pith. sign in

arxiv: 2606.12693 · v2 · pith:2FRZD4LJnew · submitted 2026-06-10 · ⚛️ physics.gen-ph

Self-Reconstructing Codazzi Defects, mathbb{CP}¹ Quantization, and the Minimal Standard-Model Carrier

Piotr Ogonowski This is my paper

Pith reviewed 2026-06-27 07:21 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords Codazzi defectsCP1 quantizationstandard model carrierself-reconstructionhypercharge normalizationSchur completionone-generation package
0
0 comments X

The pith

A filtered reconstruction of codimension-three Codazzi defects selects the separated support that yields the standard model's global symmetry and one-generation package.

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

The paper sets out a filtered local reconstruction scheme for codimension-three Codazzi defects in four-dimensional Lorentzian branches. The scheme treats the closure defect of the self-reconstruction loop as a lexicographic residual that fixes, in sequence, the projective link, Gauss-local charges, Toeplitz support, determinant carrier, finite shadow, torsor response, and Schur completion. For a worldline defect resolved by a CP1 link, faithful reconstruction of the non-scalar jet types together with CP1 Toeplitz visibility selects the separated support E3 ⊕ E2 and reduces finite visibility to the degree-one line. Once this carrier is fixed, the split top-form condition produces the familiar S(U(3) × U(2))/Z6 global form together with the standard one-generation exterior package, hypercharge normalization, and anomaly cancellation. The remaining steps keep the full Z6 finite shadow, realize its projective-color projection as a boundary torsor, and organize the low-energy sector through a B-L-filtered Schur-Kuranishi completion, treating Yukawa couplings, neutrino masses, mixing, and running as outputs of the completed branch rather than selection inputs.

Core claim

After faithful reconstruction of the Gauss-local charges and CP1 Toeplitz visibility selects the separated support E3 ⊕ E2 and the degree-one line, the split top-form condition supplies the standard global symmetry S(U(3) × U(2))/Z6 and the usual one-generation exterior package with its conventional hypercharge normalization and anomaly checks; the determinant package then serves as the structural comparison layer while the full Z6 finite shadow is retained and the locked low sector is completed by a B-L-filtered Schur-Kuranishi procedure.

What carries the argument

The filtered local reconstruction scheme for codimension-three Codazzi defects, which organizes the closure defect of the self-reconstruction loop as a lexicographic residual fixing the projective link, Gauss-local charges, Toeplitz support, determinant carrier, finite shadow, torsor response, and Schur completion in that order.

If this is right

  • The determinant package functions as the structural comparison layer for the reconstructed carrier.
  • The full Z6 finite shadow is retained and its projective-color projection is realized as a boundary torsor.
  • The locked low-energy sector is organized by a B-L-filtered Schur-Kuranishi completion.
  • Yukawa, neutrino, mixing, running, and contact coefficients are treated as completed-branch data rather than inputs to carrier selection.
  • A scale-free charged-lepton balance residual appears as a Schur-layer diagnostic whose zero-correction form recovers the Koide-type singlet-torsor balance.

Where Pith is reading between the lines

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

  • The same residual structure might be used to derive numerical values for mixing angles and mass ratios directly from the Schur completion rather than fitting them afterward.
  • The finite Schur-tensor datum left by the observed deviation from the Koide balance could be compared with precision lepton-mass data to test whether the reconstruction loop closes consistently at low energies.
  • If the lexicographic ordering of the residual proves stable under small deformations of the Lorentzian branch, the scheme could be extended to higher-generation or multi-family carriers without additional selection rules.

Load-bearing premise

The filtered local reconstruction scheme for codimension-three Codazzi defects organizes the closure defect of the self-reconstruction loop as a lexicographic residual whose entries fix the projective link, Gauss-local charges, Toeplitz support, determinant carrier, finite shadow, torsor response, and Schur completion in that order.

What would settle it

A direct computation showing that the selected E3 ⊕ E2 carrier with degree-one line fails to reproduce the observed hypercharge normalization or the standard anomaly cancellation pattern would falsify the carrier-selection step.

read the original abstract

A filtered local reconstruction scheme is formulated for codimension-three Codazzi defects in four-dimensional Lorentzian branches. The closure defect of the self-reconstruction loop is organized as a lexicographic residual whose entries fix, in order, the projective link, Gauss-local charges, Toeplitz support, determinant carrier, finite shadow, torsor response, and Schur completion. For a worldline defect with resolved link $\mathbb{CP}^1_\Gamma$, the scalar two-jet leaves two principal non-scalar types, $V_1$ and $V_2$. Faithful reconstruction of these Gauss-local charges, together with $\mathbb{CP}^1$ Toeplitz visibility, selects the separated support $E_3\oplus E_2$; reduced finite visibility fixes the degree-one line. After this carrier has been selected, the split top-form condition gives the familiar $S(U(3)\times U(2))/\mathbb Z_6$ global form and the standard one-generation exterior package, with the usual hypercharge normalization and anomaly checks. This determinant package is used as the structural comparison layer for the reconstructed carrier. The remaining construction keeps the full $\mathbb Z_6$ finite shadow, realizes its projective-color projection as a boundary torsor, and organizes the locked low sector by a $B-L$-filtered Schur-Kuranishi completion. Yukawa, neutrino, mixing, running, and contact coefficients are thereby treated as completed-branch data rather than as inputs to the carrier selection. A scale-free charged-lepton balance residual is recorded as a Schur-layer diagnostic; its zero-correction form gives the Koide-type singlet-torsor balance, while the observed deviation is left as a finite Schur-tensor datum.

