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REVIEW 5 major objections 5 minor 38 references

Embeddings of the Standard Model in $E_8$

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The Standard Model of particle physics is entirely contained in the 45-dimensional split real algebra so(7,3), a subalgebra of E8(-24), and this embedding is essentially unique.

desk verdict The so(7,3) embedding with compact SU(3) is concrete and checkable; the mass/mixing numerology in Section 4 is post hoc and should not carry the paper's weight. read the letter →

arxiv 2507.16517 v2 pith:5LHI4AIW submitted 2025-07-22 hep-ph math.GR

Robert A. Wilson This is my paper · ORCID
classification hep-phmath.GR
keywords E8(-24)so(73)StandardModelembeddingcompactSU(3)mixinganglesfrommassesthreegenerationsquantumvacuumgravity
topics Quantum Gravity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the entire Standard Model — gauge group, three generations, and even a first-order quantum gravity — sits inside the 45-dimensional split real algebra so(7,3), a subalgebra of the exceptional algebra E8(-24), and that this embedding is essentially the only physically plausible one. The key repair over earlier E8 proposals is that the colour group becomes compact SU(3) again, which answers the standard objection that a noncompact SL(3,R) colour group would turn gluons into fermions. A reinterpretation of the Dirac algebra then yields numerical predictions, including a lepton-mixing angle of about 33.024°, a weak mixing angle with sin²(ϕ/2) ≈ 0.22265, and a tau mass of 1776.841464(4) MeV/c², all derived from mass ratios rather than fitted. If the construction holds, the Standard Model's parameters are not arbitrary: mixing angles depend on masses, and masses emerge from interactions with a dynamic quantum vacuum.

What carries the argument

The load-bearing object is the split real Lie algebra so(7,3), the D5 subalgebra of the exceptional real form E8(-24); the paper fixes a complex structure so that Spin(7,3) splits as Spin(1,3)⊗Spin(6), making the Lorentz group noncompact and colour SU(3) compact. On top of this, the argument replaces Dirac spinors with pairs of complex 4-vectors and interprets the su(3,1) part of the algebra as the quantum vacuum: its 9 gauge bosons and 6 gauge fermions are the photon, gluons, and neutrinos/antineutrinos. The numerical engine is the 'generation geometry', in which the three fermion generations are projected onto the first two components of weak isospin as an equilateral triangle in U(1), yielding the mass-angle relation that produces the quoted mixing angles.

What would settle it

Measure the tau mass to better than 1 keV/c² or the solar neutrino mixing angle to better than 0.1°: if the tau mass is not within the quoted uncertainty of 1776.841464(4) MeV/c², or the neutrino mixing angle is not 33.024°, the equilateral-triangle generation geometry is ruled out.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Standard Model is contained in so(7,3), not in the whole E8(-24), and that the correct real-form splitting is Spin(7,3) → Spin(1,3)⊗Spin(6), not the earlier Spin(4)⊗Spin(3,3) that produced SL(3,R) colour. This restores compact SU(3), keeps the Lorentz group as Spin(1,3), and leaves the gauge group SU(3,1) embedded in a way that has no unwanted lepton-quark mixing. The same structure lets the three fermion generations be seen as vertices of an equilateral triangle in the charge/hypercharge plane, and the paper derives several Standard Model mixing angles and mass ratios from that geometry. The paper concludes that the rest of E8 is unnecessary: so(7,3) already contains everything that exists in the Standard Model, except spinors, which the model replaces with pairs of complex 4-vectors.

Load-bearing premise

The numerical predictions stand on the unproven geometric identification of the three generations as an equilateral triangle in the U(1) direction, with a 'well-defined mass direction' whose angle to a side is a PMNS mixing angle; if that identification fails, the angle calculations collapse even if the embedding itself is accepted.

