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REVIEW 3 major objections 4 minor 44 references

The CKM sector of the exceptional-Jordan programme: finite-Dirac mass moduli, two conditional angle estimates, and the weak-to-mass bridge

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper argues that two of the four CKM quark-mixing parameters follow, without any fitted continuous parameter, from square-root mass ratios on chains in the exceptional-Jordan programme, and that the remaining two are confined to a sin

desk verdict Serious, honest speculative model building with real algebraic content, but the two claimed no-fit CKM estimates are far weaker than the abstract suggests: one is a disclosed convention choice plus an error compensation, and the other is a 5.3% tension. read the letter →

arxiv 2608.01445 v1 pith:TCT73BFS submitted 2026-08-02 hep-ph hep-th

Tejinder P. Singh This is my paper · ORCID
classification hep-phhep-th PACS 12.15.Ff
keywords CKMmatrixquarkmassesandmixingexceptionalJordanalgebrasquare-rootmassrelationsfiniteDiracoperatorAlbertnine-linktextures
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

The paper aims to show that the CKM quark-mixing matrix is partly arithmetic of the quark mass spectrum rather than independent data. It works inside the exceptional-Jordan programme — an attempt to derive fermion properties from the algebra of $3\times 3$ Hermitian octonionic matrices — and fixes every mass-ratio input before any mixing is addressed. It then shows that two of the four CKM parameters, the Cabibbo angle and the 2–3 mixing angle, emerge from three structural postulates with no fitted continuous parameters: $|V_{us}| = 0.2371$ ($+5.3\%$ high) and $|V_{cb}| = 0.0422$ ($+0.8\%$ high) against the 2026 global fit. It also derives the exact identity $|V_{ub}|/|V_{cb}| = \sqrt{m_u/m_c}$, a factor 2.1 below the measured ratio, and proves this shortfall cannot be repaired anywhere inside the minimal model; the last two parameters ($\theta_{13}$ and $\delta_{CP}$) are fitted as one complex long-edge rotation. If the construction holds, mixing angles become genuine consequences of the mass spectrum, and the missing dynamical structure is localised to a single complex number — though the paper states plainly that this is a conditional layer, not a completed derivation.

What carries the argument

The load-bearing machinery is the occupied-chain compression of the multiplicative symmetric-cube lift, $R_f = P_{\rm occ}\,\Gamma_3(\hat X_f)\,P_{\rm occ}$, whose singular values are exactly the square-root mass monomials of the parent chain; normalizing $Q_f = R_f^\dagger R_f/\rho^2$ gives spectrum $(m_{f1}/m_{f3},\, m_{f2}/m_{f3},\, 1)$, so the mass ratios live in a finite Dirac operator. Mixing is then carried by the adjacent-edge lift $\tan\theta_{ij} = \sqrt{m_i/m_j}$ composed along each chain (the transport reading), the virtual-node amplitude rule $\sin\theta_{23} = \sin\theta_2\,\sin\theta_3$ for the down chain, and the balanced shared-edge rotor $\varphi_{12} = -2\chi = 90^\circ$.

What would settle it

Derive the virtual-node composition rule from the parent theory and check which re-unitarization it selects: polar decomposition predicts $|V_{cb}| = 0.014$ against the measured $0.04188$, a clear miss. Independently, an improved determination of $m_c/m_t$ at the matching scale that moves the parent input from $1/12.28$ toward the measured $1/16.65$ drives the predicted $|V_{cb}|$ to about $0.064$ at fixed amplitude reading — either measurement settles the central claim.

Watch

Extended reading notes

Core claim

Assembling $V = U_u^\dagger U_d$ with Postulates 1–3 and nothing else produces the paper's central object, the conditional no-fit layer: $|V_{us}| = 0.2371$ and $|V_{cb}| = 0.0422$, against the 2026 global-fit values $0.22517$ and $0.04188$, together with the exact identity $|V_{ub}|/|V_{cb}| = \tan\theta_{12}^u = \sqrt{m_u/m_c} = 0.0424$, a factor 2.1 below the measured 0.090. The remaining two CKM degrees of freedom are then fitted through a minimal long-edge insertion $R_{13}(\varepsilon,\omega)$, which yields two discrete balanced-orientation branches, $(\varepsilon,\omega) = (0.002079, 288.2^\circ)$ and $(0.005358, 202.2^\circ)$, with all other CKM observables following by unitarity. On

Load-bearing premise

The load-bearing premise is Postulate 2's amplitude reading of virtual-node composition (Sec. 4.3), $\sin\theta_{23} = \sin\theta_2\sin\theta_3$: the paper itself discloses that four plausible re-unitarizations of the same projected transport give $|V_{cb}| = 0.014$, $0.045$, $0.063$ and $0.042$, so the headline $+0.8\%$ agreement exists only in that convention and only in the all-theory frame where the parent's $\sqrt{m_c/m_t}$ overshoot partially compensates.

