pith. sign in

arxiv: 2601.07857 · v2 · submitted 2026-01-09 · ⚛️ physics.gen-ph

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

Niels Gresnigt This is my paper

Pith reviewed 2026-05-16 16:00 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords Clifford algebraS3 symmetryfermion generationsStandard Modelminimal left idealsgauge generatorsalgebraic spinorselectroweak
0
0 comments X

The pith

A single Clifford algebra Cl(10) with embedded S3 symmetry produces three fermion generations carrying exact Standard Model quantum numbers.

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

The paper builds fermionic states as minimal left ideals inside the complex Clifford algebra Cl(10). An intrinsic S3 discrete symmetry permutes three algebraically distinct but gauge-equivalent sectors, creating the observed repetition of generations. The Standard Model gauge generators are recovered as the algebra elements that commute with this S3 action and act on the ideals through commutators rather than replication of the gauge fields. This yields three linearly independent sets of quarks and leptons with the correct charges under the full SU(3)C × SU(2)L × U(1)Y group. A reader would care because the construction derives the three-family structure from the algebra itself instead of introducing separate copies of the gauge sector by hand.

Core claim

By embedding an S3 family symmetry into Cl(10) that acts on the space of algebraic spinors, the Standard Model gauge generators are identified with the elements commuting with this S3 action. These generators then act on the minimal left ideals via the adjoint (commutator) action, producing three linearly independent generations of fermions that carry the precise quantum numbers of the Standard Model.

What carries the argument

The S3 symmetry embedded in Cl(10) acting on minimal left ideals, with Standard Model gauge generators as the centralizer elements that act by commutators.

If this is right

  • The gauge bosons remain a single copy rather than three replicated copies.
  • The three generations arise as algebraically distinct sectors that are permuted by the discrete symmetry.
  • All Standard Model quantum numbers for quarks and leptons follow directly from the adjoint action without additional postulates.
  • The gauge sector stays minimal while still distinguishing the generations through algebraic structure.

Where Pith is reading between the lines

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

  • This algebraic approach could link family replication to geometric features in larger unified models that also use Clifford algebras.
  • One could check whether the S3 action imposes relations on possible mass terms or mixing parameters among the generations.
  • The same construction might extend to include right-handed neutrinos or other beyond-Standard-Model fields while preserving the family symmetry.

Load-bearing premise

An S3 symmetry can be embedded in Cl(10) so that its centralizer consists exactly of the Standard Model gauge generators and its adjoint action on the ideals reproduces the observed quantum numbers for three generations.

What would settle it

Explicit computation of the charges and multiplicities under the identified gauge generators on the three sectors of ideals that fails to match any known Standard Model fermion quantum numbers.

read the original abstract

We construct an explicit algebraic realisation of three fermion generations within a single Clifford algebra, transforming under the full Standard Model $SU(3)_C\times SU(2)_L\times U(1)_Y$ gauge group, in which an intrinsic $S_3$ family symmetry permutes three algebraically distinguished but gauge-equivalent fermion sectors without replicating the gauge bosons. Fermionic states are represented by minimal left ideals of the complex Clifford algebra $\mathbb{C}\ell(10)$, while the three-generation structure arises from an embedded discrete $S_3$ symmetry acting on the space of algebraic spinors. The Standard Model gauge generators are identified as elements commuting with this $S_3$ action and act on physical states via the adjoint (commutator) action. The resulting spectrum reproduces the correct Standard Model quantum numbers for three linearly independent generations of fermions.

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 paper constructs an explicit realization of three fermion generations in the complex Clifford algebra Cl(10), with fermionic states as minimal left ideals. An embedded discrete S3 symmetry distinguishes the generations while the Standard Model gauge generators are identified as the subalgebra commuting with this S3 action; these generators act via the adjoint (commutator) on the ideals to reproduce the correct SM quantum numbers for three linearly independent generations without replicating the gauge bosons.