Editorial extensions

If this is right

  • The compactness objection is removed: colour symmetry is compact SU(3) inside a Spin(1,3)⊗Spin(6) splitting, so all gauge bosons are anti-Hermitian.
  • The full E8(-24) is not needed; the Standard Model plus a first-order quantum gravity fits inside so(7,3), and the remaining E8 structure is surplus.
  • Mixing angles are determined by mass ratios: the sample calculations give θ ≈ 33.024° for the lepton-mixing angle, sin²(ϕ/2) ≈ 0.22265 for the weak angle, and m(τ) = 1776.841464(4) MeV/c².
  • The model's vacuum contains 9 gauge bosons and 6 gauge fermions (neutrinos/antineutrinos), which together act as a dynamic quantum background whose tidal and magnetic structures could explain dark-matter-like effects and the neutron-lifetime anomaly.
  • First-order gravitational predictions agree with general relativity, but second-order self-interaction has the opposite sign, offering a route to distinguish the two experimentally.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the mass-angle connection is real, the Standard Model's nine mixing angles would no longer be free inputs: they would be computable functions of the running masses, and a full Dirac-replacement equation would close the system.
  • The equilateral-triangle generation geometry is logically independent of the so(7,3) embedding; a future calculation could retain the embedding while abandoning that specific angle, so the two claims should be tested separately.
  • The background-vacuum picture implies measurable time- and place-dependence in particle masses (for instance in W/Z mass determinations at different latitudes), which is a sharp, falsifiable consequence beyond the paper's sample calculations.
  • The proposed second-order sign flip in gravitational self-interaction would distinguish this model from General Relativity in strong-field or interferometric tests, although the paper only states the sign qualitatively.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper proposes a modified embedding of the Standard Model into the real Lie algebra E8(-24), arguing that the Standard Model is entirely contained in the subalgebra so(7,3). The author replaces the noncompact colour group used in earlier 'octions' work with a compact SU(3), reinterprets Dirac spinors as pairs of complex 4-vectors, and claims that this structure yields a unique physically plausible embedding. The paper also derives a series of numerical relations, including a lepton mixing angle from an equilateral-triangle geometry, a weak mixing angle from a hyperbola construction, a prediction of the tau mass from a vector identity, and several angles supposedly appearing in the CKM matrix. Finally, it interprets parts of the algebra as quantum gravity and argues that masses and mixing angles depend on a dynamic background spacetime.

Significance. If the embedding claim were established, it would be a notable structural observation in the search for unified models: the restoration of compact SU(3) colour inside so(7,3), with an explicit generator-level decomposition, is concrete and checkable. The paper also makes falsifiable numerical predictions, which is a strength in principle. However, the quantitative program is not currently supported: the mass-angle relations are introduced as geometric ansatze rather than derived from the algebra, and the tau mass 'prediction' is encoded in the coordinate choice. The manuscript is honest about the absence of a replacement Dirac equation, but that absence is load-bearing for the claimed predictions. The structural Lie-algebraic part may survive a major revision; the numerical and cosmological claims as they stand are speculative.

major comments (5)
  1. [Section 4, Eqs. (19)-(20)] The derivation of theta ≈ 33.024 degrees is not a consequence of the so(7,3) embedding. The three lepton generations are asserted to form an equilateral triangle in the first two components of weak isospin, and a 'well-defined mass direction' is assumed, but no algebraic definition of this projection or direction is given. Equation (19) is therefore a geometric ansatz chosen after the fact, and the abstract's claim that 'the mixing angles depend on masses' is unsupported. The same criticism applies to Eqs. (21)-(22), where tan phi = 3/2 is simply posited and then converted by trigonometry into sin^2(phi/2) = 0.22265.
  2. [Section 4, Eqs. (23)-(26)] The tau mass prediction is encoded in the coordinate choice. The vectors e, mu, tau, p are selected so that e + mu + tau + 3p = (0,5,5,5), which is then identified with five neutrons, yielding a linear relation among the four masses. No independent derivation of these coordinate vectors from the representation theory of so(7,3) is supplied. The relation is therefore tautological rather than predictive, and the claimed precision for m(tau) in Eq. (26) is not justified by the model.
  3. [Section 3, Eqs. (15)-(18), and Section 8] The paper explicitly states that no replacement for the Dirac equation is provided ('I make no attempt to guess what it is' in Section 4; 'we need an explicit replacement for the Dirac equation' in Section 8). Since the mass spectrum and the mass-angle relations are supposed to be defined by the Dirac equation on SO(7,3)/SU(3,1), the quantitative predictions cannot be checked. The structural embedding of the gauge group may stand, but the quantitative program is incomplete.
  4. [Sections 5-6] The quantum gravity claims are not derived. The identification of SU(3,1)/SO(3,1) with the Einstein tensor is stated, but no field equations, action, or quantitative comparison with general relativity is given. The 'tidal gluons' and 'antisymmetric gluons' are introduced as interpretations without any dynamical content, and the applications to the neutron lifetime anomaly, CP violation, and variations in G are speculative. This does not support the conclusion that the model contains 'a first-order version of gravity'.
  5. [Section 2, Eq. (1)] The claim of uniqueness of the embedding is not established. The elimination of the five D5 + D3 splittings relies on empirical plausibility (e.g., absence of proton decay, absence of lepton-quark mixing) and on the requirement that the Lorentz group appear as a real form; it is not a mathematical exhaustion of all embedding possibilities. A uniqueness claim of this kind would require a precise definition of 'physically plausible' and a systematic classification.
minor comments (5)
  1. [Title page] The title contains spacing errors: 'ST ANDARD' and 'INE8' should be 'STANDARD' and 'IN E8'.
  2. [Abstract] The term 'octions' appears to be a typo for 'octonions' or should be introduced as a named model; as written it is confusing.
  3. [Section 4] The phrase 'three generations of electron' should read 'electron, muon, and tau generations'.
  4. [Section 4, Eq. (28)] The statement that the angles psi and chi 'can be found in the CKM matrix' is not substantiated by standard CKM parametrizations; the values quoted are not standard CKM parameters, so this claim needs justification or removal.
  5. [Section 6] The list 'the weather, the train timetable, the holiday season' is informal and should be replaced by concrete physical variables if these correlations are meant to be testable.