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

3 major / 1 minor

Summary. The manuscript formulates a filtered local reconstruction scheme for codimension-three Codazzi defects in four-dimensional Lorentzian branches. The closure defect of the self-reconstruction loop is organized as a lexicographic residual whose entries fix, in order, the projective link, Gauss-local charges, Toeplitz support, determinant carrier, finite shadow, torsor response, and Schur completion. For a worldline defect with resolved link CP^1_Γ, faithful reconstruction selects the separated support E3⊕E2; reduced finite visibility fixes the degree-one line. The split top-form condition then yields the S(U(3)×U(2))/Z6 global form with standard hypercharge normalization and anomaly checks. Yukawa, neutrino, mixing, running, and contact coefficients are treated as completed-branch data, with a scale-free charged-lepton balance residual recorded as a Schur-layer diagnostic giving a Koide-type singlet-torsor balance.

Significance. If the reconstruction scheme derives the SM gauge structure, hypercharge normalization, and one-generation package from Codazzi defect closure without built-in selection criteria, the result would offer a novel geometric origin for the Standard Model from Lorentzian defect theory. The framing of parameters as branch data rather than inputs, and the recording of the Koide-type balance as a diagnostic, would be strengths if the derivations are supplied. The absence of explicit steps currently prevents evaluation of these claims.

major comments (3)
  1. [Abstract] Abstract: the central claim that 'after this carrier has been selected, the split top-form condition gives the familiar S(U(3)×U(2))/Z6 global form and the standard one-generation exterior package, with the usual hypercharge normalization and anomaly checks' is asserted without any explicit equations, intermediate steps, or verification showing how the lexicographic residual entries produce these structures. This is load-bearing for the SM-selection claim.
  2. [Abstract] Abstract: the filtered local reconstruction scheme is stated to organize the closure defect 'as a lexicographic residual whose entries fix, in order, the projective link, Gauss-local charges, Toeplitz support, determinant carrier, finite shadow, torsor response, and Schur completion,' yet no derivation from the Codazzi defect closure or self-reconstruction loop is supplied to justify this exact sequence. The ordering directly enables selection of E3⊕E2 and the subsequent SM form.
  3. [Abstract] Abstract: the scheme is presented as selecting the SM structure via the residual, but the use of phrases such as 'familiar' and 'usual' together with the fixed ordering that reproduces the known global form and hypercharge indicates the target is incorporated into the selection criteria, undermining the claim of independent reconstruction.
minor comments (1)
  1. [Abstract] Abstract: terms including CP^1_Γ, V1, V2, E3⊕E2, Toeplitz visibility, Schur-Kuranishi completion, and B-L-filtered Schur-Kuranishi completion are introduced without definition or cross-reference, impeding assessment of the construction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for greater explicitness in the abstract's presentation of the reconstruction scheme. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'after this carrier has been selected, the split top-form condition gives the familiar S(U(3)×U(2))/Z6 global form and the standard one-generation exterior package, with the usual hypercharge normalization and anomaly checks' is asserted without any explicit equations, intermediate steps, or verification showing how the lexicographic residual entries produce these structures. This is load-bearing for the SM-selection claim.

    Authors: We agree that the abstract, constrained by length, presents the outcome without the intermediate equations. The full manuscript derives the split top-form condition from the residual entries via the self-reconstruction loop and CP^1 visibility, but to improve transparency we will expand the abstract or insert a short derivation paragraph in the revision. revision: yes

  2. Referee: [Abstract] Abstract: the filtered local reconstruction scheme is stated to organize the closure defect 'as a lexicographic residual whose entries fix, in order, the projective link, Gauss-local charges, Toeplitz support, determinant carrier, finite shadow, torsor response, and Schur completion,' yet no derivation from the Codazzi defect closure or self-reconstruction loop is supplied to justify this exact sequence. The ordering directly enables selection of E3⊕E2 and the subsequent SM form.

    Authors: The sequence follows from the hierarchical closure of the Codazzi defect in the Lorentzian branch, with each entry fixed by the preceding geometric data (projective link first, then Gauss charges, etc.). We will add an explicit derivation of this ordering from the self-reconstruction loop in the revised version. revision: yes

  3. Referee: [Abstract] Abstract: the scheme is presented as selecting the SM structure via the residual, but the use of phrases such as 'familiar' and 'usual' together with the fixed ordering that reproduces the known global form and hypercharge indicates the target is incorporated into the selection criteria, undermining the claim of independent reconstruction.

    Authors: The selection criteria are fixed by the CP^1 Toeplitz visibility and faithful reconstruction of the Gauss-local charges for the resolved link; they are not chosen to match the SM. The adjectives 'familiar' and 'usual' describe the resulting form, not the input criteria. We will revise the wording to make the independence of the geometric selection clearer. revision: partial

Circularity Check

1 steps flagged

Lexicographic residual ordering imposed to select Standard Model carrier by construction

specific steps
  1. self definitional [Abstract]
    "The closure defect of the self-reconstruction loop is organized as a lexicographic residual whose entries fix, in order, the projective link, Gauss-local charges, Toeplitz support, determinant carrier, finite shadow, torsor response, and Schur completion. ... After this carrier has been selected, the split top-form condition gives the familiar S(U(3)×U(2))/Z6 global form and the standard one-generation exterior package, with the usual hypercharge normalization and anomaly checks."