Editorial extensions

If this is right

  • Two of the four CKM parameters — the Cabibbo angle and the 2–3 mixing angle — would be determined by the mass spectrum plus structural rules rather than fitted: Layer 1 gives $|V_{us}| = 0.2371$ and $|V_{cb}| = 0.0422$ from no fitted continuous CKM parameter.
  • The long-standing shortfall in $|V_{ub}|/|V_{cb}|$ — here an exact identity equal to $\sqrt{m_u/m_c}$, a factor 2.1 low — is proven irreparable inside the minimal model: any relative (2,3) phase repairs $|V_{ub}|$ only by inflating $|V_{cb}|$ by the same factor, so the missing structure is confined to one complex (1,3) element.
  • The up–down asymmetry of quark mixing becomes a node-counting statement: the down chain contains an unoccupied virtual node that suppresses the 2–3 amplitude via $\sin\theta_{23} = \sin\theta_2\sin\theta_3$, while the up chain has no virtual node and is unsuppressed.
  • The same machinery predicts a matching suppression in the charged-lepton sector and, under the adopted cyclic Majorana placement, a real PMNS block with vanishing leptonic CP violation ($J_\ell = 0$, $\delta^\ell_{CP}\in\{0,\pi\}$).
  • A Yukawa block linear in six diagonal mass links plus three directed Albert links would have the support count and single-cycle topology of the bottom-up nine-link textures, with the balanced Cabibbo link contributing a $\pi/2$ holonomy — a candidate route to the observed clustering of unitarity-triangle angles at multiples of $\pi/8$.

Reading between the lines

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

  • The paper's fragility table implies a correlation it does not foreground: keeping the amplitude reading, any improvement of the parent input $\sqrt{m_c/m_t}$ toward its measured value moves the predicted $|V_{cb}|$ from 0.0422 toward 0.0635 (elasticity $-1.9$), so the $+0.8\%$ agreement and the 26% mass-hierarchy tension stand or fall together.
  • Because $|V_{us}|$ prefers $\varphi_{12}\simeq 107^\circ$ while the balanced rotor fixes $90^\circ$ and yields $+5.3\%$, a future vacuum derivation of the Cabibbo-block phase must either explain that tension or force a compensating $\sim5\%$ shift in $\sqrt{m_s/m_d}$ — the mixing and mass sectors of the programme are jointly constrained.
  • The two balanced-orientation branches, with $\varepsilon/(s^u_{12}s^u_{23})$ equal to 0.604 and 1.557 respectively, show that the apparent near-lock of 0.604 to 3/5 is orientation-dependent numerology; any test of that ratio must wait for the vacuum dynamics that selects the orientation.
  • The proposed nine-link skeleton converts future precision measurements into an either/or: if the unitarity-triangle angles lock onto the ($\pi/2, \pi/8, 3\pi/8$) cluster at the hundredth-of-a-degree level, a completed bridge must reproduce those special values — whereas the paper's own unitarity propagation ($\alpha = 92.4^\circ$) already sits outside the $88.74^\circ$–$90.03^\circ$ band the botto
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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

3 major / 4 minor

Summary. The paper continues the exceptional-Jordan-programme attempt to derive fermion mass ratios and CKM mixing from a small set of algebraic/geometric inputs. It contains four main parts. First, an operator-level 'audit': compressing the multiplicative symmetric-cube lift onto occupied chain nodes yields a diagonal root-mass operator whose positive square reproduces the proposed mass ratios; this is packaged into a finite Dirac block. Second, an explicit Albert-algebra lift of the Cabibbo bridge (the inner derivation D12(-gχ)) that, for an adopted F1-F2 Peirce-family embedding, reproduces the local Teli-Singh phase law φ12 = -2χ. Third, a transport model for the full CKM matrix: Uf = R23 R12, with rung angles from the Jordan chains, a postulated amplitude rule for the virtual-node (down-sector) composition, real (2,3) blocks, and balanced shared-edge phase |φ12|=90°. This produces a 'Layer 1' no-fit layer: |Vus|=0.2371 (+5.3%), |Vcb|=0.0422 (+0.8%), |Vts|, |Vtd|, and the exact identity |Vub|/|Vcb|=sqrt(mu/mc), which is a factor 2.1 below data. The remaining two CKM degrees of freedom are then fitted by an R13(ε,ω) insertion, with two discrete orientation branches. Fourth, a lepton-side mirror and a comparison with the nine-link texture programme. The paper repeatedly and explicitly labels its postulates [P], derived statements [D], and open items [O], and it ships reproducible numerical code.