Circularity Check

3 steps flagged · score 6.0 of 10

Section 4's mass-angle 'predictions' are encoded in asserted geometric/coordinate ansätze; the so(7,3) embedding itself is not circular.

  1. self definitional [Section 4, Eqs. (19)-(20)]
    "First we project the three lepton generations onto the first two components of weak isospin, where we can interpret the three generations of electron as the vertices of an equilateral triangle, embedded in the circle SO(2)=U(1). There is then a well-defined mass direction, and we can calculate the angle θ between this direction and one of the sides of the triangle from the equation cos(60◦ − θ)/cos θ = m(τ) − m(e)/m(τ) − m(µ), which yields an angle θ ≈ 33.024◦. Clearly this is an important angle in the lepton-mixing matrix..."

    The projection, the equilateral-triangle placement, and especially the 'well-defined mass direction' are asserted, not derived from the so(7,3) coordinates introduced in Section 3. Eq. (19) is the only definition of θ: it converts the input masses into an angle. The later sentence identifies this angle with the PMNS matrix, so the paper's headline relation 'mixing angles depend on masses' is true by construction of Eq. (19), not by E8 structure. The numerical value 33.024° is therefore an output of the chosen ansatz, not an independent prediction.

  2. fitted input called prediction [Section 4, Eqs. (21)-(22)]
    "In this case the embedding is into the group SO(1,1), which is geometrically a hyperbola, defined by the equation x2 − y2 = 1. The two branches of the hyperbola distinguish the leptons... We then calculate the angle between the charge direction and the hypercharge direction from the equation tan ϕ = 3/2, which yields sin2(ϕ/2) = 1/2 − 1/√13 ≈ .22265. It is again no surprise that this angle ϕ/2 is the same as the experimental value of the weak mixing angle."

    No derivation is given for tan ϕ = 3/2 from the so(7,3) embedding; the 'suitably scaled' square and the charge/hypercharge axes are not specified algebraically. Once tan ϕ is fixed, sin²(ϕ/2) is a forced trigonometric value, and the equality with the weak mixing angle is asserted. The 'prediction' is therefore the chosen value of ϕ, renamed as a physical angle, not a consequence of the Lie algebra.

1 more flagged steps
  1. fitted input called prediction [Section 4, Eqs. (23)-(26)]
    "Using charge and IL, JL, KL as coordinates, for transparency, we then have e = (−1, 1, 1, 0), µ = (−1, 0, 1, 1), τ = (−1, 1, 0, 1) ... p = (1, 1, 1, 1). We can then calculate the neutral combination e + µ + τ + 3p = (0, 5, 5, 5) which can only reasonably be interpreted as five neutrons. Remarkably, it is true that the total mass of these 6 particles is indeed equal to the mass of five neutrons, to within the bounds of experimental uncertainty [13]."

