y f(xf zf)) is the component of the cubic Jordan norm associ- ated with spectral asymmetry away from the reflection- symmetric caseτ f = 0. However, this decomposition does not b...

  7. [7]

    Gunaydin and F

    M. Gunaydin and F. Gursey, Quark structure and octo- nions, J. Math. Phys.14, 1651 (1973)

    1973

  8. [8]

    J. C. Baez, The Octonions, Bull. Am. Math. Soc.39, 145 (2002), [Erratum: Bull. Am. Math. Soc. 42, 213 (2005)], arXiv:math/0105155

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  9. [9]

    G. M. Dixon,Division Algebras: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics, Mathematics and Its Applications, Vol. 290 (Springer New York, New York, NY, 1994)

    1994

  10. [10]

    Towards a unified theory of ideals

    C. Furey, Unified Theory of Ideals, Phys. Rev. D86, 025024 (2012), arXiv:1002.1497 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  11. [11]

    Standard model physics from an algebra?

    C. Furey,Standard model physics from an algebra?, Ph.D. thesis, University of Waterloo (2015), arXiv:1611.09182 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  12. [12]

    G. M. Quinta, Spacetime Grand Unified Theory (2025), arXiv:2507.11564 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  13. [13]

    Ramond,Introduction to Exceptional Lie Groups and Algebras, Tech

    P. Ramond,Introduction to Exceptional Lie Groups and Algebras, Tech. Rep. (California Institute of Technology, Pasadena, California, 1977)

    1977

  14. [14]

    The Standard Model, The Exceptional Jordan Algebra, and Triality

    L. Boyle, The Standard Model, The Exceptional Jordan Algebra, and Triality (2020), arXiv:2006.16265 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  15. [15]

    T. A. Springer and F. D. Veldkamp, Exceptional Groups, inOctonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Mathematics (Springer Berlin Heidelberg, Berlin, Heidelberg, 2000) pp. 173–183

    2000

  16. [16]

    Toupin, The division algebra tower as holographic chain: Shadow symmetry, cayley-dickson doubling, and the origin of spacetime dimensions (2026)

    D. Toupin, The division algebra tower as holographic chain: Shadow symmetry, cayley-dickson doubling, and the origin of spacetime dimensions (2026)

    2026

  17. [17]

    Electroweak Structure and Three Fermion Generations in Clifford Algebra with S3 Family Symmetry

    N. Gresnigt, Electroweak Structure and Three Fermion Generations in Clifford Algebra with S3 Family Symme- try (2026), arXiv:2601.07857 [physics.gen-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  18. [18]

    Higgs and Yukawa Structure in a Clifford Algebra Model with Three Generations and $S_3$ Family Symmetry

    N. Gresnigt, Higgs Sector and Flavour Structure in an Algebraic Three-Generation Model with S3 Family Sym- metry (2026), arXiv:2604.24795 [physics.gen-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  19. [19]

    Bhatt, R

    V. Bhatt, R. Mondal, V. Vaibhav, and T. P. Singh, Ma- jorana neutrinos, exceptional Jordan algebra, and mass ratios for charged fermions, J. Phys. G49, 045007 (2022), arXiv:2108.05787 [hep-ph]

    work page arXiv 2022

  20. [20]

    T. P. Singh, Fermion mass ratios from the exceptional Jordan algebra (2025), arXiv:2508.10131 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  21. [21]

    Octonions, exceptional Jordan algebra and the role of the group F_4 in particle physics

    I. Todorov and S. Drenska, Octonions, exceptional Jor- dan algebra and the role of the groupF 4 in particle physics, Adv. Appl. Clifford Algebras28, 82 (2018), arXiv:1805.06739 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  22. [22]

    Fulton and J

    W. Fulton and J. Harris,Representation Theory: A First Course, Graduate Texts in Mathematics, Vol. 129 (Springer New York, New York, NY, 2004)

    2004

  23. [23]

    The Exceptional Jordan Eigenvalue Problem

    T. Dray and C. A. Manogue, The exceptional Jordan eigenvalue problem, Int. J. Theor. Phys.38, 2901 (1999), arXiv:math-ph/9910004

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  24. [24]

    Z.-z. Xing, H. Zhang, and S. Zhou, Updated Values of Running Quark and Lepton Masses, Phys. Rev. D77, 113016 (2008), arXiv:0712.1419 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  25. [25]

    nu-fit.org/?q=node/309(2025), global fit results for neutrino oscillation parameters

    NuFIT Collaboration, NuFIT 6.1 (2025),http://www. nu-fit.org/?q=node/309(2025), global fit results for neutrino oscillation parameters. 19

    2025

  26. [26]

    Planck 2018 results. VI. Cosmological parameters

    N. Aghanimet al.(Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  27. [27]

    Farnsworth, F

    S. Farnsworth, F. Finster, C. F. Paganini, and T. P. Singh, Causal Fermion Systems, Non-Commutative Geometry and Generalized Trace Dynamics (2026), arXiv:2603.05018 [math-ph]

    work page arXiv 2026

  28. [28]

    Trace dynamics and a noncommutative special relativity

    K. Lochan and T. P. Singh, Trace dynamics and a non- commutative special relativity, Phys. Lett. A375, 3747 (2011), arXiv:1109.0300 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  29. [29]

    Jacobson,Structure and Representations of Jordan Al- gebras, Colloquium Publications (American Mathemati- cal Society, 1968)

    N. Jacobson,Structure and Representations of Jordan Al- gebras, Colloquium Publications (American Mathemati- cal Society, 1968)

    1968

  30. [30]

    A. H. Chamseddine, A. Connes, and W. D. van Suijlekom, Noncommutativity and physics: a non- technical review, Eur. Phys. J. ST232, 3581 (2023), arXiv:2207.10901 [hep-th]

    work page arXiv 2023

  31. [31]

    Bengtsson and M

    I. Bengtsson and M. Cederwall, Particles, Twistors and the Division Algebras, Nucl. Phys. B302, 81 (1988)

    1988