Significance. If the central programme-level claim held, this would be a notable step: two of the four physical CKM parameters would be fixed, up to stated structural choices, with no continuous CKM fit, from a Jordan-algebraic spectral construction. The paper also proves an exact model identity, |Vub|/|Vcb| = sqrt(mu/mc), which is independent of the contested amplitude rule, and it demonstrates why the minimal transport model cannot repair the long-edge deficit. The operator audit (Proposition 1) and the Albert lift (Section 3.6) are self-contained algebraic results, and the numerical layer is reproducible from the included scripts. The paper is unusually honest: the fragility of |Vcb| under re-unitarization conventions and under the parent-theory sqrt(mc/mt) input is disclosed in Section 6. That honesty does not, however, remove the fact that the 'no-fit' determination of θ23 is not an algebraic output of the exceptional-Jordan construction alone, but a combination of one of several equally admissible conventions and a partially compensating mass-input error. The significance of the paper is therefore real but conditional; it is a status report and a set of sharply localized open problems, not

major comments (3)
  1. [Sec. 4.3, Eq. (64), and Sec. 6] Postulate 2 is the load-bearing step behind the claimed no-fit |Vcb|, but the paper itself shows that the same projected triangular block admits four re-unitarization conventions giving sin θ_eff = 0.0675, 0.1257, 0.1435, 0.1232 and hence |Vcb| = 0.014, 0.045, 0.063, 0.042. Nothing in the exceptional-Jordan algebra selects the 'amplitude reading' (64). Since Eq. (64) is a postulate rather than a derived rule, the central claim that Layer 1 'fixes' or 'determines' θ23 is not supported. The abstract and conclusions should be reframed so that |Vcb|=0.0422 is explicitly one value under one of several arbitrary conventions, with the spread treated as a theory-level uncertainty, unless a derivation of (64) is supplied.
  2. [Sec. 6, fragility table and Sec. 11] The +0.8% agreement for |Vcb| is obtained only in the 'all-theory frame', where the parent input sqrt(mc/mt)=0.0814 is 26% above the measured value 0.0601. Replacing it with the measured value moves the same amplitude-reading prediction to |Vcb|=0.0635 (+52%). Thus the headline agreement is a partial cancellation between a convention choice (Postulate 2) and a known 26% mass-ratio error. This is disclosed in Section 6, but the abstract still presents |Vcb|=0.0422 as one of the two no-fit angle estimates. The abstract and the final conclusions must carry the same qualification; otherwise the central quantitative claim is materially overstated.
  3. [Sec. 8, Eq. (68) and orientation-branch table] The long-edge insertion R13(ε,ω) is a two-parameter fit, and the two balanced orientations give (ε,ω)=(0.002079,288.2°) and (0.005358,202.2°), with branch-dependent ε/(s12 s23) ratios of 0.604 and 1.557. The paper discloses this, and it is correct to say that ω is convention-dependent. But the conclusion that the long edge is 'reduced to one fitted complex number' is only true after an arbitrary discrete branch choice, and the branch is not dynamically selected. The localization result is still meaningful, but the claim should be stated as a localization of the remaining two degrees of freedom, not as a reduction to a single unambiguous fitted complex number.
minor comments (4)
  1. [Abstract] The abstract's phrase 'two conditional angle estimates' is appropriate, but the immediately following sentence 'the no-fit layer gives |Vcb|=0.0422' should say 'in the all-theory frame under the amplitude reading' to avoid implying a convention-independent no-fit result.
  2. [Sec. 4.3] The notation θ2 and θ3 in Eq. (64) is introduced in words but not displayed as a small chain diagram; a one-line diagram or labels (occupied→virtual, virtual→occupied) would help the reader track which rung is which.
  3. [Table 1] The Table 1 entries |Vtd| and J are labelled 'unitarity consequence', but at pure Layer 1 there is no fitted CP phase; clarify whether the table uses δ_CP=0/π or actually shows a phase-dependent formula.
  4. [General] The list of referee-like strengths in the text (for example, 'the failure is reported, with code') is useful, but the paper would benefit from a single short subsection titled 'Numerical reproducibility' rather than scattering script references through Sections 9 and Appendix D.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the two Layer-1 angle estimates are conditional outputs of explicitly labelled postulates, not hidden fits; the disclosed fragilities and the fitted Layer-2 parameters are not dressed up as predictions.