    The coordinate vectors e, µ, τ, p are presented 'for transparency' without derivation from the so(7,3) representation theory. The linear identity e+µ+τ+3p = (0,5,5,5) is an algebraic consequence of those assigned coordinates, and the interpretation of (0,5,5,5) as five neutrons is an additional stipulation. The mass relation is then used to solve for m(τ); the predicted tau mass is thus extracted from a relation built into the coordinate assignment plus measured masses, rather than derived from the embedding. The self-citation [13] does not fill that gap, since the equation is restated here as the basis for the prediction.

full rationale

The group-theoretic embedding in Sections 2–3 has independent mathematical content: the survey of E8(−24) real forms, the choice of Spin(7,3), and the restoration of compact SU(3) are arguments internal to this paper, not a self-citation chain. The circularity is confined to Section 4, where the numerical 'predictions' are generated by geometric and coordinate ansätze that are asserted rather than derived. Eq. (19) defines a mass-dependent angle and then identifies it with the PMNS angle; Eq. (21) fixes tan ϕ = 3/2 and then identifies the resulting value with the weak mixing angle; Eqs. (23)–(25) assign particle coordinates so that e+µ+τ+3p=5n, and then use the mass version of that relation to predict the tau mass. The paper itself concedes in Section 8 that it has not written down a precise replacement for the Dirac equation, confirming that these mass-angle relations are not consequences of a derived first-principles equation. Because the central so(7,3) embedding does not reduce to these numerological relations, the overall circularity is partial rather than total, hence a score of 6.

Assumptions & free parameters 2 free parameters · 4 assumptions · 3 invented entities

The quantitative output of the paper depends on inherited E8 conventions, several hand-chosen particle coordinates, and geometric placements that are not derived from the algebra. The central embedding claim is more self-contained than the numerical predictions, which is why the ledger separates the group-theoretic axioms from the ad hoc numerical choices.

free parameters (2)
  • Particle coordinate vectors for e, mu, tau, p = e=(-1,1,1,0); mu=(-1,0,1,1); tau=(-1,1,0,1); p=(1,1,1,1)
    Section 4 chooses these coordinates so that e+mu+tau+3p=5n in the coordinate lattice; the choice is not derived, so the resulting mass relation and tau mass prediction are effectively fitted to the desired outcome.
  • Generation triangle geometry in weak-isospin plane = equilateral triangle, mass direction at theta with cos(60-theta)/cos(theta) equal to a mass ratio
    The equilateral placement and the definition of theta in Eq. (19) are assumed to match the PMNS angle; no group-theoretic derivation is supplied.
assumptions (4)
  • domain assumption The Manogue-Dray-Wilson E8(-24) octonion model and its coordinate conventions are a valid starting point.
    The paper inherits the framework of [1] and the notation without re-deriving the octonion construction.
  • ad hoc to paper Spinors are not required to describe spin; pairs of complex four-vectors can replace them.
    Section 3 argues an electron is never isolated, so the spin-1/2 property can be modeled by a string twist. This is a physical assertion, not a theorem.
  • ad hoc to paper Three lepton generations project to the first two weak-isospin components as an equilateral triangle.
    Section 4 uses this projection to compute the 33.024 degree angle; no derivation from the algebra is given.
  • domain assumption The physically plausible embedding is unique because alternative splittings fail on compactness, proton decay, or lepton-quark mixing.
    Section 2 eliminates other real forms by physical plausibility arguments; these are reasonable but not mathematical proof.
invented entities (3)
  • Gauge fermions (three neutrinos and three antineutrinos in the adjoint of SU(3,1))
    purpose: Form the quantum vacuum and give neutrinos zero Dirac mass.
    Section 3 identifies six adjoint generators of SU(3,1) as neutrinos and antineutrinos; this is an interpretive claim with no independent experimental handle.
  • Tidal gluons (symmetric gluons as a massless fluid)
    purpose: Provide gravitational effects similar to dark matter without Dirac mass.
    Section 6 proposes symmetric gluons flow as tides and act like dark matter; no quantitative prediction is given.
  • Antisymmetric gluons as tiny massless magnets forming physical magnetic field lines
    purpose: Explain why magnetic field lines behave as physical objects.
    Section 6 speculation; no falsifiable observable is derived.

