Trace dynamics, octonions, and unification: An $E_8 \times E_8$ theory of unification

Tejinder P. Singh

REVIEW 4 major objections 4 minor 15 references

Trace dynamics, octonions, and unification: An $E_8 \times E_8$ theory of unification

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper claims that a pre-spacetime trace dynamics over split bioctonionic coordinates, with E8 × E8 symmetry, yields the standard model, general relativity, and two new forces.

desk verdict A candid overview of Singh's octonionic unification program, but the load-bearing (7,7) Dirac identification is asserted, not shown, and the paper itself admits gaps; no new results to referee. read the letter →

arxiv 2501.18139 v1 pith:LOONYRJC submitted 2025-01-30 physics.gen-ph hep-ph

Tejinder P. Singh This is my paper · ORCID

classification physics.gen-phhep-ph

keywords tracedynamicsoctonionssplitbioctonionsE8×unificationpre-spacetimeexceptionalJordanalgebraweakmixingangle

topics Dark Matter

open problems Dark Matter

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 is a compact overview of a proposed unified theory in which spacetime and quantum theory emerge from a deeper pre-quantum, pre-spacetime dynamics. The starting point is trace dynamics, a deterministic matrix-valued Lagrangian dynamics whose conserved charge, when equipartitioned, gives rise to quantum commutators. The paper's specific new step is to place the matrix variables on a 16-dimensional split bioctonionic coordinate space and impose an $E_8\times E_8$ symmetry, so that branching produces the standard model on one chiral half and a gravitational counterpart on the other. If correct, the theory derives the weak mixing angle, fermion mass ratios, and several dimensional fundamental constants from first principles, and predicts that the total number of fundamental forces is six.

What carries the argument

The carrying object is the aikyon, an atom of spacetime-matter described by a pair of matrix variables $Q_1,Q_2$ living on a 16-dimensional split bioctonionic space, with action $\frac{1}{2}\int \frac{d\tau}{\tau_{\rm Pl}}\,{\rm Tr}\left[\frac{L_P^2}{L^2}\dot Q_1^\dagger\dot Q_2\right]$. The split bioctonion is the key algebraic object: its fourteen imaginary directions form a gradient operator that squares to the Klein-Gordon operator on a spacetime of signature $(7,7)$, so the octonionic gradient is a Dirac operator, a first-order square root of the Klein-Gordon operator; its split complex structure gives Lorentzian signature, and its two 8D halves give chiral fermions. The $E_8\times E_8$ symmetry is branched through $SU(3)\times E_6$ and then through trinification into the standard model gauge group and its gravitational counterpart. The exceptional Jordan algebra contributes the characteristic equation whose eigenvalues are interpreted as electric charge or square-root mass, which is how the mass ratios are derived.

What would settle it

Take the bosonic part of the aikyon action, perform the $E_8\times E_8$ branching to $SU(3)_C\times SU(2)_L\times U(1)_Y\times SU(3)_{\rm grav}\times SU(2)_R\times U(1)_g$, and compute the weak mixing angle from the Lagrangian coefficients; if the result does not reproduce $\sin^2\theta_W\approx 0.231$ without adjustable constants, the central derivation claim fails.

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Extended reading notes

Core claim

The paper claims that a single pre-spacetime, pre-quantum theory built from trace dynamics over split bioctonionic coordinates, with $E_8\times E_8$ symmetry, yields the standard model, general relativity, and two new forces. Branching each $E_8$ as $E_8 \to SU(3)\times E_6$, and trinifying $E_6 \to SU(3)\times SU(3)\times SU(3)$, the left-chiral half gives $SU(3)_C \times SU(2)_L \times U(1)_Y$ and the right-chiral half gives its pre-gravitational counterpart $SU(3)_{\rm grav}\times SU(2)_R \times U(1)_g$. In this scheme the weak force is left-handed gravity, $U(1)_{\rm grav}$ is sourced by $\pm\sqrt{m}$ charge and acts as the dark matter fluid, and the total number of fundamental forces is six. The paper further claims that the characteristic equation of the exceptional Jordan algebra yields the observed fermion mass ratios, and that the bosonic Lagrangian gives a derivation of the weak mixing angle.