    The scheme is formulated such that its residual is organized in a specific order whose entries fix the quantities that select E3⊕E2 and trigger the split top-form reproducing the known SM form; the ordering and 'faithful reconstruction' criteria are defined to yield this output by construction, with no independent principle shown forcing this sequence over alternatives.

full rationale

The paper's central derivation begins by formulating a filtered local reconstruction scheme whose closure defect is explicitly organized as a lexicographic residual with a fixed entry order that determines the projective link, Gauss-local charges, Toeplitz support, determinant carrier, and subsequent selections. This ordering directly produces the E3⊕E2 support, degree-one line, and split top-form condition that yields the familiar S(U(3)×U(2))/Z6 with usual hypercharge and anomaly structure. Because the scheme is defined to organize the residual in precisely this sequence, the reproduction of the known SM global form and one-generation package reduces to the input formulation of the scheme rather than an independent geometric derivation from Codazzi defect closure. The text acknowledges the output as 'familiar' and 'usual' while treating Yukawa coefficients as post-selection data, confirming the target is built into the selection criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies insufficient detail to enumerate free parameters, axioms, or invented entities; the scheme relies on numerous specialized geometric objects whose independence from the target SM result cannot be audited.

pith-pipeline@v0.9.1-grok · 5852 in / 1394 out tokens · 28628 ms · 2026-06-27T07:21:34.098209+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

125 extracted references · 118 canonical work pages · 3 internal anchors

  1. [1]

    Lett ers in Mathematical Physics 112, 95 (2022) https://doi.org/10.1007/s11005-022-01591-6 arXiv:2201.01981 [math.DG]

    H´ elein, F.: Dynamical mechanisms for kaluza–klein theories. Lett ers in Mathematical Physics 112, 95 (2022) https://doi.org/10.1007/s11005-022-01591-6 arXiv:2201.01981 [math.DG]

    work page doi:10.1007/s11005-022-01591-6 2022

  2. [2]

    European Journal of Phy sics 36(2), 025018 (2015) https://doi.org/10.1088/0143-0807/36/2/025018 arXiv:1503.07 802 [math-ph]

    Cariglia, M., Alves, F.K.: The eisenhart lift: a didactical introduction of modern geometri- cal concepts from hamiltonian dynamics. European Journal of Phy sics 36(2), 025018 (2015) https://doi.org/10.1088/0143-0807/36/2/025018 arXiv:1503.07 802 [math-ph]

    work page doi:10.1088/0143-0807/36/2/025018 2015

  3. [3]

    EPL 133(2), 21002 (2021) https://doi.org/10.1209/0295-5075/133/21002 arXiv:2009.0380 6 [hep-th]

    Silva, J.E.G.: A field theory in randers–finsler spacetime. EPL 133(2), 21002 (2021) https://doi.org/10.1209/0295-5075/133/21002 arXiv:2009.0380 6 [hep-th]

    work page doi:10.1209/0295-5075/133/21002 2021

  4. [4]

    https://arxiv.org/abs/2601.13491

    Czuchry, E., Gazeau, J.-P.: Finite-resolution measurement induc es topological curvature defects in spacetime (2026). https://arxiv.org/abs/2601.13491

    Pith/arXiv arXiv 2026

  5. [5]

    Physica Scripta 100(1), 015018 (2024) https://doi.org/10.1088/1402-4896/ad98ca

    Ogonowski, P., Skindzier, P.: Alena tensor in unification applications . Physica Scripta 100(1), 015018 (2024) https://doi.org/10.1088/1402-4896/ad98ca

    work page doi:10.1088/1402-4896/ad98ca 2024

  6. [6]

    not so dark with alena tensor

    Ogonowski, P.: The halo effect and quantum vortices. not so dark with alena tensor. Physica Scripta 101(15), 155002 (2026) https://doi.org/10.1088/1402-4896/ae59ca

    work page doi:10.1088/1402-4896/ae59ca 2026

  7. [7]

    Volume 1: Two-S pinor Calculus and Relativistic Fields

    Penrose, R., Rindler, W.: Spinors and Space-Time. Volume 1: Two-S pinor Calculus and Relativistic Fields. Cambridge University Press, Cambridge (1984). https://doi.org/10.1017/CBO9780511564048

    work page doi:10.1017/cbo9780511564048 1984

  8. [8]

    Raymond O.: Twistor Geometry and Field Theory

    Ward, R.S., Wells, J. Raymond O.: Twistor Geometry and Field Theory . Cambridge University Press, Cambridge (1990). https://doi.org/10.1017/CBO9780511524493

    work page doi:10.1017/cbo9780511524493 1990

  9. [9]

    https://doi.org/10.48550/arXiv.2104.05099

    Woit, P.: Euclidean Twistor Unification (2021). https://doi.org/10.48550/arXiv.2104.05099 . https://arxiv.org/abs/2104.05099

    work page doi:10.48550/arxiv.2104.05099 2021

  10. [10]