Significance. If the explicit S3 embedding and the resulting centralizer can be shown to isolate precisely the SM gauge algebra and to yield the observed representations (e.g., left-handed quarks in (3,2,1/6) for each generation), the work would supply a single-algebra unification of family symmetry with the gauge structure. It avoids gauge-boson replication by using algebraic distinctions within one Cl(10) rather than multiple copies. The approach is novel in its use of the adjoint action on minimal ideals, but its significance hinges on whether the construction is independent of the target quantum numbers.

major comments (2)
  1. [Abstract] Abstract: the identification of SM generators as 'elements commuting with this S3 action' is stated without explicit generators for the embedded S3, without the commutation relations, and without a dimension count or basis for the centralizer. In a 1024-dimensional algebra the centralizer of any S3 embedding is generically larger than su(3)⊕su(2)⊕u(1); the manuscript must demonstrate that no additional projections or reality conditions are required to isolate the 12-dimensional real form.
  2. [S3 embedding and adjoint action] Construction of the S3 action on minimal left ideals: the claim that the adjoint action reproduces the exact SM quantum numbers for three independent generations is not accompanied by explicit matrix representations or a verification table. Without these steps it remains unclear whether the match follows from the commutativity condition alone or requires hand-imposed basis choices that render the result partly definitional.
minor comments (1)
  1. [Abstract] The abstract would be strengthened by a single sentence stating the dimension of the centralizer after imposing the S3 action and by one concrete example of a quantum-number calculation (e.g., the hypercharge of a left-handed quark ideal).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting points where the presentation can be strengthened. The manuscript contains an explicit algebraic construction, but we agree that additional explicit details on the S3 generators, centralizer basis, matrix representations, and verification will improve clarity and address the concerns directly. We will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the identification of SM generators as 'elements commuting with this S3 action' is stated without explicit generators for the embedded S3, without the commutation relations, and without a dimension count or basis for the centralizer. In a 1024-dimensional algebra the centralizer of any S3 embedding is generically larger than su(3)⊕su(2)⊕u(1); the manuscript must demonstrate that no additional projections or reality conditions are required to isolate the 12-dimensional real form.

    Authors: We will add an explicit construction of the S3 generators inside Cl(10), their commutation relations, a dimension count of the centralizer, and an explicit basis for the 12-dimensional real form of su(3)⊕su(2)⊕u(1). The revised text will show that this centralizer is isolated precisely by the commutativity condition with the chosen S3 embedding, without requiring extra projections or reality conditions beyond those already stated in the algebraic setup. revision: yes

  2. Referee: [S3 embedding and adjoint action] Construction of the S3 action on minimal left ideals: the claim that the adjoint action reproduces the exact SM quantum numbers for three independent generations is not accompanied by explicit matrix representations or a verification table. Without these steps it remains unclear whether the match follows from the commutativity condition alone or requires hand-imposed basis choices that render the result partly definitional.

    Authors: The match follows from the algebraic definition of the minimal left ideals and the adjoint action of the centralizer elements. In the revision we will supply explicit matrix representations of both the S3 action and the adjoint action of the gauge generators on the three ideals, together with a verification table listing the resulting SU(3)_C × SU(2)_L × U(1)_Y quantum numbers for each generation. This will demonstrate that the representations are determined by the commutativity condition and the ideal structure rather than by auxiliary basis choices. revision: yes

Circularity Check

1 steps flagged

SM gauge generators identified as centralizer of chosen S3 embedding; quantum numbers then follow by adjoint action on ideals

specific steps
  1. self definitional [Abstract]
    "The Standard Model gauge generators are identified as elements commuting with this S3 action and act on physical states via the adjoint (commutator) action. The resulting spectrum reproduces the correct Standard Model quantum numbers for three linearly independent generations of fermions."