full rationale

I walked the claimed derivation chain from the mass-ratio inputs to the CKM layer. The no-fit layer's inputs are the parent Jordan chain data (delta = sqrt(3/8), occupied-node monomials, Sec. 2 and Prop. 1), the transport extension (Postulate 1, Eq. 63), the virtual-node amplitude reading (Postulate 2, Eq. 64), and the balanced shared-edge rotor (Postulate 3, Sec. 5). None of these is defined using the CKM targets. |Vus| is computed from the first-rung mass ratios; |Vcb| is the difference-form combination of the 23 angles (Eq. 66), not a refit of |Vcb|. Postulate 2 is explicitly labelled [P], and the paper discloses four alternative re-unitarizations giving |Vcb| in 0.014-0.063 (Sec. 4.3) and the strong sensitivity to sqrt(mc/mt) (Sec. 6 fragility table). Choosing and clearly stating an axiom among disclosed alternatives is not circularity; the paper does not call the other conventions predictions. The exact relation |Vub|/|Vcb| = sqrt(mu/mc) is a derived consequence of the transport block structure (Lemmas 1-2) plus the defined tan theta12^u = sqrt(mu/mc), not a redefinition of the measured target. Layer 2 explicitly fits (epsilon, omega) to |Vub| and delta_CP (Sec. 8), and the lower block of Table 2 is explicitly described as a unitarity consistency check, not independent prediction. The paper's reliance on the author's prior work [1,2,4] is substantial, but those cited results are parameter-free mass-ratio and phase-frame constructions that are externally falsifiable (against mass data and the octonionic transport algebra) and do not include the CKM observables as inputs; hence they are independent support rather than a load-bearing self-citation chain. The conditional theorems in Appendices A and B are self-contained. The disclosed fragilities (Postulate 2 convention spread, sqrt(mc/mt) sensitivity, balanced-orientation branch ambiguity, and the assumptions ledger in Sec. 10.1) are correctly identified by the paper itself as limitations; they are correctness risks, not circular reductions.

Assumptions & free parameters 5 free parameters · 11 assumptions · 4 invented entities

The central numbers rest on a chain of programme inputs [P] inherited from the author's earlier papers: delta = sqrt(3/8), chain and node assignments, sector scales, balanced quadrature, and Postulate 2's amplitude reading. The algebraic sections (Prop. 1, the Albert lift, Lemmas 1-2) add no extra hidden parameters. Layer 2 carries two fitted continuous parameters plus a discrete orientation choice. The invented entities (chiral-soldering channel, virtual node, family kernel, nine-link skeleton) are proposed structures without independent falsifiable handles at this stage.