    , year =

    Bott, R.: Homogeneous vector bundles. Annals of Mathematics 66(2), 203–248 (1957) https://doi.org/10.2307/1969996

    work page doi:10.2307/1969996 1957

  11. [11]

    Nuclear Physics B 107(3), 365–380 (1976) https://doi.org/10.1016/0550-3213(76)90143-7

    Wu, T.T., Yang, C.N.: Dirac monopole without strings: Monopole har monics. Nuclear Physics B 107(3), 365–380 (1976) https://doi.org/10.1016/0550-3213(76)90143-7

    work page doi:10.1016/0550-3213(76)90143-7 1976

  12. [12]

    Graduate Texts in Mathematics, vol

    Fulton, W., Harris, J.: Representation Theory: A First Course. Graduate Texts in Mathematics, vol

  13. [13]

    Harris, J

    Springer, New York, NY (1991). https://doi.org/10.1007/978-1-4612-0979-9

    work page doi:10.1007/978-1-4612-0979-9 1991

  14. [14]

    Communications in Mathematical Physics 23(3), 199–230 (1971) https://doi.org/10.1007/BF01877742 40

    Doplicher, S., Haag, R., Roberts, J.E.: Local observables and pa rticle statistics I. Communications in Mathematical Physics 23(3), 199–230 (1971) https://doi.org/10.1007/BF01877742 40

    work page doi:10.1007/bf01877742 1971

  15. [15]

    Communications in Mathematical Physics 165(2), 281–296 (1994) https://doi.org/10.1007/BF02099772

    Bordemann, M., Meinrenken, E., Schlichenmaier, M.: Toeplitz quan tization of k¨ ahler manifolds and gl( n), n → ∞ limits. Communications in Mathematical Physics 165(2), 281–296 (1994) https://doi.org/10.1007/BF02099772

    work page doi:10.1007/bf02099772 1994

  16. [16]

    Nuclear Physics B 600(3), 531–547 (2001) https://doi.org/10.1016/S0550-3213(00)00743-4

    Alexanian, G., Pinzul, A., Stern, A.: Generalized coherent state a pproach to star prod- ucts and applications to the fuzzy sphere. Nuclear Physics B 600(3), 531–547 (2001) https://doi.org/10.1016/S0550-3213(00)00743-4

    work page doi:10.1016/s0550-3213(00)00743-4 2001

  17. [17]

    Bu lletin of the American Mathematical Society 47(3), 483–552 (2010) https://doi.org/10.1090/S0273-0979-10-01294-2

    Baez, J.C., Huerta, J.: The algebra of Grand Unified Theories. Bu lletin of the American Mathematical Society 47(3), 483–552 (2010) https://doi.org/10.1090/S0273-0979-10-01294-2

    work page doi:10.1090/s0273-0979-10-01294-2 2010

  18. [18]

    Advances in Applied Clifford Algebras 28(3), 52 (2018) https://doi.org/10.1007/s00006-018-0869-4

    Stoica, O.C.: Leptons, quarks, and gauge from the complex cliffo rd algebra Cℓ6. Advances in Applied Clifford Algebras 28(3), 52 (2018) https://doi.org/10.1007/s00006-018-0869-4

    work page doi:10.1007/s00006-018-0869-4 2018

  19. [19]

    The European Physical Journal C 83(8), 747 (2023) https://doi.org/10.1140/epjc/s10052-023-11923-y

    Gresnigt, N.G., Gourlay, L., Varma, A.: Three generations of colo red fermions with S3 family symmetry from Cayley–Dickson sedenions. The European Physical Journal C 83(8), 747 (2023) https://doi.org/10.1140/epjc/s10052-023-11923-y

    work page doi:10.1140/epjc/s10052-023-11923-y 2023

  20. [20]

    Frontiers in Physics 11, 1264925 (2023) https://doi.org/10.3389/fphy.2023.1264925

    Ogonowski, P.: Developed method: interactions and their quant um picture. Frontiers in Physics 11, 1264925 (2023) https://doi.org/10.3389/fphy.2023.1264925

    work page doi:10.3389/fphy.2023.1264925 2023

  21. [21]

    International Journal of Modern Physics D 32(03), 2350010 (2023) https://doi.org/10.1142/S0218271823500104

    Ogonowski, P.: Proposed method of combining continuum mechan ics with einstein field equations. International Journal of Modern Physics D 32(03), 2350010 (2023) https://doi.org/10.1142/S0218271823500104

    work page doi:10.1142/s0218271823500104 2023

  22. [22]

    Physica Scripta 100(10), 105026 (2025) https://doi.org/10.1088/1402-4896/ae12e2

    Ogonowski, P.: Gravitational waves and higgs-like potential fro m alena tensor. Physica Scripta 100(10), 105026 (2025) https://doi.org/10.1088/1402-4896/ae12e2

    work page doi:10.1088/1402-4896/ae12e2 2025

  23. [23]

    Co mmunications in Mathematical Physics 62(3), 213–234 (1978) https://doi.org/10.1007/BF01202525

    Callias, C.: Axial anomalies and index theorems on open spaces. Co mmunications in Mathematical Physics 62(3), 213–234 (1978) https://doi.org/10.1007/BF01202525

    work page doi:10.1007/bf01202525 1978

  24. [24]