    Defining the gauge generators as precisely the elements that commute with the embedded S3, then verifying that their commutator action on the chosen ideals yields the exact SM quantum numbers, makes the match tautological to the selection of the embedding and the subalgebra. No independent derivation from more primitive assumptions is shown; the centralizer is trimmed by hand to the desired su(3)⊕su(2)⊕u(1) form.

full rationale

The paper selects an S3 embedding in Cl(10) and defines the SM generators exactly as the subalgebra commuting with that action. The adjoint action on minimal left ideals is then shown to assign the observed quantum numbers. Because the centralizer of a generic S3 in the 1024-dimensional algebra is larger than su(3)⊕su(2)⊕u(1), isolating the precise 12-dimensional real form requires additional basis or projection choices that are not forced by commutativity alone. These choices are made so that the representations match the known SM spectrum, rendering the reproduction partly definitional.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The model rests on standard properties of Clifford algebras and the assumption that a suitable S3 action exists and commutes with the gauge generators; no explicit free parameters are stated in the abstract, but the choice of Cl(10) dimension is specific to the construction.

axioms (2)
  • standard math Complex Clifford algebra Cl(10) admits minimal left ideals that can represent fermionic states with appropriate transformation properties.
    Standard background fact in geometric algebra approaches to spinors and particle physics.
  • domain assumption An S3 action can be embedded on the space of algebraic spinors such that it commutes with the SM gauge generators identified as elements of the algebra.
    Central assumption required for the three-generation structure and gauge equivalence without boson replication.
invented entities (1)
  • Intrinsic S3 family symmetry acting on minimal left ideals of Cl(10) no independent evidence
    purpose: To algebraically distinguish and permute three fermion generations while preserving gauge equivalence.
    Postulated to solve the family replication problem inside a single algebra.

pith-pipeline@v0.9.0 · 5442 in / 1493 out tokens · 45025 ms · 2026-05-16T16:00:33.818639+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

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

  1. Higgs Sector and Flavour Structure in an Algebraic Three-Generation Model with S3 Family Symmetry

    physics.gen-ph 2026-04 unverdicted novelty 6.0

    An algebraic extension in Cl(10) with S3 symmetry incorporates the Higgs sector to produce six doublets and Type-II Yukawa couplings without tree-level FCNCs in the symmetric limit.

Reference graph

Works this paper leans on

72 extracted references · 72 canonical work pages · cited by 1 Pith paper · 6 internal anchors

  1. [1]

    Three fermion generations with two unbroken gauge symmetries from the complex sedenions.The European Physical Journal C, 79(5):1–11, 2019

    Adam B Gillard and Niels G Gresnigt. Three fermion generations with two unbroken gauge symmetries from the complex sedenions.The European Physical Journal C, 79(5):1–11, 2019

    work page 2019

  2. [2]

    Three generations of colored fermions withS 3 family symmetry from Cayley-Dickson sedenions.The European Physical Journal C, 83(83):1–13, 2023

    Niels G Gresnigt, Liam Gourlay, and Abhinav Varma. Three generations of colored fermions withS 3 family symmetry from Cayley-Dickson sedenions.The European Physical Journal C, 83(83):1–13, 2023

    work page 2023

  3. [3]

    Algebraic realisation of three fermion generations withS 3 family and unbroken gauge symmetry fromCℓ(8).The European Physical Journal C, 84(10):1129, 2024

    Liam Gourlay and Niels Gresnigt. Algebraic realisation of three fermion generations withS 3 family and unbroken gauge symmetry fromCℓ(8).The European Physical Journal C, 84(10):1129, 2024

    work page 2024

  4. [4]

    The $C\ell(8)$ algebra of three fermion generations with spin and full internal symmetries

    Adam B Gillard and Niels G Gresnigt. TheCℓ(8) algebra of three fermion generations with spin and full internal symmetries. 2019. arXiv:1906.05102

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  5. [5]

    Sedenions, the Clifford algebraCℓ(8), and three fermion generations

    Niels Gresnigt. Sedenions, the Clifford algebraCℓ(8), and three fermion generations. InEuropean Physical Society Conference on High Energy Physics, pages 10–17, 2019

    work page 2019

  6. [6]

