Loading article...
Loading article...
Stream/Area:
-
Subject:
-
Paper ID:
IJSREM67145
Volume:
10 |
9 |
Sep 2026
Pages:
1-75
Publication Frequency:
Monthly (12 issues per year)
License:International Journal of Scientific Research in Engineering and Management follows the Creative Commons Attribution 4.0 (CC BY 4.0) License, allowing free use and distribution with proper attribution.
Independent Researcher
Independent Researcher
We define mass as the modulus-squared of a complex rotor — a quantised circulation in a source/sink medium — and show that this definition, together with one axiom, that the imaginary unit acts as an integral rotation of the plane, organises the fermion mass spectrum in a way that is derived where it is derived and labelled as input where it is not. The Koide constant emerges as a scale-invariant angle, so its constancy is geometry and its value a single condition; each of the four fermion sectors is generated by one clock, one phase and one amplitude. The charged-lepton hierarchy angle δ = 0.2222212(20), consistent with 2/9 radians at 0.5σ — an observation due to Brannen and Rosen — and with no rational fraction of a turn, is reached in this work by two roads: a phase law θ = u(1−u) with 1/u = N_c = 3, and the ratio of E₆’s Weyl invariant degrees 2 and 9. The central result of this paper is that the two roads are one. The crystallographic restriction selects the binary tetrahedral group 2T as the largest finite subgroup of SU(2) with only crystallographic element orders; McKay’s correspondence turns 2T into the affine E₆ diagram, on which triality is tensoring with a character of Ab(2T) = Z₃ and N_c is the branch-node mark; deleting the branch node gives the trinification SU(3)³, and E₆ is the unique simple Lie algebra admitting SU(N)^N at maximal rank. The product of the three SU(3) Coxeter elements is the ζ₃-regular element of W(E₆), whose centraliser is the Hessian group G₂₅ = 3^{1+2} ⋊ 2T. The 27 weights of E₆ form a torsor for the Heisenberg group 3^{1+2} and project three-to-one onto the nine inflection points of the Hesse pencil, with quarks, antiquarks and leptons of one generation landing as one of the four Hesse triangles; the phase law’s N_c² and the degree 9 of G₂₅ are the order of the Heisenberg quotient. Every step is verified by deposited scripts. A pre-registered blind protocol over the canonical integer emissions of ℂ⊗𝕆 finds 2/9 as the unique hit among eighteen ratios; a no-go theorem excludes vacuum-alignment origins; a radian theorem licenses reading a dimensionless degree ratio as an angle. Sector clocks fix the scales as inputs, one unification clock at 2×10¹⁶ GeV threading the four sectors. The neutrino sector predicts normal ordering, Σm_ν ≈ 58 meV, and a scale-free ratio Δm²₃₁/Δm²₂₁ = 31.88 against 33.56 ± 0.64 measured, 2.6σ low after JUNO’s first release; JUNO’s forthcoming atmospheric determination decides it. We state what is not derived: χ = π/4, all mass scales, three generations, the breaking of E₆, the gauge couplings, and the quark angles.
KEYWORDS
From a single definition — mass as the modulus-squared of a complex rotor, 𝑚 = (𝑘/2𝑐2) 𝑧𝑧‾, the elastic energy of a trapped circulation, with the phase cancelling so that the modulus is phase-independent (an algebraic property of the norm form, not a dynamical statement) — we have assembled a coherent account of the fermion mass spectrum in which structure is forced by geometry while the scales and the amplitude remain inputs, the phase being derived at the lepton point only. The headline is a bridge from the lightest fermion to the unification scale: the neutrino sector, read in its proper inverse form, places a clock at 2 × 1016 GeV — the grand-unification scale, obtained from the direct inverse-form fit — and a descending chain organises the scales between it and the charged-lepton clock, with the measured nucleon mass inserted at the soliton step; the sub-percent proximity of mN/3 to the lepton clock is the known numerical coincidence, equivalently Σmℓ ≈ 2mN. The results are concrete. The Koide constant is a scale-invariant angle, 𝑄 = 1/(3cos2𝜙), so its constancy is geometry and its value the single condition 𝜙 = 45∘. Each of the four sectors is generated by one clock, one phase, and one amplitude, and every member’s mass is reproduced from that triple (Sec. 19): the charged leptons to 0.01%, the up quarks (heavy masses; light mass scheme-limited) to 0.03%, the down quarks to 0.1%, and the neutrinos exactly in their inverse form. The four clocks organise into two tiers joined by known physics: the neutrino clock is the unification scale itself (2 × 1016 GeV, emergent, coinciding with the supersymmetric unification scale), and the charged-lepton clock descends from it through a single chain — transmutation to 𝛬𝑄𝐶𝐷, the rotor soliton to 𝑚𝑁, the forced 𝑁𝑐 = 3 to the per-rotor share 𝑚𝑁/𝑁𝑐 — reaching 313 MeV to better than half a percent, with the measured nucleon mass inserted at the soliton step alongside the unification scale and its coupling.