    Geometric an d Functional Analysis 3(5), 431–438 (1993) https://doi.org/10.1007/BF01896237

    Anghel, N.: On the index of Callias-type operators. Geometric an d Functional Analysis 3(5), 431–438 (1993) https://doi.org/10.1007/BF01896237

    work page doi:10.1007/bf01896237 1993

  25. [25]

    Classics in Mathematics

    Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin Heidelberg (1995). https://doi.org/10.1007/978-3-642-66282-9

    work page doi:10.1007/978-3-642-66282-9 1995

  26. [26]

    Quantal phase factors accompanying adiabatic changes,

    Berry, M.V.: Quantal Phase Factors Accompanying Adiabatic Ch anges. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 392(1802), 45–57 (1984) https://doi.org/10.1098/rspa.1984.0023

    work page doi:10.1098/rspa.1984.0023 1984

  27. [27]

    Physical Review Letters 52(24), 2111–2114 (1984) https://doi.org/10.1103/PhysRevLett.52.2111

    Wilczek, F., Zee, A.: Appearance of Gauge Structure in Simple Dyn amical Systems. Physical Review Letters 52(24), 2111–2114 (1984) https://doi.org/10.1103/PhysRevLett.52.2111

    work page doi:10.1103/physrevlett.52.2111 1984

  28. [28]

    Bismut, J.-M., Freed, D.S.: The analysis of elliptic families. II. Dirac o perators, eta invariants, and the holonomy theorem. Communications in Mathematical Physics 107(1), 103–163 (1986) https://doi.org/10.1007/BF01206955

    work page doi:10.1007/bf01206955 1986

  29. [29]

    Physical Re view Letters 10(12), 531–533 (1963) https://doi.org/10.1103/PhysRevLett.10.531

    Cabibbo, N.: Unitary symmetry and leptonic decays. Physical Re view Letters 10(12), 531–533 (1963) https://doi.org/10.1103/PhysRevLett.10.531

    work page doi:10.1103/physrevlett.10.531 1963

  30. [30]

    Progress of Theoretical Physics 49(2), 652–657 (1973) https://doi.org/10.1143/PTP.49.652

    Kobayashi, M., Maskawa, T.: CP-violation in the renormalizable the ory of weak interaction. Progress of Theoretical Physics 49(2), 652–657 (1973) https://doi.org/10.1143/PTP.49.652

    work page doi:10.1143/ptp.49.652 1973

  31. [31]

    Physical Review Letters 51(21), 1945–1947 (1983) https://doi.org/10.1103/PhysRevLett.51.1945

    Wolfenstein, L.: Parametrization of the Kobayashi-Maskawa ma trix. Physical Review Letters 51(21), 1945–1947 (1983) https://doi.org/10.1103/PhysRevLett.51.1945

    work page doi:10.1103/physrevlett.51.1945 1945

  32. [32]

    Physical Review Letter s 55(10), 1039–1042 (1985) https://doi.org/10.1103/PhysRevLett.55.1039

    Jarlskog, C.: Commutator of the quark mass matrices in the sta ndard electroweak model and a measure of maximal CP nonconservation. Physical Review Letter s 55(10), 1039–1042 (1985) https://doi.org/10.1103/PhysRevLett.55.1039

    work page doi:10.1103/physrevlett.55.1039 1985

  33. [33]

    and others

    Navas, S., et al. : Review of Particle Physics. Physical Review D 110, 030001 (2024) https://doi.org/10.1103/PhysRevD.110.030001 . 2025 update 41

    work page doi:10.1103/physrevd.110.030001 2024

  34. [34]

    Physical Review D 28(1), 252–254 (1983) https://doi.org/10.1103/PhysRevD.28.252

    Koide, Y.: New view of quark and lepton mass hierarchy. Physical Review D 28(1), 252–254 (1983) https://doi.org/10.1103/PhysRevD.28.252

    work page doi:10.1103/physrevd.28.252 1983

  35. [35]

    Geometry & Topology 14(2), 903–966 (2010) https://doi.org/10.2140/gt.2010.14.903

    Freed, D.S., Lott, J.: An index theorem in differential K-theory. Geometry & Topology 14(2), 903–966 (2010) https://doi.org/10.2140/gt.2010.14.903

    work page doi:10.2140/gt.2010.14.903 2010

  36. [36]

    Topolo gy 24(1), 89–95 (1985) https://doi.org/10.1016/0040-9383(85)90047-3

    Quillen, D.: Superconnections and the chern character. Topolo gy 24(1), 89–95 (1985) https://doi.org/10.1016/0040-9383(85)90047-3

    work page doi:10.1016/0040-9383(85)90047-3 1985

  37. [37]

    Classics in Mathematics

    Besse, A.L.: Einstein Manifolds. Classics in Mathematics. Springer , Berlin Heidelberg (2007). https://doi.org/10.1007/978-3-540-74311-8

    work page doi:10.1007/978-3-540-74311-8 2007

  38. [38]

    a review of results

    Schlichenmaier, M.: Berezin-Toeplitz quantization for compact K ¨ ahler manifolds. a review of results. Advances in Mathematical Physics 2010, 927280 (2010) https://doi.org/10.1155/2010/927280

    work page doi:10.1155/2010/927280 2010

  39. [39]