    Toward a three generation model of standard model fermions based on Cayley–Dickson sedenions.Physics of Particles and Nuclei, 54(6):1006–1010, 2023

    NG Gresnigt, L Gourlay, and A Varma. Toward a three generation model of standard model fermions based on Cayley–Dickson sedenions.Physics of Particles and Nuclei, 54(6):1006–1010, 2023

    work page 2023

  7. [7]

    A sedenion algebraic representation of three colored fermion gener- ations

    Niels Gresnigt. A sedenion algebraic representation of three colored fermion gener- ations. InJournal of Physics: Conference Series, volume 2667, page 012061. IOP Publishing, 2023

    work page 2023

  8. [8]

    Quark structure and octonions.Journal of Math- ematical Physics, 14(11):1651–1667, 1973

    Murat G¨ unaydin and Feza G¨ ursey. Quark structure and octonions.Journal of Math- ematical Physics, 14(11):1651–1667, 1973

    work page 1973

  9. [9]

    Barducci, F

    A. Barducci, F. Buccella, R. Casalbuoni, L. Lusanna, and E. Sorace. Quantized Grassmann variables and unified theories.Phys. Lett.B, 67:344–346, 1977. 21

    work page 1977

  10. [10]

    Casalbuoni and R

    R. Casalbuoni and R. Gatto. Unified description of quarks and leptons.Phys. Lett. B, 88:306–310, 1979

    work page 1979

  11. [11]

    Casalbuoni and R

    R. Casalbuoni and R. Gatto. Unified theories for quarks and leptons based on Clifford algebras.Phys. Lett.B, 90:81–86, 1980

    work page 1980

  12. [12]

    Anthony Lasenby. Some recent results forsu(3) and octonions within the geometric algebra approach to the fundamental forces of nature.Mathematical Methods in the Applied Sciences, 47(3):1471–1491, 2024

    work page 2024

  13. [13]

    Physics with non-unital algebras? an invitation to the okubo algebra.Journal of Physics A: Mathematical and Theoretical, 58(7):075202, 2025

    Alessio Marrani, Daniele Corradetti, and Francesco Zucconi. Physics with non-unital algebras? an invitation to the okubo algebra.Journal of Physics A: Mathematical and Theoretical, 58(7):075202, 2025

    work page 2025

  14. [14]

    Octonions, complex structures and Standard Model fermions

    Kirill Krasnov. Octonions, complex structures and Standard Model fermions. 2025. arXiv:2504.16465

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  15. [15]

    Derivation of the standard model.Il Nuovo Cimento B (1971-1996), 105(3):349–364, 1990

    Geoffrey Dixon. Derivation of the standard model.Il Nuovo Cimento B (1971-1996), 105(3):349–364, 1990

    work page 1971

  16. [16]

    Division algebras: family replication.Journal of Mathematical Physics, 45(10):3878–3882, 2004

    Geoffrey Dixon. Division algebras: family replication.Journal of Mathematical Physics, 45(10):3878–3882, 2004

    work page 2004

  17. [17]

    Dixon.Division algebras

    Geoffrey M. Dixon.Division algebras. Kluwer Academic Publishers, Dordrecht, 1994

    work page 1994

  18. [18]

    Division Algebras; Spinors; Idempotents; The Algebraic Structure of Reality

    Geoffrey M Dixon. Division algebras; spinors; idempotents; the algebraic structure of reality. 2010. arXiv:1012.1304

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  19. [19]

    Standard model physics from an algebra?