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Cite this review

Pith. "Pith review of Embeddings of the Standard Model in $E_8$." pith.science (2026). https://pith.science/paper/5LHI4AIW

@misc{pith2026250716517,
  author       = {Pith},
  title        = {Pith review of: Embeddings of the Standard Model in $E_8$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5LHI4AIW}},
  note         = {Machine review of arXiv:2507.16517}
}
abstract

I present a modified version of the Manogue-Dray-Wilson `octions' model of elementary particles, that overcomes some of the objections to that model that have been raised. In particular, I restore the compactness of the Standard Model gauge group, and show how the symmetry-breaking of the weak $SU(2)$ relates to the symmetry-breaking between the three generations of elementary fermions. In the process of attempting to implement a Dirac equation for three generations of fermions simultaneously, it turns out that some parts of the $E_8$ model are not required for the Standard Model, which is entirely contained in the subalgebra $\mathfrak{so}(7,3)$. In particular a re-interpretation of the Dirac spinors allows us to interpret part of the model as quantum gravity, which is then compared to General Relativity. The general structure of the model shows that the mixing angles depend on masses, and that the masses emerge from quantum interactions with the dynamic background spacetime (vacuum). Some sample calculations are given to support these predictions.

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Works this paper leans on

38 extracted references · 31 canonical work pages

  1. [1]

    C. A. Manogue, T. Dray and R. A. Wilson (2022), Octions: an E8 description of the standard model, J. Math. Phys. 63, 081703

    work page 2022

  2. [2]

    Distler and S

    J. Distler and S. Garibaldi (2010), There is no E8 theory of everything, Communications in Math. Phys. 298 (2), 419–436

    work page 2010

  3. [3]

    A. G. Lisi (2007), An exceptionally simple theory of everything, arXiv:0711.0770

    arXiv 2007

  4. [4]

    R. A. Wilson (2022), Chirality in an E8 model of elementary particles, arXiv:2210.06029

    work page Pith review arXiv 2022

  5. [5]

    Chester, A

    D. Chester, A. Marrani and M. Rios (2020), Beyond the standard model with six-dimensional spacetime, arXiv:2002.02391

    arXiv 2020

  6. [6]

    Penrose (1967), Twistor algebra, J

    R. Penrose (1967), Twistor algebra, J. Math. Phys. 8 (2), 345–366

    work page 1967

  7. [7]

    Adamo (2017), Lectures on twistor theory

    T. Adamo (2017), Lectures on twistor theory . arXiv:1712.02196

    arXiv 2017

  8. [8]

    Georgi and S

    H. Georgi and S. Glashow (1974), Unity of all elementary-particle forces, Physical Review Letters 32 (8), 438

    work page 1974

Show all 38 references
  1. [9]

    J. C. Pati and A. Salam (1974), Lepton number as the fourth ‘color’, Phys. Rev. D 10 (1), 275–289

    1974

  2. [10]

    Pontecorvo (1958), Inverse beta processes and non-conservation of lepton charge, Soviet Physics JETP 7, 172

    B. Pontecorvo (1958), Inverse beta processes and non-conservation of lepton charge, Soviet Physics JETP 7, 172

    1958

  3. [11]

    Z. Maki, M. Nakagawa and S. Sakata (1962), Remarks on the unified model of elementary particles, Progress of Theoretical Physics 28 (5), 870

    1962

  4. [12]

    Jansson (2025), Electroweak quantum numbers in the D4 root system, The European Phys

    H. Jansson (2025), Electroweak quantum numbers in the D4 root system, The European Phys. J. C 85, 76

    2025

  5. [13]

    R. A. Wilson (2022), Remarks on the group-theoretical foundations of particle physics,Intern. J. of Geom. Methods in Modern Phys. 19, 2250164

    2022

  6. [14]

    Cabibbo (1963), Unitary symmetry and leptonic decays, Physical Review Letters 10 (12), 531–533

    N. Cabibbo (1963), Unitary symmetry and leptonic decays, Physical Review Letters 10 (12), 531–533

    1963

  7. [15]

    Kobayashi and T

    M. Kobayashi and T. Maskawa (1973), CP-violation in the renormalizable theory of weak interaction, Progress of Theoretical Physics 49 (2), 652–657

    1973

  8. [16]

    ’t Hooft (2001), Introduction to general relativity , Rinton

    G. ’t Hooft (2001), Introduction to general relativity , Rinton

    2001

  9. [17]

    Einstein (1916), Die Grundlage der allgemeinen Relativit¨ atstheorie, Annalen der Physik 49 (7), 769–822

    A. Einstein (1916), Die Grundlage der allgemeinen Relativit¨ atstheorie, Annalen der Physik 49 (7), 769–822

    1916

  10. [18]

    Einstein (1955), The meaning of relativity , 5th ed., Princeton UP

    A. Einstein (1955), The meaning of relativity , 5th ed., Princeton UP

    1955

  11. [19]

    Leclerc (2006), The Higgs sector of gravitational gauge theories, Annals of Physics 321, 708–743