    Proceedings of the London Mathematical Society 47(1), 15–26 (1983) https://doi.org/10.1112/plms/s3-47.1.15

    Derdzi´ nski, A., Shen, C.-L.: Codazzi tensor fields, curvatur e and pontryagin forms. Proceedings of the London Mathematical Society 47(1), 15–26 (1983) https://doi.org/10.1112/plms/s3-47.1.15

    work page doi:10.1112/plms/s3-47.1.15 1983

  40. [40]

    General Relativity and Gravitation 55(4), 62 (2023) https://doi.org/10.1007/s10714-023-03106-7

    Mantica, C.A., Molinari, L.G.: Codazzi tensors and their space-tim es, and cotton gravity. General Relativity and Gravitation 55(4), 62 (2023) https://doi.org/10.1007/s10714-023-03106-7

    work page doi:10.1007/s10714-023-03106-7 2023

  41. [41]

    General Relativity and Gravitation 39, 343–359 (2007) https://doi.org/10.1007/s10714-006-0388-9

    Ferrando, J.J., S´ aez, J.A.: On the space-times admitting two sh ear-free geodesic null congruences. General Relativity and Gravitation 39, 343–359 (2007) https://doi.org/10.1007/s10714-006-0388-9

    work page doi:10.1007/s10714-006-0388-9 2007

  42. [42]

    Volume 2: Spino r and Twistor Methods in Space-Time Geometry

    Penrose, R., Rindler, W.: Spinors and Space-Time. Volume 2: Spino r and Twistor Methods in Space-Time Geometry. Cambridge University Press, Cam bridge (1986). https://doi.org/10.1017/CBO9780511524486

    work page doi:10.1017/cbo9780511524486 1986

  43. [43]

    Commun ications in Mathematical Physics 83(1), 11–29 (1982) https://doi.org/10.1007/BF01947068

    Uhlenbeck, K.K.: Removable singularities in yang–mills fields. Commun ications in Mathematical Physics 83(1), 11–29 (1982) https://doi.org/10.1007/BF01947068

    work page doi:10.1007/bf01947068 1982

  44. [44]

    and Trudinger, N.S

    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations o f Second Order, 2nd edn. Classics in Mathematics. Springer, Berlin (2001). https://doi.org/10.1007/978-3-642-61798-0 . Reprint of the 1998 edition

    work page doi:10.1007/978-3-642-61798-0 2001

  45. [45]

    Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K.: Quantum The ory of Angular Momentum, p

  46. [46]

    https://doi.org/10.1142/0270

    World Scientific, Singapore (1988). https://doi.org/10.1142/0270

    work page doi:10.1142/0270 1988

  47. [47]

    Re views of Modern Physics 52(2), 299–339 (1980) https://doi.org/10.1103/RevModPhys.52.299

    Thorne, K.S.: Multipole expansions of gravitational radiation. Re views of Modern Physics 52(2), 299–339 (1980) https://doi.org/10.1103/RevModPhys.52.299

    work page doi:10.1103/revmodphys.52.299 1980

  48. [48]

    Communications in Mat hematical Physics 131(1), 51–107 (1990) https://doi.org/10.1007/BF02097680

    Doplicher, S., Roberts, J.E.: Why there is a field algebra with a comp act gauge group describing the superselection structure in particle physics. Communications in Mat hematical Physics 131(1), 51–107 (1990) https://doi.org/10.1007/BF02097680

    work page doi:10.1007/bf02097680 1990

  49. [49]

    Journal of High Energy Physics 2014(1), 114 (2014) https://doi.org/10.1007/JHEP01(2014)114

    Jante, R., Schroers, B.J.: Dirac operators on the taub–nut sp ace, monopoles and SU(2) representations. Journal of High Energy Physics 2014(1), 114 (2014) https://doi.org/10.1007/JHEP01(2014)114

    work page doi:10.1007/jhep01(2014)114 2014

  50. [50]

    Branes, Quantization and Fuzzy Spheres

    S¨ amann, C., Szabo, R.J.: Branes, quantization and fuzzy sphe res. Proceedings of Science CNCFG2010, 005 (2010) https://doi.org/10.22323/1.127.0005 arXiv:1101.5987 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.22323/1.127.0005 2010

  51. [51]

    Journal of High Energy Physics 2013(8), 115 (2013) https://doi.org/10.1007/JHEP08(2013)115

    Aharony, O., Seiberg, N., Tachikawa, Y.: Reading between the line s of four-dimensional gauge theories. Journal of High Energy Physics 2013(8), 115 (2013) https://doi.org/10.1007/JHEP08(2013)115

    work page doi:10.1007/jhep08(2013)115 2013

  52. [52]

    The European Physical Journa l C 80(6), 583 (2020) https://doi.org/10.1140/epjc/s10052-020-8141-1

    Gresnigt, N.G.: The Standard Model particle content with comple te gauge symmetries from the minimal ideals of two clifford algebras. The European Physical Journa l C 80(6), 583 (2020) https://doi.org/10.1140/epjc/s10052-020-8141-1

    work page doi:10.1140/epjc/s10052-020-8141-1 2020

  53. [53]

    Journal of High Energy Physics 2021(4), 164 (2021) https://doi.org/10.1007/JHEP04(2021)164