free parameters (5)
  • delta = sqrt(3/8) (Jordan spread) = 0.612372
    Sets all rung tangents and both Layer-1 CKM estimates; adopted [P] from refs [1,2], not derived here.
  • sector scales s_d = 1, s_u = 2/3 = 1, 2/3
    Normalize the chain monomials and fix the up/down rung angle hierarchy.
  • balanced shared-edge phase phi12 = 90 degrees
    Selected as the distinguished quadrature (Postulate 3); |Vus| would prefer about 107 degrees, so this is a chosen value, not a fit outcome.
  • balanced-orientation branch = A or B
    Two discrete orientations give different (epsilon, omega) pairs with identical targets; no dynamics selects one.
  • long-edge parameters (epsilon, omega) = A: (0.00207936, 288.1915 deg); B: (0.00535850, 202.1844 deg)
    Explicitly fitted to |Vub| and delta_CP; the only continuous CKM parameters beyond Layer 1 in this construction.
assumptions (11)
  • standard math Power associativity of one-element Jordan subalgebras and spectral idempotents map to ordinary projections (Sec. 3).
    Used in Proposition 1 to turn the Jordan spectral element X_f into an ordinary operator on C^3; standard Jordan algebra fact.
  • standard math Representation-theoretic branchings Spin(5,5) -> Spin(3,3) x Spin(2,2), 16 = (4,2,1) + (4*,1,2), and the invariant form of the Albert cubic (Sec. 3.4).
    Underpins the chiral-soldering channel claim; taken as established representation theory.
  • domain assumption Parent minimal chains, monomial masses, delta = sqrt(3/8), occupied nodes (1,2,4) and (1,2,3) from refs [1,2].
    All mass-ratio and rung-angle inputs are inherited unmodified; the CKM outputs inherit their deviations.
  • domain assumption Two isomorphic Albert copies J_L + J_R and branch pairing (Sigma_L, J_L) <-> flavour, (Sigma_R, J_R) <-> mass (Sec. 3.5).
    The weak-to-mass bridge presupposes this identification; the gluing kernel is not calculated.
  • ad hoc to paper The F1-F2 Peirce pair represents the first two physical families (Sec. 3.6).
    Required for the exact lift to carry the physical Cabibbo transport; author labels it [P].
  • ad hoc to paper Postulate 1: transport extension U_f = R23 R12 with CKM = U_u^dagger U_d.
    The three-generation unitary is assumed as the ordered product of rung rotations; Hamiltonian-texture reading fails in Appendix E.
  • ad hoc to paper Postulate 2: virtual-node amplitude reading sin theta23 = sin theta2 sin theta3 (Sec. 4.3).
    One of four re-unitarizations of a non-unitary triangular block; alternatives give |Vcb| = 0.014 to 0.063.
  • ad hoc to paper Postulate 3: phases live only on the shared Cabibbo edge, with balanced rotor |phi12| = 90 degrees and real (2,3) blocks.
    Selects the distinguished phase point; |Vus| prefers about 107 degrees, so this is a choice with a disclosed 5.3% cost.
  • ad hoc to paper Layer-2 insertion R13(epsilon, omega) in U_u to fit |Vub| and delta_CP (Sec. 8).
    The remaining two CKM parameters are explicitly fitted, with two discrete orientation branches.
  • ad hoc to paper Factorized bridge Y_f = B tensor Xi_f with common nonzero B (Sec. 3.4).
    Conditional ansatz needed for the cancellation VCKM = U_Xi_u^dagger U_Xi_d; sector-dependent B would invalidate it.
  • ad hoc to paper Cyclic Majorana Peirce placement and real-linear projection (Sec. 10.2).
    Gives the CP-safe support (e4,e3,e6) and J_l = 0 with delta_CP = 0 or pi; changes with a complex or identity-mixed bridge.
invented entities (4)
  • Universal chiral-soldering channel B (Peirce complement in J2(O_s))
    purpose: Pairs opposite Spin(3,3) Weyl modules in the Albert cubic; candidate physical Higgs or finite-Dirac channel.
    Exact algebraic component, but the localization to a physical field is a programme postulate with no external prediction.
  • Virtual ac2 node on the down chain
    purpose: Unoccupied node that mediates the 2-3 transition with sin theta2 sin theta3 suppression.
    No independent observable; the amplitude rule is one convention among several.
  • Family kernel Xi_f
    purpose: Dynamical family-space factor that would carry CKM mixing after cancelling the universal B.
    Introduced so that B does not fix mixing; no equation determines Xi_f.
  • Nine-link skeleton of six diagonal and three directed Peirce links
    purpose: Candidate ultraviolet realization of Arkani-Hamed et al. textures with one cycle and pi/2 holonomy.
    Paper states it is a candidate skeleton only if the Yukawa block is linear in projected links; exponentiating the generator is generally dense.

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Cite this review

Pith. "Pith review of The CKM sector of the exceptional-Jordan programme: finite-Dirac mass moduli, two conditional angle estimates, and the weak-to-mass bridge." pith.science (2026). https://pith.science/paper/TCT73BFS

@misc{pith2026260801445,
  author       = {Pith},
  title        = {Pith review of: The CKM sector of the exceptional-Jordan programme: finite-Dirac mass moduli, two conditional angle estimates, and the weak-to-mass bridge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TCT73BFS}},
  note         = {Machine review of arXiv:2608.01445}
}
abstract