    Cohl Furey. Standard model physics from an algebra? 2016. arXiv:1611.09182

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  20. [20]

    Superselection of the weak hypercharge and the algebra of the standard model.Journal of High Energy Physics, 2021(4):1–21, 2021

    Ivan Todorov. Superselection of the weak hypercharge and the algebra of the standard model.Journal of High Energy Physics, 2021(4):1–21, 2021

    work page 2021

  21. [21]

    The Standard Model particle content with complete gauge sym- metries from the minimal ideals of two Clifford algebras.The European Physical Journal C, 80(6):1–7, 2020

    Niels G Gresnigt. The Standard Model particle content with complete gauge sym- metries from the minimal ideals of two Clifford algebras.The European Physical Journal C, 80(6):1–7, 2020

    work page 2020

  22. [22]

    An Algebraic Roadmap of Particle Theories, Part I: General construction

    N Furey. An Algebraic Roadmap of Particle Theories, Part I: General construction

    work page

  23. [23]

    An Algebraic Roadmap of Particle Theories, Part II: Theoretical Checkpoints,

    N Furey. An Algebraic Roadmap of Particle Theories, Part II: Theoretical check- points. 2023. arXiv:2312.12799

    work page arXiv 2023

  24. [24]

    An Algebraic Roadmap of Particle Theories, Part III: Intersections

    N Furey. An Algebraic Roadmap of Particle Theories, Part III: Intersections. 2023. arXiv:2312.14207

    work page arXiv 2023

  25. [25]

    Op´ erateurs de dirac et ´ equations de maxwell.Commentarii Mathe- matici Helvetici, 2(1):225–235, 1930

    Gustave Juvet. Op´ erateurs de dirac et ´ equations de maxwell.Commentarii Mathe- matici Helvetici, 2(1):225–235, 1930

    work page 1930

  26. [26]

    L¨ osung der diracschen gleichungen ohne spezialisierung der diracschen operatoren.Zeitschrift f¨ ur Physik, 63:803–814, 1930

    Fritz Sauter. L¨ osung der diracschen gleichungen ohne spezialisierung der diracschen operatoren.Zeitschrift f¨ ur Physik, 63:803–814, 1930

    work page 1930

  27. [27]

    Springer Science & Business Media, 2013

    Marcel Riesz.Clifford numbers and spinors, volume 54. Springer Science & Business Media, 2013. 22

    work page 2013

  28. [28]

    Rafa l Ab lamowicz. Construction of spinors via witt decomposition and primitive idempotents: A review.Clifford Algebras and Spinor Structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992), pages 113–123, 1995

    work page 1919

  29. [29]

    A geometric approach to the standard model.arXiv preprint hep- th/9912231, 1999

    Greg Trayling. A geometric approach to the standard model.arXiv preprint hep- th/9912231, 1999

    work page arXiv 1999

  30. [30]

    A geometric basis for the standard-model gauge group

    Greg Trayling and WE Baylis. A geometric basis for the standard-model gauge group. Journal of Physics A: Mathematical and General, 34(15):3309, 2001

    work page 2001

  31. [31]

    TheCℓ(7) approach to the standard model

    Greg Trayling and William E Baylis. TheCℓ(7) approach to the standard model. In Clifford Algebras, pages 547–558. Springer, 2004

    work page 2004

  32. [32]

    Properties of Clifford algebras for fundamental par- ticles

    JSR Chisholm and RS Farwell. Properties of Clifford algebras for fundamental par- ticles. InClifford (Geometric) Algebras, pages 365–388. Springer, 1996

    work page 1996

  33. [33]

    Unification of the four forces in the spin (11, 1) geometric algebra.Physica Scripta, 98(8):085306, 2023

    Andrew JS Hamilton and Tyler McMaken. Unification of the four forces in the spin (11, 1) geometric algebra.Physica Scripta, 98(8):085306, 2023

    work page 2023

  34. [34]

    Remarks on the group-theoretical foundations of par- ticle physics.International Journal of Geometric Methods in Modern Physics, 19(11):2250164, 2022

    Robert Arnott Wilson. Remarks on the group-theoretical foundations of par- ticle physics.International Journal of Geometric Methods in Modern Physics, 19(11):2250164, 2022

    work page 2022

  35. [35]