    Todorov, I.: Superselection of the weak hypercharge and the algebra of the Standard Model. Journal of High Energy Physics 2021(4), 164 (2021) https://doi.org/10.1007/JHEP04(2021)164

    work page doi:10.1007/jhep04(2021)164 2021

  54. [54]

    The European Physical Journal C 78(5), 375 (2018) https://doi.org/10.1140/epjc/s10052-018-5844-7 42

    Furey, C.: 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), 375 (2018) https://doi.org/10.1140/epjc/s10052-018-5844-7 42

    work page doi:10.1140/epjc/s10052-018-5844-7 2018

  55. [55]

    Ph ysics Letters B 831, 137186 (2022) https://doi.org/10.1016/j.physletb.2022.137186

    Furey, N., Hughes, M.J.: Division algebraic symmetry breaking. Ph ysics Letters B 831, 137186 (2022) https://doi.org/10.1016/j.physletb.2022.137186

    work page doi:10.1016/j.physletb.2022.137186 2022

  56. [56]

    Annalen de r Physik 537(4), 2400322 (2025) https://doi.org/10.1002/andp.202400322

    Furey, N.: An algebraic roadmap of particle theories. Annalen de r Physik 537(4), 2400322 (2025) https://doi.org/10.1002/andp.202400322

    work page doi:10.1002/andp.202400322 2025

  57. [57]

    Princeton Un iversity Press, Princeton, NJ (1974)

    Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Princeton Un iversity Press, Princeton, NJ (1974). https://doi.org/10.1515/9781400881826

    work page doi:10.1515/9781400881826 1974

  58. [58]

    Annals of Mathematics 99(1), 48–69 (1974) https://doi.org/10.2307/1971013

    Chern, S.-S., Simons, J.: Characteristic forms and geometric inv ariants. Annals of Mathematics 99(1), 48–69 (1974) https://doi.org/10.2307/1971013

    work page doi:10.2307/1971013 1974

  59. [59]

    Jackiw, R., Rebbi, C.: Solitons with fermion number 1/2. Phys. Rev . D 13(12), 3398–3409 (1976) https://doi.org/10.1103/PhysRevD.13.3398

    work page doi:10.1103/physrevd.13.3398 1976

  60. [60]

    Physics Letters B 288(3-4), 342–347 (1992) https://doi.org/10.1016/0370-2693(92)91112-M arXiv:hep-lat/ 9206013 [hep-lat]

    Kaplan, D.B.: A method for simulating chiral fermions on the lattice . Physics Letters B 288(3-4), 342–347 (1992) https://doi.org/10.1016/0370-2693(92)91112-M arXiv:hep-lat/ 9206013 [hep-lat]

    work page doi:10.1016/0370-2693(92)91112-m 1992

  61. [61]

    , TITLE =

    Federer, H., Fleming, W.H.: Normal and integral currents. Anna ls of Mathematics 72(3), 458–520 (1960) https://doi.org/10.2307/1970227

    work page doi:10.2307/1970227 1960

  62. [62]

    Proceeding s of the Centre for Mathematical Anal- ysis, Australian National University, vol

    Simon, L.: Lectures on Geometric Measure Theory. Proceeding s of the Centre for Mathematical Anal- ysis, Australian National University, vol. 3. Australian National Univ ersity, Centre for Mathematical Analysis, Canberra (1983)

    1983

  63. [63]

    Pr ogress in Nonlinear Differ- ential Equations and Their Applications, vol

    Bethuel, F., Brezis, H., H’elein, F.: Ginzburg-Landau Vortices. Pr ogress in Nonlinear Differ- ential Equations and Their Applications, vol. 13. Birkh”auser Bosto n, Boston, MA (1994). https://doi.org/10.1007/978-1-4612-0287-5

    work page doi:10.1007/978-1-4612-0287-5 1994

  64. [64]

    Calculus of Variations and Partial Differential Equations 14(2), 151–191 (2002) https://doi.org/10.1007/s005260100093

    Jerrard, R.L., Soner, H.M.: The jacobian and the Ginzburg-Land au energy. Calculus of Variations and Partial Differential Equations 14(2), 151–191 (2002) https://doi.org/10.1007/s005260100093

    work page doi:10.1007/s005260100093 2002

  65. [65]

    In: SEESA W 2 5: Proceedings of the Inter- national Conference on the Seesaw Mechanism, pp

    Senjanovi´ c, G.: See-saw and grand unification. In: SEESA W 2 5: Proceedings of the Inter- national Conference on the Seesaw Mechanism, pp. 45–64. World S cientific, ??? (2005). https://doi.org/10.1142/9789812702210 0004

    work page doi:10.1142/9789812702210 2005

  66. [66]

    i: The higgs boson in the standard model

    Djouadi, A.: The anatomy of electro-weak symmetry breaking. i: The higgs boson in the standard model. Physics Reports 457(1–4), 1–216 (2008) https://doi.org/10.1016/j.physrep.2007.10.004

    work page doi:10.1016/j.physrep.2007.10.004 2008

  67. [67]

    In: SUS Y09: The 17th International Conference on Supersymmetry and the Unification of Fundamental Interactio ns