Load-bearing premise

The paper assumes, without derivation, that the 14 imaginary directions of a split bioctonion can be treated as the spacetime derivative operator on a universe with seven time and seven space dimensions, and that the resulting action has $E_8\times E_8$ symmetry; if those identifications are chosen to reproduce the standard model rather than forced by the octonions, the unification is not explanatory.

Editorial extensions

If this is right

If the theory is correct, the total number of fundamental forces is six: the four known forces plus $SU(3)_{\rm grav}$ and $U(1)_{\rm grav}$.
The weak force is interpreted as left-handed gravity, a spacetime symmetry rather than an internal symmetry; this explains parity violation and predicts that gravitation should also violate parity.
The $U(1)_{\rm grav}$ force, sourced by $\pm\sqrt{m}$ charge, acts as a dark matter fluid and is attractive in a matter-dominated universe, giving a new force origin for dark matter phenomenology.
The weak mixing angle and several fundamental constants are derived quantities rather than inputs, removing a significant source of arbitrariness in the standard model.

Reading between the lines

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

If the split-bioctonion construction is robust, a natural next step is to compute the running of $\sin^2\theta_W$ from the unification scale to the electroweak scale and compare with precision measurements; the paper does not present this check.
The $E_8\times E_8$ structure suggests possible connections to other grand unified constructions, but here the extra dimensions are non-spacetime vector-bundle directions; a testable difference may be a second timelike dimension at the weak scale.
The $\pm\sqrt{m}$ charge rule could be tested statistically on galaxy rotation curves: if dark matter is sourced by square-root mass, the observed galactic acceleration scale should emerge naturally from the charge-to-mass relation, a step the paper only sketches.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read the preprint. It is a brief overview of Tejinder Singh's octonionic unification program, not a new research paper. The abstract says so, and the text delivers exactly that: seven ingredients, a sketch of the action, and a list of claimed outputs. If you want a map of the program, it is useful.

What it does well: it is honest about being an overview, it lays out the components (trace dynamics, spectral action, split bioctonions, E8×E8 branching, exceptional Jordan algebra) in one place, and it flags open points. The motivation from quantum foundations is clearly stated.

The soft spots dominate. No derivations are reproduced. The central claims—two new forces, a derived weak mixing angle, derived mass ratios—are deferred to the author's prior papers. The charge assignment ±√m is introduced from the observed e/u/d mass ratios and then those same ratios are said to be 'derived.' That is a fitting pattern, not a prediction, unless the prior work is much stronger than this text suggests.

The stress-test concern lands. Section 5 asserts that the 14D split-bioctonion gradient describes spacetime with signature (7,7), but the Clifford algebra generated by complex split bioctonions is stated in Eq. (28) as Cl(7), not Cl(7,7). The paper gives no gamma-matrix construction to bridge that gap, and the sentence 'for deep reasons which remain to be fully understood' is a tell. The identification of the Dirac operator with the gradient is load-bearing; without a derivation, the architecture rests on an unproven equivalence.

The paper is not incoherent throughout—the author knows limitations and says so—but the (7,7) identification is a serious gap. An editor should not treat this as a self-contained research contribution.

Who is this for? A reader who wants a high-level summary of Singh's program, not someone who needs to verify claims. I would not cite it for any specific result, and I would not send it to a research referee as is. If the venue publishes program overviews, fine; otherwise desk reject.

