From Sources and Sinks to the Standard Model: A Rotor–Division-Algebra Construction of Particle Structure, with a Computational Ledger of What Is Forced and What Is Not

From a single premise—a source, a sink, and the rotor between them—together with the division-algebra tower and its exceptional Jordan closure, a substantial and parameter-free block of the Standard Model’s discrete structure follows: the gauge group and color count, the electric-charge spectrum, the full hypercharge pattern of a generation as a rigid flux-count, spin-1⁄2, one generation, the nucleon as a soliton with a derived stabilizer, and the structural origin of CP violation as a geometric triangle area. Two integers organize the whole picture: 𝑁𝑐 = 3, which appears as the color count, the charge sum, both anomaly traces, and the fermion-statistics sign; and the quaternionic 𝑆𝑈(2), which supplies every 𝜋3 winding, including those inside the octonionic groups. What the geometry does not reach—the QCD scale, the Yukawa sector, the orientation of parity—it fails to reach for reasons we state precisely, and in each case the Standard Model fails identically. The claim is not that geometry is the complete origin of the Standard Model, but that it is the demonstrable origin of its discrete skeleton, with a boundary that is itself a theorem.

E. Wigner, “On an algebraic generalization of the quantum mechanical formalism,” Ann. of Math. 35, 29 (1934). A. A. Albert, “On a certain algebra of quantum mechanics,” Ann. of Math. 35, 65 (1934). M. Dubois-Violette, “Exceptional quantum geometry and particle physics,” Nucl. Phys. B 912, 426 (2016). I. Todorov, M. Dubois-Violette, “Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra,” Int. J. Mod. Phys. A 33, 1850118 (2018). I. Todorov, “Octonions, exceptional Jordan algebra and the role of the group 𝐹4 in particle physics,” Adv. Appl. Clifford Algebras 28, 82 (2018). L. Boyle, “The Standard Model, the exceptional Jordan algebra, and triality,” arXiv:2006.16265 (2020). T. H. R. Skyrme, “A non-linear field theory,” Proc. R. Soc. A 260, 127 (1961). G. S. Adkins, C. R. Nappi, E. Witten, “Static properties of nucleons in the Skyrme model,” Nucl. Phys. B 228, 552 (1983). E. Witten, “Global aspects of current algebra,” Nucl. Phys. B 223, 422 (1983); “Current algebra, baryons, and quark confinement,” ibid. 223, 433 (1983). N. S. Manton, “Geometry of Skyrmions,” Commun. Math. Phys. 111, 469 (1987). J. Goldstone, F. Wilczek, “Fractional quantum numbers on solitons,” Phys. Rev. Lett. 47, 986 (1981). H. Georgi, C. Jarlskog, “A new lepton–quark mass relation in a unified theory,” Phys. Lett. B 86, 297 (1979). Y. Koide, “A new view of quark and lepton mass hierarchy,” Phys. Rev. D 28, 252 (1983). F. Gürsey, P. Ramond, P. Sikivie, “A universal gauge theory model based on 𝐸6,” Phys. Lett. B 60, 177 (1976).