The Standard Model does not determine quark masses or the CKM matrix. In the exceptional-Jordan programme, square-root masses occupy short $\operatorname{Sym}^{3}(\mathbf{3})$ chains. Compressing the symmetric-cube lift onto occupied nodes gives a root-mass operator whose square yields the proposed mass ratios, packaging that spectrum and relative left frames in one finite Dirac operator without deriving the frames. Conditional on the transport, virtual-node-amplitude, real-$(2,3)$ and balanced-quadrature choices, the no-fit layer gives $|V_{us}|=0.2371$ ($5.3\%$ high) and $|V_{cb}|=0.0422$ ($0.8\%$ high) at $M_Z$. The relation $|V_{ub}|/|V_{cb}|=\sqrt{m_u/m_c}$ is a factor two low; one complex long edge is fitted. The two balanced orientations give branches $(\varepsilon,\omega)=(0.002079,288.2^\circ)$ and $(0.005358,202.2^\circ)$, so the former $\varepsilon$ ratio is not robust and raw $\omega$ is convention-dependent. For an adopted $F_1$--$F_2$ family embedding, we construct an exact Peirce-changing Albert lift in $\mathfrak f_4(\mathbb C)$, reproducing conjugate up/anti-down transport and $\varphi_{12}=-2\chi$. For an adopted cyclic Majorana placement and real-linear projection, its completion has real $(e_4,e_3,e_6)$ support while the $e_1$ quadrature vanishes; $J_\ell=0$ and $\delta^\ell_{CP}=0$ or $\pi$ are compatibility results in this class. The minimal radial-quartic cyclic truncation has equal-magnitude full-rank extrema or a flat direction; a mixed Albert cubic gives stable alignment only in a chosen three-edge subspace. Six diagonal mass links plus three directed Peirce links would form a connected one-cycle nine-link graph if the Yukawa block is linear in the projected bridge. This is a candidate Arkani-Hamed et al. texture skeleton, not a derivation: family-vacuum selection, chiral projection and right-frame locking remain open.

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Works this paper leans on

44 extracted references · 34 canonical work pages

  1. [1]

    Fermion mass ratios from the exceptional Jordan algebra,

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

    arXiv

  2. [2]

    Fermion mass hierarchies and the exceptional Jordan algebra,

    B. G. Teli and T. P. Singh, “Fermion mass hierarchies and the exceptional Jordan algebra,” arXiv:2605.24866 [hep-ph]

    arXiv

  3. [3]

    Fermion Mixing Matrices and the Exceptional Jordan Algebra

    B. G. Teli and T. P. Singh, “Fermion mixing matrices and the exceptional Jordan algebra,” arXiv:2607.00412 [hep-ph]

    work page Pith review arXiv

  4. [4]

    Leptonic CP conservation and the quark CP phase from octonionic flavor structure,

    B. G. Teli and T. P. Singh, “Leptonic CP conservation and the quark CP phase from octonionic flavor structure,” arXiv:2606.27836 [hep-ph]

    arXiv

  5. [5]

    Unitary symmetry and leptonic decays,

    N. Cabibbo, “Unitary symmetry and leptonic decays,” Phys. Rev. Lett.10, 531 (1963)

    work page 1963

  6. [6]

    CP violation in the renormalizable theory of weak interaction,

    M. Kobayashi and T. Maskawa, “CP violation in the renormalizable theory of weak interaction,” Prog. Theor. Phys.49, 652 (1973)

    work page 1973

  7. [7]

    Weak self-masses, Cabibbo angle, and broken SU(2)×SU(2),

    R. Gatto, G. Sartori and M. Tonin, “Weak self-masses, Cabibbo angle, and broken SU(2)×SU(2),” Phys. Lett. B28, 128 (1968)

    work page 1968

  8. [8]

    Calculating the Cabibbo angle,

    H. Fritzsch, “Calculating the Cabibbo angle,” Phys. Lett. B70, 436 (1977); “Quark masses and flavor mixing,” Nucl. Phys. B155, 189 (1979)

    work page 1977

Show all 44 references
  1. [9]

    The problem of mass,

    S. Weinberg, “The problem of mass,” Trans. New York Acad. Sci.38, 185 (1977)

    1977

  2. [10]

    Discrete flavor symmetries and a formula for the Cabibbo angle,

    F. Wilczek and A. Zee, “Discrete flavor symmetries and a formula for the Cabibbo angle,” Phys. Lett. B70, 418 (1977). 31

    1977

  3. [11]

    A new lepton-quark mass relation in a unified theory,

    H. Georgi and C. Jarlskog, “A new lepton-quark mass relation in a unified theory,” Phys. Lett. B86, 297 (1979)

    1979

  4. [12]

    Nearest neighbor interactions and the physical content of Fritzsch mass matrices,

    G. C. Branco, L. Lavoura and F. Mota, “Nearest neighbor interactions and the physical content of Fritzsch mass matrices,” Phys. Rev. D39, 3443 (1989)