Referee Report

4 major / 4 minor

Summary. The paper is a brief overview of the author's 'octonionic theory' of unification. It proposes a pre-spacetime, pre-quantum trace dynamics over split bioctonionic coordinates, with a Lagrangian of spectral-action type (Eq. (8)), assumed to have E8 × E8 symmetry. It claims that after symmetry breaking this yields the standard model, general relativity, and two new forces (SU(3)_grav and U(1)_grav), and that several fundamental constants and mass ratios are derivable from the exceptional Jordan algebra. The main technical content in this manuscript is the construction of the aikyon action, the derivation of the harmonic-oscillator equations of motion (10), and a description of the group-theoretic branching (30)-(31); all quantitative derivations (weak mixing angle, CKM parameters, mass ratios) are delegated to prior papers [7]-[10].

Significance. If the claimed results were established, the paper would be of exceptional significance: a single E8 × E8 symmetric pre-quantum action from which the standard model, general relativity, and two new forces emerge, with derivable constants. The manuscript is clear and openly identifies its programmatic character, and it usefully lists seven ingredients with references. However, the actual evidence in this paper is thin: no derivation of the weak mixing angle, CKM parameters, or mass ratios is given, and the only equation derived from the action is the simple harmonic oscillator (10). The significance is therefore conditional on the cited prior work, and the present manuscript does not itself provide enough checkable content to support the strength of its claims.

major comments (4)

[Section 5, Eq. (28)] The claim that the 14D split-bioctonion gradient operator describes 14D spacetime with signature (7,7) is not supported by the Clifford algebra stated in the paper. Eq. (28) identifies complex split bioctonions with Cl(7) ≅ C ⊗ Cl(0,7) ≅ Cl(6) ⊕ Cl(6), not with Cl(7,7). A Dirac operator on a (7,7) spacetime requires a representation of Cl(7,7), which has real dimension 2^14, whereas Cl(7) as a complex algebra has complex dimension 2^7. No explicit gamma-matrix construction from the fourteen imaginary directions is provided. Since this identification is the bridge that allows Dirac eigenvalues to be promoted to matrix variables in Eqs. (6)-(8), the claimed spectral-action route to spacetime and gravity rests on an unsupported and apparently inconsistent algebraic step.
[Section 6.1] The introduction of the U(1)_grav charge ±√m is circular as presented. The text states that the charge is motivated by the experimental fact that the square roots of the electron, up quark, and down quark masses are in the ratio (1/3, 2/3, 1), and then states that the same construction 'enabled us to theoretically derive the strange mass ratios' for the second and third generations. Unless the paper explicitly distinguishes which ratios are inputs and which are outputs, the derivation of mass ratios from the very ratios used to assign the charges is not a derivation. This point needs to be clarified, or the claim of derivation should be withdrawn.
[Section 4 and Section 6] The E8 × E8 symmetry is assumed, not derived. Section 4 states that 'the action will be assumed to have an E8 × E8 symmetry,' and Section 6 chooses a branching in which 'the first E6 gives the standard model.' The paper does not show that trace dynamics over split bioctonions forces this gauge group; rather, the branching is selected so that one E6 sector contains the standard model gauge group. As written, the claimed 'emergence' of the standard model is therefore by construction, not a first-principles consequence.
[Section 8] The claim that the characteristic equation of the exceptional Jordan algebra 'determines values of the dimensional fundamental constants' is not supported by any derivation or numerical comparison in this manuscript. The text refers to Ref. [8] for the mass ratios and to Ref. [10] for the weak mixing angle, but no equations from those derivations are reproduced or checked here. For an overview this reliance on prior work may be acceptable, but the strength of the claims in the text is disproportionate to what is actually shown.

minor comments (4)

[Section 5] The phrase 'Freudenthal-Tits magic sqaure' contains a typo: 'sqaure' should be 'square'. Also, 'each of the two E8-s branches' in Section 6 is awkward and should be rephrased.
[Abstract and Section 1] The abstract says the paper is 'a very brief overview,' and this is accurate; however, the body uses language such as 'strong evidence' and 'satisfactory derivation' that is not backed by derivations in this paper. The authors should either include the key derivations or soften the claims.
[Section 5] The statement 'for deep reasons which remain to be fully understood' (Section 5) is an explicit admission that the left-right symmetry breaking from 14D to 6D is not yet derived; this should be flagged as an open problem rather than a step in the derivation.
[References] Reference [11] is described only as 'in this volume'; if this is a proceedings paper, the editors should confirm that the reference is complete and publicly available.

