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The results presented here are based on the concepts of the Cauchy continuum and, the elementary cell at the Planck scale. The symmetrization of quaternion relations and the postulate of quaternion velocity have been crucial in driving significant advancements. They allowed considering the momentum of the expanding Cauchy continuum, \ (u̇₀ (t, x) \). The momentum expansion/compression is the apparent result of the scalar potential of the expansion/compression: \ (₀ (t, x) \). The key new results are listed below: The vectorial \ (G₀ (m) (₀ +) \), \ (G₀ (m) \) and scalar: \ (G₀ (m) ₀, \) \ (G₀ (m) ^*, \) propagators are postulated and used to generate the 2nd order PDE systems for the proton, electron and neutron. The scrupulous assessment of the 2nd order PDE systems allows postulating the two 2nd order PDE systems for the ᵤ_ and d_ quarks from the ᵤp_ and down_ groups. It was verified that both the proton and the neutron obey experimental findings and are formed by three quarks. The proton and neutron are formed by d-u-u_ and d-d-u_ complexes, respectively. All particle systems comply with the Cauchy equation of motion and can be considered as stable particles. The u and d quarks do not meet the Cauchy relations. The inconsistencies of the quarks’ PDE with the quaternion forms of the Cauchy equation of motion account for their lifetime and the observed Quarks Chains. That is, explain the Wilczek phenomenological paradox: “Quarks are Born Free, but Everywhere They are in Chains”. Symmetrizing the variables led to the derivation of the Maxwell’s equations at the macro-scale and the quarks at the Planck scale.
Danielewski et al. (Mon,) studied this question.