Prior work by the author. Chaudhary, R. (2026) Grading the Hydrodynamic Forms of Quantum Mechanics, and the Scale and Basis of the Stochastic Exit, Int. J. Quantum Found., https://ijqf.org/archives/8515. Chaudhary, R. (2026) The Schrödinger Equation as the Shadow of the Rotor Medium, Int. J. Quantum Found. [confirm venue and link]. Chaudhary, R. Iota as Rotation: Unified Thesis v3.1, Zenodo [DOI to add]. Chaudhary, R. Complex Numbers as Rotational Planes, Zenodo [DOI to add]. Chaudhary, R. Rotor Mechanics — Vortex-Flow Formulation v3.1, Zenodo [DOI to add]. Chaudhary, R. Rotor Geometry and Koide (rev. 2), Zenodo [DOI to add]. Chaudhary, R. Renormalization Flow of the Koide Angle v3, Zenodo [DOI to add]. Chaudhary, R. The Collapse Scale as the Deck Fixed Point v3, Zenodo [DOI to add]. Chaudhary, R. Rotors as the Standard Model v14, Zenodo [DOI to add]. Chaudhary, R. From Geometry to Clocks to Scales V8.12 and The Lepton Angle as the Unique Exportable Angle of E₆ v0.8, Zenodo [DOIs to add] — superseded by the present paper. Koide relation and the phase. Koide, Y. (1982) Lett. Nuovo Cim. 34, 201; (1983) Phys. Rev. D 28, 252; (1990) Mod. Phys. Lett. A 5, 2319; (2006) hep-ph/0605074. Brannen, C. A. (2006) The Lepton Masses, brannenworks.com/MASSES2.pdf; APS NW Section Meeting B3, 00014. Rosen, G. (2007) Mod. Phys. Lett. A 22, 283. Żenczykowski, P. (2012) arXiv:1210.4125; (2013) arXiv:1301.4143; (2018) Open Phys. 16, 10.1515/phys-2018- 0058. Foot, R. (1994) hep-ph/9402242. Sumino, Y. (2009) Phys. Lett. B 671, 477. Rivero, A. and Gsponer, A. (2005) hep-ph/0505220. Kocik, J. (2012) arXiv:1201.2067. Singh, T. P. (2020) The characteristic equation of the exceptional Jordan algebra: its eigenvalues, and their relation with the mass ratios of quarks and leptons, arXiv:2006.16274; (2021) Quantum theory without classical time: octonions, and a theoretical derivation of the fine structure constant 1/137, arXiv:2110.07548. Makaryev, D. (2026) Can the Koide formula be derived from topology? and associated preprint, ResearchGate, February 2026. Binary tetrahedral group in physics. Case, K. M., Karplus, R. and Yang, C. N. (1956) Phys. Rev. 101, 874. Frampton, P. H. and Kephart, T. W. (1995) Int. J. Mod. Phys. A 10, 4689. Eby, D. A., Frampton, P. H. and Matsuzaki, S. (2009) arXiv:0907.3425. Eby, D. A. (2013) Binary Tetrahedral Flavor Symmetry, arXiv:1304.4193. Natale, A. (2016) arXiv:1608.06999. Unification and the 27. Glashow, S. L. (1984) in Fifth Workshop on Grand Unification, World Scientific. Babu, K. S., He, X.-G. and Pakvasa, S. (1986) Phys. Rev. D 33, 763. Rosner, J. L. (2014) arXiv:1404.5198. Distler, J. and Garibaldi, S. (2010) Commun. Math. Phys. 298, 419. Mathematics. Hesse, O. (1844) J. reine angew. Math. 28, 68. Maschke, H. (1889) Math. Ann. 33, 317. Coble, A. B. (1929) Algebraic Geometry and Theta Functions, AMS. Shephard, G. C. and Todd, J. A. (1954) Canad. J. Math. 6, 274. Springer, T. A. (1974) Invent. Math. 25, 159. McKay, J. (1980) Proc. Symp. Pure Math. 37, 183. Kronheimer, P. B. (1989) J. Diff. Geom. 29, 665. Douglas, M. R. and Moore, G. (1996) hep-th/9603167. Artebani, M. and Dolgachev, I. (2009) L’Enseignement Math. 55, 235, arXiv:math/0611590. On the geometry of ℂ³/Δ₂₇ and del Pezzo surfaces (2010) arXiv:1001.2893. Borel, A. and de Siebenthal, J. (1949) Comment. Math. Helv. 23, 200. Bourbaki, N., Lie Groups and Lie Algebras, Ch. 4–6. Hurwitz, A. (1896) Nachr. Ges. Wiss. Göttingen, 313. Conway, J. H. and Smith, D. A. (2003) On Quaternions and Octonions, A K Peters. Korkine, A. and Zolotarev, G. (1873) Math. Ann. 6, 366. Musin, O. R. (2008) Ann. Math. 168, 1. Niven, I. (1956) Irrational Numbers, MAA. Physics of the medium. Sadoc, J.-F. and Mosseri, R. (1999) Geometrical Frustration, CUP. Alexander, S. and McTague, J. (1978) Phys. Rev. Lett. 41, 702. Takabayasi, T. (1952) Prog. Theor. Phys. 8, 143. Wallstrom, T. C. (1994) Phys. Rev. A 49, 1613. Abrikosov, A. A. (1957) Sov. Phys. JETP 5, 1174. Data. JUNO Collaboration (2025) First measurement of reactor neutrino oscillations at JUNO, arXiv:2511.14590. Particle Data Group (2024) Phys. Rev. D 110, 030001. Esteban, I. et al., NuFIT 6.x (2025–26), www.nu-fit.org. The remaining references of the two source papers are retained in their sections.
Publication Date
9/7/2026
Peer Review Type
Double-Blind Peer Review
Review Rounds: 2
Documents
Article Views
0
Total views since publication
PDF Downloads
7
Total downloads since publication
Geographic Reach
5
Digital Preservation: This article is permanently archived on Zenodo and complies with all data preservation standards across major indexing platforms.
Plagiarism Screening
This manuscript has been checked for plagiarism using TurnitinPlagiarismChecker software.
Conflict of Interest
The authors declare no conflict of interest related to this research work.
Funding Information
No funding information available for this research.
Related research in the same subject area
Loading similar articles...