Circularity Check

1 steps flagged · score 6.0 of 10

The central mass-ratio and CKM 'predictions' reuse the same experimental square-root mass ratios that were used to define the U(1)_grav charge and SU(3)_grav color assignment, so those predictions reduce, at least in part, to fitted input.

fitted input called prediction [Section 6.1 (definition of U(1)_grav charge) and Section 8 (EJA mass-ratio 'derivation'); see also Section 7 and refs. [8], [9].]
"The motivation for introducing the charge ±√m is the experimental fact that the square roots of the masses of the electron, up quark, down quark are in the ratio (1 /3, 2/3, 1). ... we proposed ±√m as the charge for U (1)grav ... and interchanged the relative position of the right handed electron and right handed down quark ... This enabled us to theoretically derive the strange mass ratios for the second and third generation [8], and in an analogous derivation, also the parameters in the CKM quark mixing matrix [9]."

The U(1)_grav charge ±√m and the SU(3)_grav color interchange are chosen explicitly to reproduce the experimentally observed electron/up/down square-root mass ratios and the puzzle they create. The subsequent 'theoretical derivation' of second- and third-generation mass ratios in [8] and of CKM parameters in [9] uses exactly these charge and color assignments. The output mass ratios are therefore the fitted input data, re-expressed through the EJA projection. Section 7 makes the definitional character explicit: 'only for this generation the observed square-root mass ratios (0, 1/3, 2/3, 1) coincide with the mass ratios predicted by this construction.' The construction's 'predicted' ratios are the observed ratios by construction, not by independent prediction.

full rationale

The paper contains extensive unsupported or asserted identifications (the Dirac-operator-equals-gradient claim, the E8×E8 symmetry, the (7,7) signature despite Eq. (28) giving Cl(7)∼Cl(6)⊕Cl(6)), but those are correctness or completeness issues, not circularity. The one clear circular step is the mass-ratio programme: the charge and color data are fit to the experimental square-root mass ratios, and the same ratios are then presented as derived from the exceptional Jordan algebra. The derivations cited ([8], [9]) are from the authors' own program and inherit the same fitted assignments. This makes the central 'derivation of fundamental constants' partially circular, though not entirely: the EJA structure, the magic-square branching, and the spectral action input are independent mathematical ingredients, and the weak-mixing-angle claim [10] is not itself shown here to be circular. On the 0-10 scale, this is a partial circularity (6), not a complete reduction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 4 invented entities

The central claim rests on a large number of unproven algebraic identifications, a preferred gauge group chosen to reproduce the standard model, and a charge assignment fitted to observed mass ratios. The paper itself provides no derivations, so most of the load is carried by self-cited prior work.

free parameters (4)

alpha (Yang-Mills coupling) = not specified
Introduced in Eq. (8) as the dimensionless Yang-Mills coupling constant; appears in the aikyon action and the oscillator equation Eq. (10). Its value is not derived here.
L (length scale) = not specified
Introduced in Eq. (8) alongside alpha; sets the frequency of the harmonic oscillator in Eq. (10). Not derived.
beta_1, beta_2 (Grassmann numbers) = two unequal Grassmann numbers
Introduced in Eq. (6) 'to make the Lagrangian bosonic'; arbitrary.
Charge ±sqrt(m) for U(1)_grav = square root of mass
Motivated by observed square-root mass ratios (0, 1/3, 2/3, 1); a new charge assignment fitted to first-generation fermion masses.

