Magnetism, Gravity and Inversion Breaking in Four Dimensions (INDYNA Research Note 42)

Abstract

In the INDYNA vertex action the drift sector (spin 1, magnetism) and the form sector (spin 2, gravitation) of a radius field couple statically only through a lock-in deformation. In three dimensions the cube lock-in xyz provides this coupling. We prove that in four dimensions every radius field that is even under the antipodal map u↦−u gives exactly zero coupling, and identify u↦−u with the inversion of the observer (three isoclinic Hurwitz steps). Among the regular 4D polytopes only the 5-cell is not centrally symmetric. We show that the tesseract and the 600-cell contain no regular 5-cell, while the 120-cell contains 840 of them in two symmetry orbits (120 and 720): observer and inversion-breaking pair sit on the two sides of the golden dual pair. The canonical odd invariant of a simplex, P(u)=Σₖ (u·vₖ)³, reproduces the three-dimensional lock-in exactly (24 xyz for the tetrahedron) and yields the first non-vanishing four-dimensional magnetism–gravity coupling: initial slope 0.03394920 in both orbits, κ = 0.07318514 and 0.07533183 at the crossing points. We also correct the three-dimensional coupling at the true crossing point (κ=0.2138083; the earlier register value 0.1748 mixed two amplitude conventions). None of the rules for α fixed so far reproduces the fine-structure constant; α remains open, and all attempts are listed.