    1989

  5. [13]

    Stitching the Yukawa quilt,

    P. Ramond, R. G. Roberts and G. G. Ross, “Stitching the Yukawa quilt,” Nucl. Phys. B 406, 19 (1993)

    1993

  6. [14]

    A complete survey of texture zeros in general and symmetric quark mass matrices,

    P. O. Ludl and W. Grimus, “A complete survey of texture zeros in general and symmetric quark mass matrices,” Phys. Lett. B744, 38 (2015)

    2015

  7. [15]

    On the four-zero texture of quark mass matrices and its stability,

    Z.-z. Xing and Z.-h. Zhao, “On the four-zero texture of quark mass matrices and its stability,” Nucl. Phys. B897, 302 (2015)

    2015

  8. [16]

    Mass and flavor mixing schemes of quarks and leptons,

    H. Fritzsch and Z.-z. Xing, “Mass and flavor mixing schemes of quarks and leptons,” Prog. Part. Nucl. Phys.45, 1 (2000)

    2000

  9. [17]

    Hierarchy of quark masses, Cabibbo angles and CP violation,

    C. D. Froggatt and H. B. Nielsen, “Hierarchy of quark masses, Cabibbo angles and CP violation,” Nucl. Phys. B147, 277 (1979)

    1979

  10. [18]

    On the generality of certain predictions for quark mixing,

    L. J. Hall and A. Rasin, “On the generality of certain predictions for quark mixing,” Phys. Lett. B315, 164 (1993)

    1993

  11. [19]

    Texture zeros and weak basis transformations,

    G. C. Branco, D. Emmanuel-Costa and R. Gonz´ alez Felipe, “Texture zeros and weak basis transformations,” Phys. Lett. B477, 147 (2000)

    2000

  12. [20]

    Quark structure and octonions,

    M. G¨ unaydin and F. G¨ ursey, “Quark structure and octonions,” J. Math. Phys.14, 1651 (1973)

    1973

  13. [21]

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

    C. Furey, “SU (3)C ×SU (2)L ×U (1)Y (×U (1)X ) as a symmetry of division algebraic ladder operators,” Eur. Phys. J. C78, 375 (2018)

    2018

  14. [22]

    Exceptional quantum geometry and particle physics,

    M. Dubois-Violette, “Exceptional quantum geometry and particle physics,” Nucl. Phys. B 912, 426 (2016); M. Dubois-Violette and I. Todorov, “Exceptional quantum geometry and particle physics II,” Nucl. Phys. B938, 751 (2019)

    2016

  15. [23]

    The standard model, the exceptional Jordan algebra, and triality,

    L. Boyle, “The standard model, the exceptional Jordan algebra, and triality,” J. Math. Phys.67, 071701 (2026), arXiv:2006.16265

    2026 arXiv

  16. [24]

    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 symmetry,” arXiv:2601.07857 [physics.gen-ph]

    arXiv

  17. [25]

    Review of particle physics,

    F. Takahashiet al.(Particle Data Group), “Review of particle physics,” Int. J. Mod. Phys. A41, 2630011 (2026); A. Ceccucci, Z. Ligeti and Y. Sakai, “CKM quark-mixing matrix,” revised March 2026

    2026

  18. [26]

    CP violation and the CKM matrix: assessing the impact of the asymmetricBfactories,

    J. Charleset al.(CKMfitter Group), “CP violation and the CKM matrix: assessing the impact of the asymmetricBfactories,” Eur. Phys. J. C41, 1 (2005)

    2005

  19. [27]

    The 2004 UTfit collaboration report on the status of the unitarity triangle in the standard model,

    M. Bonaet al.(UTfit Collaboration), “The 2004 UTfit collaboration report on the status of the unitarity triangle in the standard model,” JHEP07, 028 (2005)

    2004

  20. [28]

    Spacetime and internal symmetry from split bioctonions and the two extra SU (3)s of E8 ×ωE 8,

    T. P. Singh, “Spacetime and internal symmetry from split bioctonions and the two extra SU (3)s of E8 ×ωE 8,” Preprints.org (2026), manuscript 202510.0437, version 2, doi:10.20944/preprints202510.0437.v2. 32

    2026

  21. [29]

    Gravity and electroweak sector from symmetry breaking of anSO(3,3) BF theory,

    P. S. Wesley, T. P. Singh and J. M. Isidro, “Gravity and electroweak sector from symmetry breaking of anSO(3,3) BF theory,” arXiv:2602.19151v2 [hep-th] (2026)