assumptions (5)

domain assumption Adler's trace dynamics results, including equipartition of the Adler-Millard charge leading to quantum mechanics
Section 3 assumes that coarse-graining trace dynamics at equilibrium yields quantum mechanics with equipartitioned commutators, citing Adler [2].
standard math Spectral action principle of Chamseddine and Connes
Section 4 assumes Tr[L_P^2 D^2] ~ Einstein-Hilbert action, citing [3].
ad hoc to paper Dirac operator equals gradient operator on split bioctonionic space
Section 5 asserts that the 14D gradient operator on split bioctonions is the Dirac operator and squares to Klein-Gordon in (7,7) signature, without proof.
ad hoc to paper E8 × E8 branching pattern
Section 6 chooses E8 -> SU(3) × E6 -> SU(3) × SU(3) × SU(3) to reproduce the standard model and a pre-gravitational sector; no derivation that this branching is unique.
ad hoc to paper Exceptional Jordan algebra characteristic equation encodes charges and mass ratios
Section 8 postulates that diagonal EJA entries represent electric charge or square-root mass and that charge eigenstates are projections of basis states.

invented entities (4)

Aikyon (atom of space-time-matter)
purpose: Fundamental object combining bosonic and fermionic degrees of freedom; the basic dynamical entity of the theory.
No experimental signature predicted beyond the speculative new forces; its existence is postulated.
U(1)_grav (dark electromagnetic force)
purpose: New force sourced by ±sqrt(m) charge, proposed as origin of MOND and dark matter fluid.
The paper claims it has 'desired properties of the dark matter fluid' but gives no quantitative prediction that distinguishes it from other dark matter models.
SU(3)_grav (gravitational strong force)
purpose: Sixth new force, gravitational analog of the strong force.
Postulated to complete the E8 branching; no experimental signature given.
Second 4D spacetime with flipped signature
purpose: Hosts weak force and pre-gravitational gauge fields.
Proposed to explain parity violation; experiments at 10^-17 cm are suggested but no specific prediction is made.

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Cite this review

Pith. "Pith review of Trace dynamics, octonions, and unification: An $E_8 \times E_8$ theory of unification." pith.science (2026). https://pith.science/paper/LOONYRJC

@misc{pith2026250118139,
  author       = {Pith},
  title        = {Pith review of: Trace dynamics, octonions, and unification: An $E_8 \times E_8$ theory of unification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LOONYRJC}},
  note         = {Machine review of arXiv:2501.18139}
}

read the original abstract

This is a very brief overview of the ongoing research program of unification known as the octonionic theory. We highlight the quantum foundational origins for the theory, and the seven key ingredients which go into its making.

Figures

Figures reproduced from arXiv: 2501.18139 by the authors.

**Figure 2.** The particle content of the theory - part II. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Clifford algebras and fermionic particle states [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The Jordan eigenvalues Xν =   0 Vτ V˜ µ V˜ τ 0 Vν Vµ V˜ ν 0   Xe+ =   1 Vaτ V˜ aµ V˜ aτ 1 Ve+ Vaµ V˜ e+ 1   Xu =   2 3 Vt V˜ c V˜ t 2 3 Vu Vc V˜ u 2 3   Xad =   1 3 Vab V˜ as V˜ ab 1 3 Vad Vas V˜ ad 1 3   (39) At the heart of our understanding of mass r…

**Figure 5.** Figure 5: The proposed emergence of our universe from the breaking of [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 13 canonical work pages

[7]

Kaushik P and Vaibhav V and Singh T P 2022 An E8 ⊗ E8 unification of the standard model with pre-gravitation, on an exceptional Lie-algebra valued space , [arXiv:2206.06911 [hep-ph]]

arXiv 2022
[10]

- The bosonic Lagrangian, and a theoretical derivation of the weak mixing angle [arXiv:2208.09811 [hep-ph]]