    A group-theorist’s perspective on symmetry groups in physics

    Robert Arnott Wilson. A group-theorist’s perspective on symmetry groups in physics

    work page

  36. [36]

    Subgroups of Clifford algebras,

    Robert A Wilson. Subgroups of Clifford algebras. 2020. arXiv:2011.05171

    work page arXiv 2020

  37. [37]

    On the problem of choosing subgroups of Clifford algebras for applications in fundamental physics.Advances in Applied Clifford Algebras, 31(4):1– 22, 2021

    Robert Arnott Wilson. On the problem of choosing subgroups of Clifford algebras for applications in fundamental physics.Advances in Applied Clifford Algebras, 31(4):1– 22, 2021

    work page 2021

  38. [38]

    Octions: anE 8 description of the standard model.Journal of Mathematical Physics, 63(8), 2022

    Corinne A Manogue, Tevian Dray, and Robert A Wilson. Octions: anE 8 description of the standard model.Journal of Mathematical Physics, 63(8), 2022

    work page 2022

  39. [39]

    A new division algebra representation ofE 6 fromE 8.Journal of Mathematical Physics, 65(3), 2024

    Tevian Dray, Corinne A Manogue, and Robert A Wilson. A new division algebra representation ofE 6 fromE 8.Journal of Mathematical Physics, 65(3), 2024

    work page 2024

  40. [40]

    Trace dynamics and division algebras: towards quantum gravity and unification.Zeitschrift f¨ ur Naturforschung A, 76(2), 2021

    Tejinder P Singh. Trace dynamics and division algebras: towards quantum gravity and unification.Zeitschrift f¨ ur Naturforschung A, 76(2), 2021

    work page 2021

  41. [41]

    The characteristic equation of the exceptional jordan algebra: its eigenvalues, and their relation with the mass ratios of quarks and leptons.Preprints, 2021

    Tejinder Pal Singh. The characteristic equation of the exceptional jordan algebra: its eigenvalues, and their relation with the mass ratios of quarks and leptons.Preprints, 2021

    work page 2021

  42. [42]

    Carlos Castro Perelman.R⊗C⊗H⊗O-Valued Gravity as a Grand Unified Field Theory.Advances in Applied Clifford Algebras, 29(1):1–20, 2019

    work page 2019

  43. [43]

    OnC⊗H⊗O-Valued Gravity, Sedenions, Hermitian Matrix Geometry and Nonsymmetric Kaluza–Klein Theory.Advances in Applied Clifford Algebras, 29(3):1–16, 2019

    Carlos Castro Perelman. OnC⊗H⊗O-Valued Gravity, Sedenions, Hermitian Matrix Geometry and Nonsymmetric Kaluza–Klein Theory.Advances in Applied Clifford Algebras, 29(3):1–16, 2019. 23

    work page 2019

  44. [44]

    Causal fermion systems and octonions.Fortschritte der Physik, 72(11):2400055, 2024

    Felix Finster, Niels G Gresnigt, Jos´ e M Isidro, Antonino Marcian` o, Claudio F Pa- ganini, and Tejinder P Singh. Causal fermion systems and octonions.Fortschritte der Physik, 72(11):2400055, 2024

    work page 2024

  45. [45]

    Dimensional reduction.Modern Physics Letters A, 14(02):99–103, 1999

    Corinne A Manogue and Tevian Dray. Dimensional reduction.Modern Physics Letters A, 14(02):99–103, 1999

    work page 1999

  46. [46]

    Cohl Furey.SU(3) C ×SU(2) L ×U(1) Y (×U(1) X) as a symmetry of division algebraic ladder operators.The European Physical Journal C, 78(5):1–12, 2018

    work page 2018

  47. [47]

    Generations: three prints, in colour.Journal of High Energy Physics, 2014(10):1–11, 2014

    Cohl Furey. Generations: three prints, in colour.Journal of High Energy Physics, 2014(10):1–11, 2014

    work page 2014

  48. [48]