    M. Leclerc (2006), The Higgs sector of gravitational gauge theories, Annals of Physics 321, 708–743

    2006

  12. [20]

    Ivanenko and G

    D. Ivanenko and G. Sardanashvily (1983), The gauge treatment of gravity, Physics Reports 94, 1–45

    1983

  13. [21]

    E. R. Cohen and B. N. Taylor (1973), The 1973 least-squares adjustments of the fundamental constants, J. Phys. Chem. Ref. Data 2 (4), 663–734

    1973

  14. [22]

    B. N. Taylor, W. H. Parker and D. N. Langenberg (1969), Determination of e/h, using macroscopic quantum phase coherence in superconductors: implications for quantum electro- dynamics and the fundamental physical constants,Reviews of modern physics 41(3), 375–496. 18 ROBERT ARN...

    1969

  15. [23]

    R. A. Wilson (2022), Is there a universal concept of mass in fundamental physics? arXiv:2205.05443

    2022 arXiv

  16. [24]

    A. Einstein (1919), Do gravitational fields play an essential role in the structure of ele- mentary particles of matter? Sitzungsberichte der K¨ oniglichen Preußischen Akademie der Wissenschaften, 349–356

    1919

  17. [25]

    G. T. Gillies (1997), The Newtonian gravitational constant: recent measurements and related studies, Reports on progress in physics 60 (2), 151–225

    1997

  18. [26]

    Rothleitner and S

    C. Rothleitner and S. Schlamminger (2017), Invited review article: Measurements of the Newtonian constant of gravitation, G, Review of Scientific Instruments 88, 111101

    2017

  19. [27]

    Aaltonen et al

    T. Aaltonen et al. (CDF Collaboration) (2022), High precision measurement of the W boson mass with the CDF II detector, Science 376:6589, 170–176

    2022

  20. [28]

    R. A. Wilson (2022), A toy model for the W/Z mass ratio, arXiv:2204.07970

    2022 arXiv

  21. [29]

    J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay (1964), Evidence for the 2 π decay of the K20 meson, Phys. Rev. Lett. 13, 138

    1964

  22. [30]

    Pontecorvo (1968), Neutrino experiments and the problem of conservation of leptonic charge, Soviet Phys

    B. Pontecorvo (1968), Neutrino experiments and the problem of conservation of leptonic charge, Soviet Phys. JETP 26, 984–988

    1968

  23. [31]

    Gribov and B

    V. Gribov and B. Pontecorvo (1969), Neutrino astronomy and lepton charge, Phys. Rev. D 22 (9), 2227–2235

    1969

  24. [32]

    Bellerive et al

    A. Bellerive et al. (SNO collaboration) (2016), The Sudbury neutrino observatory, Nuclear Phys. B 908, 30–51. arXiv:1602.02469

    2016 arXiv

  25. [33]

    Dixon (1999), Division algebras: 26 dimensions, 3 families, arXiv:hep-th/9902016

    G. Dixon (1999), Division algebras: 26 dimensions, 3 families, arXiv:hep-th/9902016

    1999 arXiv

  26. [34]

    Dixon (2010), Division algebras; spinors; idempotents; the algebraic structure of reality, arXiv:1012.1304

    G. Dixon (2010), Division algebras; spinors; idempotents; the algebraic structure of reality, arXiv:1012.1304

    2010 arXiv

  27. [35]

    Furey (2018), SU (3)c × SU (2)L × U (1)Y (×U (1)X ) as a symmetry of division algebraic ladder operators

    C. Furey (2018), SU (3)c × SU (2)L × U (1)Y (×U (1)X ) as a symmetry of division algebraic ladder operators. Eur. Phys. J. C , 78, 375

    2018

  28. [36]

    Furey (2019), Three generations, two unbroken gauge symmetries and one eight- dimensional algebra, Phys

    N. Furey (2019), Three generations, two unbroken gauge symmetries and one eight- dimensional algebra, Phys. Lett. B 785, 84–89

    2019

  29. [37]

    Woit (2021), Euclidean twistor unification, arXiv:2104.05099

    P. Woit (2021), Euclidean twistor unification, arXiv:2104.05099

    2021 arXiv

  30. [38]

    Woit (2023), Spacetime is right-handed, arXiv:2311.00608

    P. Woit (2023), Spacetime is right-handed, arXiv:2311.00608. Queen Mary University of London Email address : r.a.wilson@qmul.ac.uk

    2023 arXiv

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