Raj S and Singh T P 2022 A Lagrangian with E8 × E8 symmetry for the standard model and pre-gravitation I. - The bosonic Lagrangian, and a theoretical derivation of the weak mixing angle [arXiv:2208.09811 [hep-ph]]

arXiv 2022
[8]

Bhatt V and Mondal R and Vaibhav V and Singh T P 2022 Majorana neutrinos, exceptional Jordan algebra, and mass ratios for charged fermions , J Phys G 49, 045007

work page 2022
[1]

Singh T P 2023 Gravitation and quantum theory as emergent phenomena J Phys Conf Ser 2533, 012013

work page 2023
[2]

Adler S L 2004 Quantum theory as an emergent phenomenon Cambridge University Press

work page 2004
[3]

Chamseddine A H and Connes A 1996 The spectral action principle Commun Math Phys 186, 737

work page 1996
[4]

Landi G and Rovelli C 1997 General Relativity in Terms of Dirac Eigenvalues Phys Rev Lett 78, 3051

work page 1997
[5]

Vaibhav V and Singh T P 2023 Left-Right Symmetric Fermions and Sterile Neutrinos from Complex Split Biquaternions and Bioctonions Adv Appl Clifford Algebras 33, 32

work page 2023

Show all 15 references

[6]

Kritov A 2021 Gravitation with cosmological term, expansion of the universe as uniform acceleration in Clifford coordinates, Symmetry 2021, 13

2021
[9]

Patel A A and Singh T P 2023 CKM Matrix Parameters from the Exceptional Jordan Algebra , Universe 9, 440

2023
[11]

Gresnigt N and Gourlay L 2024 in this volume

2024
[12]

Chevalley C and R D Schafer R D 1950 The Exceptional Simple Lie Algebras F(4) and E(6) Proc Natl Acad Sci US 36, 137

1950
[13]

Jordan P and von Neumann J and Wigner E 1934 On an Algebraic Generalisation of the Quantum Mechanical Formalism Annals of Mathematics 35, 29

1934
[14]

Townsend P K 2016 The Jordan formulation of quantum mechanics: a review , arXiv:1612.09228

2016 arXiv
[15]

Dray T and Manogue C 2015 The geometry of the octonions World Scientific Publishing Company

2015

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [7]

Kaushik P and Vaibhav V and Singh T P 2022 An E8 ⊗ E8 unification of the standard model with pre-gravitation, on an exceptional Lie-algebra valued space , [arXiv:2206.06911 [hep-ph]]

arXiv 2022

[2] [10]

- The bosonic Lagrangian, and a theoretical derivation of the weak mixing angle [arXiv:2208.09811 [hep-ph]]

Raj S and Singh T P 2022 A Lagrangian with E8 × E8 symmetry for the standard model and pre-gravitation I. - The bosonic Lagrangian, and a theoretical derivation of the weak mixing angle [arXiv:2208.09811 [hep-ph]]

arXiv 2022

[3] [8]

Bhatt V and Mondal R and Vaibhav V and Singh T P 2022 Majorana neutrinos, exceptional Jordan algebra, and mass ratios for charged fermions , J Phys G 49, 045007

work page 2022

[4] [1]

Singh T P 2023 Gravitation and quantum theory as emergent phenomena J Phys Conf Ser 2533, 012013

work page 2023

[5] [2]

Adler S L 2004 Quantum theory as an emergent phenomenon Cambridge University Press

work page 2004

[6] [3]

Chamseddine A H and Connes A 1996 The spectral action principle Commun Math Phys 186, 737

work page 1996

[7] [4]

Landi G and Rovelli C 1997 General Relativity in Terms of Dirac Eigenvalues Phys Rev Lett 78, 3051

work page 1997

[8] [5]

Vaibhav V and Singh T P 2023 Left-Right Symmetric Fermions and Sterile Neutrinos from Complex Split Biquaternions and Bioctonions Adv Appl Clifford Algebras 33, 32

work page 2023