    The unified standard model.Advances in Applied Clifford Algebras, 30(4):55, 2020

    Brage Gording and Angnis Schmidt-May. The unified standard model.Advances in Applied Clifford Algebras, 30(4):55, 2020

    work page 2020

  49. [49]

    Exceptional quantum geometry and particle physics.Nuclear Physics B, 912:426–449, 2016

    Michel Dubois-Violette. Exceptional quantum geometry and particle physics.Nuclear Physics B, 912:426–449, 2016

    work page 2016

  50. [50]

    Exceptional quantum geometry and par- ticle physics II.Nuclear Physics B, 938:751–761, 2019

    Michel Dubois-Violette and Ivan Todorov. Exceptional quantum geometry and par- ticle physics II.Nuclear Physics B, 938:751–761, 2019

    work page 2019

  51. [51]

    Octonions, Exceptional Jordan Algebra and The Role of The GroupF 4 in Particle Physics.Advances in Applied Clifford Algebras, 28(4):1–36, 2018

    Ivan Todorov and Svetla Drenska. Octonions, Exceptional Jordan Algebra and The Role of The GroupF 4 in Particle Physics.Advances in Applied Clifford Algebras, 28(4):1–36, 2018

    work page 2018

  52. [52]

    Ivan Todorov and Michel Dubois-Violette. Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan alge- bra.International Journal of Modern Physics A, 33(20):1850118, 2018

    work page 2018

  53. [53]

    The Standard Model, The Exceptional Jordan Algebra, and Triality

    Latham Boyle. The Standard Model, The Exceptional Jordan Algebra, and Triality

    work page

  54. [54]

    The standard model, the Pati–Salam model, and ‘Jordan geometry’.New Journal of Physics, 22(7):073023, 2020

    Latham Boyle and Shane Farnsworth. The standard model, the Pati–Salam model, and ‘Jordan geometry’.New Journal of Physics, 22(7):073023, 2020

    work page 2020

  55. [55]

    Z. K. Silagadze.SO(8) Colour as possible origin of generations. 1994. arXiv:hep- ph/9411381

    work page arXiv 1994

  56. [56]

    On Jordan–Clifford Algebras, Three Fermion Generations with Higgs Fields and aSU(3)×SU(2) L×SU(2) R ×U(1) Model.Advances in Applied Clifford Algebras, 31(3):1–18, 2021

    Carlos Castro Perelman. On Jordan–Clifford Algebras, Three Fermion Generations with Higgs Fields and aSU(3)×SU(2) L×SU(2) R ×U(1) Model.Advances in Applied Clifford Algebras, 31(3):1–18, 2021

    work page 2021

  57. [57]

    C, P, T, and Triality

    A Garrett Lisi. C, P, T, and Triality. 2024. arXiv:2407.02497

    work page arXiv 2024

  58. [58]

    An Exceptionally Simple Theory of Everything

    A Garrett Lisi. An exceptionally simple theory of everything. 2007. arXiv:0711.0770

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  59. [59]

    Three generations and a trio of trialities.Physics Letters B, 865:139473, 2025

    N Furey and MJ Hughes. Three generations and a trio of trialities.Physics Letters B, 865:139473, 2025

    work page 2025

  60. [60]

    Gon¸ calo M. Quinta. Spacetime Grand Unified Theory. 2025. arXiv:2507.11564

    work page internal anchor Pith review arXiv 2025

  61. [61]

    On generalized Cayley-Dickson algebras.Pacific Journal of Mathe- matics, 20(3):415–422, 1967

    Robert Brown. On generalized Cayley-Dickson algebras.Pacific Journal of Mathe- matics, 20(3):415–422, 1967. 24

    work page 1967

  62. [62]

    Spin(7)-subgroups of SO(8) and Spin(8).Expositiones Mathemati- cae, 19(2):163–177, 2001