    2026

  22. [30]

    Towards deriving the Standard Model coupled to gravity from Gener- alized Trace Dynamics via the spectral action principle,

    T. P. Singh, “Towards deriving the Standard Model coupled to gravity from Gener- alized Trace Dynamics via the spectral action principle,” Universe12, 205 (2026), doi:10.3390/universe12070205

    2026 doi

  23. [31]

    Gravity and the standard model with neutrino mixing,

    A. H. Chamseddine, A. Connes and M. Marcolli, “Gravity and the standard model with neutrino mixing,” Adv. Theor. Math. Phys.11, 991 (2007), arXiv:hep-th/0610241

    2007 arXiv

  24. [32]

    CKM matrix parameters from the exceptional Jordan algebra,

    A. A. Patel and T. P. Singh, “CKM matrix parameters from the exceptional Jordan algebra,” Universe9, 440 (2023), arXiv:2305.00668

    2023 arXiv

  25. [33]

    The standard model, the Pati–Salam model, and Jordan geometry,

    L. Boyle and S. Farnsworth, “The standard model, the Pati–Salam model, and Jordan geometry,” New J. Phys.22, 073023 (2020), arXiv:1910.11888 [hep-th]

    2020 arXiv

  26. [34]

    T. A. Springer and F. D. Veldkamp,Octonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Mathematics (Springer, Berlin, 2000)

    2000

  27. [35]

    Fermion masses, mass-mixing and the almost commutative geometry of the Standard Model,

    L. Dabrowski and A. Sitarz, “Fermion masses, mass-mixing and the almost commutative geometry of the Standard Model,” JHEP02, 068 (2019), arXiv:1806.07282 [hep-th]

    2019 arXiv

  28. [36]

    The very nearly right theory of flavor,

    N. Arkani-Hamed, C. Figueiredo, L. J. Hall and C. A. Manzari, “The very nearly right theory of flavor,” arXiv:2607.27315 [hep-ph]

    arXiv

  29. [37]

    Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal CP nonconservation,

    C. Jarlskog, “Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal CP nonconservation,” Phys. Rev. Lett.55, 1039 (1985)

    1985

  30. [38]

    The neutrino sector of the E8 ×ωE 8 octonionic programme: bridge triage, degeneracy lifting, and a TeV-scale seesaw under the desert,

    T. P. Singh, “The neutrino sector of the E8 ×ωE 8 octonionic programme: bridge triage, degeneracy lifting, and a TeV-scale seesaw under the desert,” preprint (2026), Zenodo, doi:10.5281/zenodo.21196194

    2026 doi

  31. [39]

    Right unitarity triangles, stable CP-violating phases and approximate quark- lepton complementarity,

    Z.-z. Xing, “Right unitarity triangles, stable CP-violating phases and approximate quark- lepton complementarity,” Phys. Lett. B679, 111 (2009)

    2009

  32. [40]

    Solving the strong CP problem with discrete symmetries and the right unitarity triangle,

    S. Antusch, M. Holthausen, M. A. Schmidt and M. Spinrath, “Solving the strong CP problem with discrete symmetries and the right unitarity triangle,” Nucl. Phys. B877, 752 (2013)

    2013

  33. [41]

    Nearly right unitarity triangle and CP phase in quark and lepton flavor mixings,

    Y. Mimura, “Nearly right unitarity triangle and CP phase in quark and lepton flavor mixings,” arXiv:1805.07773 [hep-ph] (2018)

    2018 arXiv

  34. [42]

    Unitarity triangle angles explained: a predictive new quark mass matrix texture,

    P. F. Harrison and W. G. Scott, “Unitarity triangle angles explained: a predictive new quark mass matrix texture,” JHEP07, 155 (2025), arXiv:2501.18508

    2025 arXiv

  35. [43]

    Left–right symmetry breaking in E6L ×E 6R occurs only in spacetime—with possible implications for strong CP,

    T. P. Singh, “Left–right symmetry breaking in E6L ×E 6R occurs only in spacetime—with possible implications for strong CP,” arXiv:2607.04132 [hep-ph]

    arXiv

  36. [44]

    Parametrization of the Kobayashi–Maskawa matrix,

    L. Wolfenstein, “Parametrization of the Kobayashi–Maskawa matrix,” Phys. Rev. Lett.51, 1945 (1983). 33

    1945

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Reviewed August 6, 2026 · model on record in the stance chip above.