    Senjanovi´ c, G.: Proton decay and grand unification. In: SUS Y09: The 17th International Conference on Supersymmetry and the Unification of Fundamental Interactio ns. AIP Conference Proceedings, vol. 1200, pp. 131–141. AIP, ??? (2010). https://doi.org/10.1063/1.3327552

    work page doi:10.1063/1.3327552 2010

  68. [68]

    Nuclear Physics B - Proceedings Supplements 18(2), 29–47 (1991) https://doi.org/10.1016/0920-5632(91)90120-4

    Connes, A., Lott, J.: Particle models and noncommutative geome try. Nuclear Physics B - Proceedings Supplements 18(2), 29–47 (1991) https://doi.org/10.1016/0920-5632(91)90120-4

    work page doi:10.1016/0920-5632(91)90120-4 1991

  69. [69]

    The Spectral Action Principle

    Chamseddine, A.H., Connes, A.: The spectral action principle. Co mmunications in Mathematical Physics 186(3), 731–750 (1997) https://doi.org/10.1007/s002200050126 arXiv:hep-th/9606001 [hep- th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s002200050126 1997

  70. [70]

    European Physical Journal Special Topics 232, 3581–3588 (2023) https://doi.org/10.1140/epjs/s11734-023-00842-4

    Chamseddine, A.H., Connes, A., van Suijlekom, W.D.: Noncommutat ivity and physics: A non-technical review. European Physical Journal Special Topics 232, 3581–3588 (2023) https://doi.org/10.1140/epjs/s11734-023-00842-4

    work page doi:10.1140/epjs/s11734-023-00842-4 2023

  71. [71]

    Journal of K-Theory 8(3), 387–417 (2011) https://doi.org/10.1017/is010011014jkt132

    Kottke, C.: An index theorem of Callias type for pseudodifferent ial operators. Journal of K-Theory 8(3), 387–417 (2011) https://doi.org/10.1017/is010011014jkt132

    work page doi:10.1017/is010011014jkt132 2011

  72. [72]

    T ransactions of the American Mathematical Society 27(1), 106–136 (1925) https://doi.org/10.1090/S0002-9947-1925-1501302-6

    Rainich, G.Y.: Electrodynamics in the general relativity theory. T ransactions of the American Mathematical Society 27(1), 106–136 (1925) https://doi.org/10.1090/S0002-9947-1925-1501302-6

    work page doi:10.1090/s0002-9947-1925-1501302-6 1925

  73. [73]

    General Relativity and Gravitation 39, 2039–2051 (2007) https://doi.org/10.1007/s10714-007-0500-9 43

    Ferrando, J.J., S´ aez, J.A.: Rainich theory for type D aligned eins tein–maxwell solutions. General Relativity and Gravitation 39, 2039–2051 (2007) https://doi.org/10.1007/s10714-007-0500-9 43

    work page doi:10.1007/s10714-007-0500-9 2039

  74. [74]

    Classical and Quantum Gra vity 34(22), 225014 (2017) https://doi.org/10.1088/1361-6382/aa9039

    Mart ´ ın-Moruno, P., Visser, M.: Generalized rainich conditions, g eneralized stress-energy condi- tions, and the hawking–ellis classification. Classical and Quantum Gra vity 34(22), 225014 (2017) https://doi.org/10.1088/1361-6382/aa9039

    work page doi:10.1088/1361-6382/aa9039 2017

  75. [75]

    Journal of Mathematical Physics 18(12), 2511–2520 (1977) https://doi.org/10.1063/1.523215

    Pleba´ nski, J.F.: On the separation of einsteinian substructure s. Journal of Mathematical Physics 18(12), 2511–2520 (1977) https://doi.org/10.1063/1.523215

    work page doi:10.1063/1.523215 1977

  76. [76]

    Physical Review Letters 106(25), 251103 (2011) https://doi.org/10.1103/PhysRevLett.106.251103

    Krasnov, K.: Pure connection action principle for general relat ivity. Physical Review Letters 106(25), 251103 (2011) https://doi.org/10.1103/PhysRevLett.106.251103

    work page doi:10.1103/physrevlett.106.251103 2011

  77. [77]

    Krasnov, K.: A gauge-theoretic approach to gravity. Proc. R . Soc. A 468(2144), 2129–2173 (2012) https://doi.org/10.1098/rspa.2011.0638

    work page doi:10.1098/rspa.2011.0638 2012

  78. [78]

    Cambridge University Press, Cambridge (2020)

    Krasnov, K.: Formulations of General Relativity: Gravity, Spino rs and Differential Forms. Cambridge University Press, Cambridge (2020). https://doi.org/10.1017/9781108772723

    work page doi:10.1017/9781108772723 2020

  79. [79]

    Journal o f High Energy Physics 2021(8), 082 (2021) https://doi.org/10.1007/JHEP08(2021)082

    Krasnov, K., Skvortsov, E.: Flat self-dual gravity. Journal o f High Energy Physics 2021(8), 082 (2021) https://doi.org/10.1007/JHEP08(2021)082

    work page doi:10.1007/jhep08(2021)082 2021

  80. [80]

    Fermions, differential forms and doubled geometry

    Krasnov, K.: Fermions, differential forms and doubled geometr y. Nuclear Physics B 936, 36–75 (2018) https://doi.org/10.1016/j.nuclphysb.2018.09.006 arXiv:1803.06160 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysb.2018.09.006 2018

Showing first 80 references.