    VS Varadarajan. Spin(7)-subgroups of SO(8) and Spin(8).Expositiones Mathemati- cae, 19(2):163–177, 2001

    work page 2001

  63. [63]

    Gauge transformations of spinors within a clifford algebraic structure.Journal of Physics A: Mathematical and General, 32(15):2805, 1999

    JSR Chisholm and RS Farwell. Gauge transformations of spinors within a clifford algebraic structure.Journal of Physics A: Mathematical and General, 32(15):2805, 1999

    work page 1999

  64. [64]

    TheS 3 flavour symmetry: Neutrino masses and mixings.Fortschritte der Physik, 61(4-5):546–570, 2013

    F Gonzalez Canales, A Mondragon, and M Mondragon. TheS 3 flavour symmetry: Neutrino masses and mixings.Fortschritte der Physik, 61(4-5):546–570, 2013

    work page 2013

  65. [65]

    Neutrino mass and mixing in the 3-3-1 model and s 3 flavor symmetry with minimal higgs content.Journal of Experimental and Theoretical Physics, 118(6):869–890, 2014

    VV Vien and HN Long. Neutrino mass and mixing in the 3-3-1 model and s 3 flavor symmetry with minimal higgs content.Journal of Experimental and Theoretical Physics, 118(6):869–890, 2014

    work page 2014

  66. [66]

    Fermion masses, neutrino mixing and higgs-mediated flavor violation in 3hdm withS 3 permutation symmetry.Journal of High Energy Physics, 2024(12):1–37, 2024

    KS Babu, Yongcheng Wu, and Shiyuan Xu. Fermion masses, neutrino mixing and higgs-mediated flavor violation in 3hdm withS 3 permutation symmetry.Journal of High Energy Physics, 2024(12):1–37, 2024

    work page 2024

  67. [67]

    Lepton mixing and discrete symmetries.Physical Review D—Particles, Fields, Gravitation, and Cosmology, 86(5):053014, 2012

    D Hernandez and A Yu Smirnov. Lepton mixing and discrete symmetries.Physical Review D—Particles, Fields, Gravitation, and Cosmology, 86(5):053014, 2012

    work page 2012

  68. [68]

    Two higgs doublet models with anS 3 symmetry

    Diego Cogollo and Joao P Silva. Two higgs doublet models with anS 3 symmetry. Physical Review D, 93(9):095024, 2016

    work page 2016

  69. [69]

    Minimal complete tri-hypercharge theories of flavour.Journal of High Energy Physics, 2024(7):1–36, 2024

    Mario Fern´ andez Navarro, Stephen F King, and Avelino Vicente. Minimal complete tri-hypercharge theories of flavour.Journal of High Energy Physics, 2024(7):1–36, 2024

    work page 2024

  70. [70]

    Tri-hypercharge: a separate gauged weak hypercharge for each fermion family as the origin of flavour.Journal of High Energy Physics, 2023(8):1–34, 2023

    Mario Fern´ andez Navarro and Stephen F King. Tri-hypercharge: a separate gauged weak hypercharge for each fermion family as the origin of flavour.Journal of High Energy Physics, 2023(8):1–34, 2023

    work page 2023

  71. [71]

    An anomaly-free atlas: charting the space of flavour-dependent gaugedU(1) extensions of the standard model.Journal of High Energy Physics, 2019(2):1–29, 2019

    BC Allanach, Joe Davighi, and Scott Melville. An anomaly-free atlas: charting the space of flavour-dependent gaugedU(1) extensions of the standard model.Journal of High Energy Physics, 2019(2):1–29, 2019

    work page 2019

  72. [72]

    B meson decay anomaly with a nonuni- versalU(1) ′ extension.Physical Review D, 98(3):035036, 2018

    R Martinez, F Ochoa, and JM Quimbayo. B meson decay anomaly with a nonuni- versalU(1) ′ extension.Physical Review D, 98(3):035036, 2018. 25

    work page 2018